CHAPTER-4 BOND VALUATION - Helping Hand Institute

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Lesson from this topic

If we both exchange one rupee, we both have one rupee each, but if we exchange one good thought, we both have two thoughts.

Comparison is the best way to judge your progress. But don’t compare with others, just compare your yesterday with your today.

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If we both exchange one rupee, we both have one rupee each, but if we exchange one good thought, we both have two thoughts.

the best way to judge your progress. But don’t compare with others, just compare your yesterday with your today.

M11 = May-2011

N11=Nov-2011

M12 N12 M13 N13 M14

IIOONN

If we both exchange one rupee, we both have one rupee each, but if we exchange one

the best way to judge your progress. But don’t compare with others, just

N14 M15

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CHAPTER-4 BOND VALUATION 4.1

4.1 Bond

A negotiable certificates evidencing indebtness. It is normally unsecured. A company, municipality or government agency generally issues a debt security.

Bond issues are considered fixed income securities because they impose fixed financial

obligations on the issuers. The issuers agrees to

(a) Pay fixed amount of interest periodically to the holder of bond.

(b) Repay a fixed amount of principal at the date of maturity.

4.1.1 Some Basics of Bond

(a) Face Value: This is the value stated on the face of the bond and is also known as par value. It represents the amount of borrowing by the firm which will be repaid after a

specific period of time.

(b) Redemption value of Bond: The Face value of bond is repaid at the end of maturity

period, is known as redemption value of bond. A bond may be redeemed at par, at premium or at discount.

(c) Market value: A bond may be traded in a stock exchange. Market value is the price at

which the bond is usually bought or sold. Market value may be different from par value.

(d) Coupon Rate or Interest: A bond carries a specific rate of interest which is also called a

coupon rate.

Interest Amt = Face value of Bond x Coupon Rate

(e) Maturity: It is the number of years after which Redemption value is paid.

(f) Call Date: Bonds which can be redeemed prior to maturity. The call date represents the date at which the bond can be called.

(g) Call Price: It is a price at which Bond can be called back before maturity.

4.2 Bond Value and Yield to Maturity [From Q-1 to 17]

ICAI ICSI ICWA

Nov-2003 M-8 June-2002 M-6 Dec-2002

Nov-2007 M-6 Dec-2003

Nov-2008 M-5 Dec-2004

Nov-2009 M-4

Nov-2010 M-2.5+2.5

Nov-2010 M-5

Nov-2011 M-8

Nov-2013 M-5

May-2015 M-2+2+2

4.2.1 VALUE OF BOND

(a) Bonds with Maturity B0 = Intt x PVAF(n years, RR) + RV x PVF(nth years, RR)

(b) Perpetual Bond B0 = Intt/RR

(c) Zero Coupon Bond B0 = RV x PVF(nth years, RR)

4.2.2 YIELD TO MATURITY (YTM)

(a) The rate of return which makes the discounted value of cash flows equal to the bond's market value is known as the YTM of the bond. So, a bond’s YTM may be defined as the IRR


for a given level of risk.

(b) At YTM

Market value of Bond = PV of Cash flow of Bond at YTM

(c) Approximate YTM

I (l-t)+(RV - NP)/N

Kd = (RV+NP)/2

(d) Find Actual YTM

First we should find two NPV at two different rates, then apply

YTM = LR + [NPVLR x (Diff of Rate)]/(NPVLR – NPVHR)

(e) YTM is also known as yield means actual return of bond.

(f) Yield to Maturity

(i) Bonds with Maturity YTM = LR + [NPVLR]*Diff of rate/[NPVLR – NPVHR]

(ii) Perpetual Bond YTM = Intt/Actual Market Price

(iii) Zero Coupon Bond YTM = LR + [NPVLR]*Diff of rate/[NPVLR – NPVHR]

4.2.3 Pricing of BOND

Conditions Pricing Action

(a) IF B0 > Actual price of Bond Overpriced It should not be purchased or it can be sold

(b) IF B0 < Actual price of Bond Underpriced It should be purchased or it should not be sold

(c) IF B0 = Actual price of Bond Correctly It may be purchased

Note: The appropriate discount rate would depend upon the risk of the bond. The risk in holding

the government bond is less than the risk associated with debentures issued by a company.

BBoonnddss wwiitthh MMaattuurriittyy

Question-1 A bond of Rs.1,000 bearing a coupon rate of 12% is redeemable at par in 10 years. Find out the value of the bond if:

(a) Required rate of return of Ram-12%, Shyam-10% and Mohan-14%. [Ans: Rs.1,000; Rs.1,123.40; Rs.895.92]

(b) If actual Price of Bond is Rs.1050, whether it should be bought or sold.

(c) What is actual YTM for Ram, Shyam & Mohan.

Imp Concept

(a) Value of bond is depended on the required rate of return. Higher the required rate of return

lower will be the value of the Bond and vice versa.

(b) Required rate of return depends on Investor and it may vary from investor to investor.

(c) Same bond may have different fair value for different investor depending on their required rate of return.

(d) YTM is same for all investors because cash outflow, cash inflow and period are same for all

investor.

Question-1A A Rs.5,000 bond with a 10% coupon rate matures in 8 years.

(a) Calculate value of bond for an investor whose required rate of return is 11 %.

(b) If bond is selling at 97%. What is YTM. [Ans: PV = Rs.4,743; MP = Rs.4,850]


Question-1B [Nov-2009] [N] [M-4] [SP] An investors is considering the purchase of the following Bond:

Face value Rs.100

Coupon rate 11 %

Maturity 3 years

(a) If he wants a yield of 13% what is the maximum price he should be ready to pay for?

(b) If the Bond is selling for Rs.97.60, what would be his yield?

Question-1C [June-02-CS] [M-6]

Question-1D [Nov-2010] [M-5]

Question-1E [May-2015] [M-2]

4.3 BOND VALUATION BEHAVIOR

[If Redemption value is equal to Face value]

(a) If RR = CR then Bond Value = Par Value.

(b) If RR > CR then Bond Value < Par Value.

(c) If RR < CR then Bond Value > Par Value.

4.4 Bond Value in case of Semi-Annual Interest

Bo = Intt/2 x PVAF(2n years, DR/2) + RV x PVF(2nth years, DR/2)

Question-2 [Nov-2010] [M-2.5] Calculate Market Price of a bond with 7.5% coupon interest, Face value Rs.10,000 & Term to maturity of 2 years, presently yielding 6%. Interest payable half yearly.

Question-2A [ICWA-Dec-2002] A Company invested in a 5 year bond issue of another company in 2010 carrying a coupon rate of 10% p.a. The interest payable at half yearly rests and

the principal repayable after 5 years in 2014 end. The current market yield has fallen to 9% during 2011. The investor company wanted to take advantage of fall in market yield by selling the bond to any willing buyer. Compute the value of the bond at the end of 2011. Assume par value of each bond Rs.1000

Question-2B [SM]

Question-2C [May-2015] [M-2]

4.5 Calculation of value of Bond if Coupon rate changes from year to year

With the change in coupon rate, only interest amt p.a. will change and other part will remain same.

Question-3 [Nov-2003] [M-8] M/s Agfa Industries is planning to issue a debenture series on the following terms:

Face value Rs.100

Term of maturity 10 years

Yearly coupon rate

Years

1-4 9%

5-8 10%


9-10 14%

The current market rate on similar debentures is 15% per annum. The Company proposes to price the issue in such a manner that it can yield 16% compounded rate of return to the

investors. The Company also proposes to redeem the debentures at 5% premium on maturity. Determine the issue price of the debentures.

Question-3A [CS-Dec-2004] [SP] Blue Ltd is contemplating a debenture issue on the following terms:

Face value Rs.100

Term of maturity 7 years

Yearly coupon rate

Years

1-2 8%

3-4 12%

5-7 15%

The current market rate on similar debentures is 15% per annum. The Company proposes to

price the issue in such a manner that it can yield 16% compounded rate of return to the investors. The Company also proposes to redeem the debentures at 5% premium on maturity. Determine the issue price of the debentures.

4.6 Bond value if required rate of return changes every year

1.26 Calculation of PVF

1) If Discount Rate for n years are same

PVF1 = 1/(1+DR)1

PVF2 = 1/(1+DR)2

PVF3 = 1/(1+DR)3

Example

Same Method 1 Method 2

Year Discount Rate PVF PVF PVF PVF

1 10% 1/(1.1) = 0.909 1/(1.1) = 0.909

2 10% 1/(1.1)(1.1) = 0.826 0.909/(1.1) = 0.826

3 10% 1/(1.1)(1.1)(1.1) = 0.751 0.826/(1.1) = 0.751

4 10% 1/(1.1)(1.1)(1.1)(1.1) = 0.683 0.751/(1.1) = 0.683

2) If Discount Rate for n years are not same

PVF1 = 1/(1+DR1)

PVF2 = 1/(1+DR1)*(1+DR2)

PVF3 = 1/(1+DR1)*(1+DR2)*(1+DR3)

Example

Different Method 1 Method 2

Year Discount Rate PVF PVF

1 10% 1/(1.1) = 0.909 1/(1.1) = 0.909

2 11% 1/(1.1)(1.11) = 0.819 0.909/(1.11) = 0.819

3 12% 1/(1.1)(1.11)(1.12) = 0.731 0.819/(1.12) = 0.731

4 13% 1/(1.1)(1.11)(1.12)(1.13) = 0.647 0.731/(1.13) = 0.647


Question-4 Consider a 10% bond having maturity of 4 years and face value of Rs.1000. The

interest rates in the market expected to be 8% in year 1, 9% in year 2, 11% in year 3 and 12%

in year 4. Calculate value of the bond. Also calculate YTM of the bond if the actual market price of the bond is equal to value calculated above.

Question-4A [SP] In an economy, the prices of Bonds reveal the following pattern of forward rates:

Year Forward rates

1 7%

2 8%

3 9%

Suppose you are interested in purchasing a 6%Bond of Rs.1,000, maturity 3 years, what should

be the price?

Question-5 [Nov-2013] [M-5] ABC Ltd issued 9% bonds of Rs.1000 each having a maturity of

3 years. The present interest rate is 12% for one year tenure. It is expected that forward rate of interest for one year tenure is going to fall by 75 basis points and further by 50 basis points for very next year in further for the same tenure. This bond has a beta value of 1.02 and is more popular in the market dues to less credit risk.

Calculate

(a) Intrinsic Value of Bond

(b) Expected price of bond in the market

4.7 Bond price is cum interest

(a) Generally Bond price is ex interest. Bond Price cum Interest means it includes Interest amt accrued till the date of its valuation

(b) First we will calculate interest amt accrued

(c) Then we will calculate Bond Price ex interest

Bond Price ex Interest = Bond Price cum Interest – Interest accrued

Ex: 12% debentures of Rs.100 on which interest is payable annually on 31 Dec.

Bond price cum interest = Rs.110 as on 30/03/2102



30/03/2012 30/06/2012 31/12/2012

Bond Price cum Interest Rs.110 Rs.111 Rs.114

Interest Accrued but not due 3% 6% 12%

Interest amt included in Bond Price Rs.3 Rs.6 Rs.12

Bond Price Ex Interest Rs.107 Rs.105 Rs.102

Question-6 A company has outstanding 8% debentures of Rs.10,00,000 on which interest is

payable annually on 31 Dec. The debentures are due for redemption at par on 1.1.1993. The market price of debenture on 31.12.1989 was Rs.103 cum interest. Ignore tax. What do you estimate to be current market rate of interest? (This is also called yield to maturity.)


4.8 Current Yield

(a) Current Yield: Yield to maturity is not the same as the current yield. Current yield is the

annual interest divided by the bond’s current value. Current yield considers only the annual interest and does not account for the capital gain or loss.

(b) Current Yield = Annual Interest / Current Market Price

Question-7 [Nov-2008] [M-5] [Nov-2011] [M-8] Based on the credit rating of the bonds, Mr. X has decided to apply the following discount rates for valuing bonds:

Credit Rating Discount rate

AAA T - bill rate + 3% spread

AA AAA + 2% spread

A AAA+ 3% spread

He is considering to invest in a AA rated, Rs.1,000 face value bond currently selling at

Rs.1025.86. The bond has five years to maturity and the coupon rate on the bond is 15% p.a. payable annually. The next interest payment is due one year from today and the bond is redeemable at par. (Assume the T-bill rate to be 9%). You are required to

(c) Calculate the intrinsic value of the bond for Mr. X. Should he invests in the bond: [Ans: Rs.1033.95]

(d) Calculate the current yield and the yield to maturity (YTM) of the bond. [Ans: 14.62%, 14.23%]

Imp Concept

In calculation of YTM, interest and capital gain both are considered. While in calculation of

Current Yield, only interest is considered. Capital gain is ignored.

4.8 Calculation of Value or YTM if bond is redeemed in installment

Question-8 ABC Ltd. has raised Rs.50 crore through an issue of 9% bond. Each bond has a face

value of Rs.500 and 10 years term to maturity. As per the terms of the issue each bond is redeemable in four equal installment starting from the end of 7th year. You are required to find out the price of the bond if YTM is 13%. And what is the Current Yield? [Ans: Rs.401.16; 11.21%]

Question-8A [SP] A PSU is proposing to sell a 8 years bond of Rs.1,000 at 10% coupon rate p.a. The bond amount will be amortized equally over its life. If an investor has a minimum required rate of return of 8%. What is the bond's present value/ Issue Price? [Ans: 1070.33]

Question-9 [SP] Bonds of MIL of FV of Rs.100 with 10% coupon paid semi annually, are selling at 5% discount on the face value. These bonds will be redeemed at par by equal installments at

the end of 5th and 6th years. MIL has an effective tax rate of 40%. What is the YTM of the Bond.

Question-9A [CS-Dec-2003]

4.9 Holding Period Return

(a) Holding Period is the period for which a Bond is held. The holding period may be few months or few years. If nothing is specified, we can assume 1 year period as holding

period.

(b) Holding period return on investment

= [(B1 – B0) + Interest] x 100/B0


(c) (B1 – B0) is known as Capital Gain and (B1 – B0)/B0 is known as return due to capital gain

(d) interest/B0 is due to interest and known as current yield

Question-10 A had purchased a bond at a price of Rs.800 with a coupon payment of Rs.150 and

sold it for Rs.1,000.

(a) What is his holding return? [Ans: 43.75%]

(b) If the bond is sold for Rs.750 after receiving Rs.150 as coupon payment, then what is his holding period return? [Ans: 12.5%]

4.10 Calculation of Bond value at the end of each year till expiration

(a) Bonds with Maturity B0 = Intt x PVAF(n years, RR) + RV x PVF(nth years, RR)

B1 = Intt x PVAF(n-1 years, RR) + RV x PVF([n-1]th years, RR)

B2 = Intt x PVAF(n-2 years, RR) + RV x PVF([n-2]th years, RR)

(b) Perpetual Bond B0 = Intt/RR

B1 = Intt/RR

B2 = Intt/RR

(c) Zero Coupon Bond B0 = RV x PVF(nth years, RR)

B1 = RV x PVF([n-1]th years, RR)

B2 = RV x PVF([n-2]th years, RR)

Question-11 A Company has issued a bond on 1.4.2003 having nominal value of Rs.100 and a coupon rate of 15% redeemable after 5 years at a premium of Rs.25. An investor whose desired rate of return is 18% wishes to know the issue price of bond at the end of each year till maturity.

Question-11A [SP] A company invested in a 5 year bond issue of another company in 2002

carrying coupon rate of 10% per annum, the interest payable at half- yearly rests and the principal repayable after 5 years in 2006 end. The current market yield has fallen to 9% during 2003. The Investor Company wanted to take advantage of the fall in market yield by selling the bond to any willing buyer. Compute the value of the bond at the end of 2003. Assume par value

of each bond Rs.1,000. [Ans: Rs.1025.90]

4.11 Yield to Call

Yield to Call: A number of companies issue bond with buy back of call provision. Thus a bond

can be redeemed or called before maturity. Yield for call is calculated in the same way as yield to maturity.

Question-12 Mr. X purchased a bond with Rs.1000 face value. It has 10% coupon rate and 4 years to maturity. Interest is payable annually. Mr. X paid Rs.1032.40 for bond.

(a) What is the bonds yield to maturity? [Ans: 8.98%]

(b) If bond can be called 2 years from now at a price of 1100, what is its yield to call?. [Ans: 12.75%]

Question-12A [SP] A bond is currently traded at Rs.950. Its face value is Rs.1,000. Coupon

rate is 10%. It is redeemable at par after 5 years from today. However the company has an

option of calling it after 3 years from today at 5% premium. Find Yield to Call.


4.12 Value and YTM of Perpetual Bond/ Irredeemable Bond

Value of Irredeemable Bond B0 = Intt/Required rate of return = Intt/YTM

YTM of irredeemable Bond YTM = Intt/Bo

Question-13 A 10% bond of FV of Rs.100 is irredeemable.

(a) Calculate value of Bond if required rate of return is 12%

(b) Calculate YTM if actual price of Bond is Rs.95

Question-13A A Company has sold Rs.1,000, 12% perpetual debentures 10 years ago. Interest rates have risen since then, so that debentures of this company are now selling at 15% yield basis.

Determine the current indicated/ expected market price of the debentures. Would you like to buy the debentures for Rs.700? [Ans: Yes]

Quesiton-14 [Nov-2010] [M-2.5] [N] Calculate Market Price of 10% Government of India

security currently quoted at Rs.110, but interest rate is expected to go up by 1%.

4.13 PURE DISCOUNTS BOND OR DEEP DISCOUNT BONDS (DDB) or Zero Coupon Bond

[May-2012] [M-4] Write short notes on Zero Coupon Bond

Solution

As name indicates these bonds do not pay interest during the life of the bonds. Instead, zero

coupon bonds are issued at discounted price to their face value, which is the amount a bond will be worth when it matures. When a zero coupon bond matures, the investor will receive one lump sum (face value) equal to the initial investment plus interest that has been accrued on the

investment made. The maturity dates on zero coupon bonds are usually long term. These maturity dates allow an investor for a long range planning. Zero coupon bonds issued by banks, government and private sector companies.

YTM of Zero Coupon Bond YTM =

LR + [NPVLR]*Diff of rate/[NPVLR – NPVHR]

Value of Zero coupon Bond B0 RV*PVF(Required Rate, nth Yr)

Question-15 A DDB is issued for a maturity period of ten years and having a par value of Rs.

25,000. Find out the value of the DDB given that the required rate of return is 15%.

Question-15A [SP] A Deep Discount Bond was issued by financial institution for a maturity

period of 10 years and having a par value of Rs.25000. Find out the value of the Bond if the required rate of return is 16%.

Question-15B [May-2015] [M-2]

4.13.1 YTM of zero coupon Bond

Question-16 A company issues Zero coupon bonds of 10 years maturity. Issue price Rs.260.

Maturity value Rs.1000. Ignore tax. Calculate YTM?


Question-16A [SP] The following is the list of prices of zero coupon bonds of various maturities. Calculate the yields to maturity of each bond.

Maturity (Years) Market Price of Rs.1000 face value bond

1 Rs.952.38

2 Rs.890.00

3 Rs.816.30

4.13.2 Imputed Interest income in case of Deep Discount Bond

Imputed Interest first year = B1 – B0

Imputed Interest Second year = B2 – B1

Imputed Interest Third year = B3 – B2 So on

Value of Zero coupon bond at the end of 1st year B1 = B0 + Intt of One year

Question-17 A company issues a 10 years maturity zero coupon bond having face value

(maturity value) of Rs. 1000 on basis of YTM of 10%. What is the imputed interest income in the first year, second year and 10th year?

4.14 Commercial Paper [Q-18 to 20]

ICAI RTP ICSI ICWA

Nov-2003 M-4 June-2007 M-5

May-2005 M-2 Nov-2011 02

May-2007 M-4 Nov-2014 17

May-2009 M-4

Nov-2012 M-5+4

May-2015 M-2

[Nov-2012] [M-4] Write short notes on commercial paper.

Solution

A Commercial paper is an unsecured money market instrument issued in the form of a promissory note. The CP comes under the regulation of RBI which issued guidelines in 1990 on

the basis of the recommendations of the Vaghul working group.

These guidelines were aimed at:

(i) Enabling the highly rated corporate borrowers to diversify their sources of short term borrowings; and

(ii) To Provide additional instrument to the short term investors;

It can be issued for maturities between 7 days and a maximum upto one year from the date of

issue. These can be issued in denominations of Rs.5 lacs or multiples thereof.

Eligibility Criteria for issuer of commercial paper

(i) The tangible net worth of the company is Rs.5 Cr or more as per audited balance sheet of the company;

(ii) The fund based working capital limit is not less than Rs.5 cr;

(iii) The company is required to get the credit rating from rating agencies such as CRISIL and

ICRA.

(iv) At the time of issue of CP, credit rating should not be more than 2 months old.

(v) The minimum Current Ratio should be 1.33:1

(vi) For public sector company, there are no listing requirement but for companies other than

PSU, the same should be listed on one or more stock exchanges.

(vii) All issue exp shall be born by Company issuing CP.


Imp Concept

(a) Value or YTM of Commercial Paper may be calculated in the same way as of Zero Coupon Bond.

(b) Only difference between two is that maturity period of zero coupon bond is more than one

year and while maturity period of commercial paper is up to 1 year

(c) It can be issued for maturities between 7 days and a maximum upto one year from the date

of issue. These can be issued in denominations of Rs.5 lacs or multiples thereof

(d) Issue Price/ Purchase Price at t0 = RV/(I+PIR)1

PIR = IR p.a.*No of days of CP/365 or IR p.a.*No of months of CP/12

(e) Value of CP at any day = Purchase Price + Interest due till valuation day

(f) Value of CP at any day = Redemption value – Interest to be due till maturity

Question-18 [June-2007-CS] [M-5] Kastor Ltd issued CP as per following conditions:

Date of issue 19/10/2006

Date of maturity 17/01/2007

Interest rate 7.25% p.a.

FV of commercial paper 10 crs

What was the net amount received by the company on issue of commercial paper.

Question-18A [May-2009] [M-4] Z Co. Ltd issued commercial paper worth Rs.10

crores as per following details:

Date of issue: 16 January, 2009

Date of maturity 17 April, 2009

No. of days: 91

Interest rate 12.04% p.a

What was the net amount received by the company on issue of CP? (Charges of intermediary

may be ignored)

Question-18B [Nov-2003] [M-4] [RTP-Nov-2014-17] [SP] M Ltd. has to make a payment

on 30th January, 2004 of Rs.80 lakhs. It has surplus cash today, i.e. 31st October, 2003; and has decided to invest sufficient cash in a bank's Certificate of Deposit scheme offering an yield of 8% p.a. on simple interest basis. What is the amount to be invested now?

Question-18C [RTP-Nov-2011-2] Suppose Govt. pays Rs.5,000 at maturity for 91 days

Treasury Bill. If Mr. Y is desirous to earn an annualized discount rate of 3.5%, then how he can pay for it

1.27 Calculation of Effective annual interest rate from periodical interest rate

If Periodic Interest rate is given, and Compounding is periodically

Effective annual Interest = (1+Periodic Interest rate)No of Periods in a year – 1

If Periodic Interest rate is given, and Compounding Annually

Effective annual Interest = Periodic Interest rate*12/Period


Question-19 [Nov-2012] [M-5] Calculate the current price and the bond equivalent yield [using simple compounding] of a money market instrument with the face value of Rs.100 and

discount yield of 8% in 90 days. Take 1 year = 360 days.

Question-19A [May-2007] [M-4] A money market instrument with face value of Rs.100 and discount yield of 6% will mature in 45 days. You are required to calculate:

(a) Current price of the instrument.

(b) Bond equivalent yield

(c) Effective annual return.

Question-19B [May-2005] [M-2] [SP] RBI sold a 91 day T-bill of face value of Rs.100 at an yield of 6%. What was the issue price? Calculate effective annual interest rate.

4.15 Calculation of Effective interest rate if Current Market Price and Redemption value and maturity period is given

(a) Interest amount for the period of CP = Redemption price – Issue Price

(b) Interest rate for the period = Interest/Issue Price

Question-20 Ram has deposited Rs.100 and received back Rs.108 after 90 days. Calculate

effective rate of interest p.a.

Question-20A Calculate the effective annual interest rates of the following two securities:

(a) 3 months maturity treasury bill, face value Rs. 1000, currently selling at Rs. 973.

(b) 10 years maturity Government security selling at par; coupon rate is 10% payable half-yearly.

Question-20B [RTP-Nov-2011-2] [SP] Suppose Mr. X purchase Treasury Bill for Rs.9,940 maturing in 91 days for Rs.10,000. Then would be annualized investment rate for Mr. X and

annualized discount rate for the Govt. Investment.


4.16 Calculation of Total Cost of Fund [Q-21 to Q-22]

ICAI ICSI ICWA

May-2006 M-6

May-2012 M-5

May-2014 M-5

(a) Total Cost of Fund in % = Effective Interest p.a. + Issue Exp p.a. [Like, Brokerage fees, Stamp Duty]

Question-21 [May-2006] [M-6] From the following particulars calculate the effective interest

p.a. as well as the total cost of funds to ABC Ltd. which is planning a Commercial Paper issue

Issue price of CP – Rs.97,350

Face value – Rs.1,00,000

Maturity period – 3 months

Issue expenses:

Brokerage: 0.125% for 3 month

Rating: 0.5% P.A

Stamp duty: 0.125% for 3 month


Question-21A [May-2012] [M-5] [SP] LMN & Co. plans to issue Commercial Paper (CP) of Rs.1,00,000 at a price of Rs.98,000.

Maturity Period: 4 Months

Expenses for issue of CP are :

(i) Brokerage 0.10%

(ii) Rating Charges 0.60% and

(iii) Stamp Duty 0.15%

Find the Effective Interest Rate per annum and the cost of Fund.

(b) If Cost of Issue/ Fund is given in amount then

Cost of Issue in % = Cost of Fund in Amt/Net amount utilized for Operation

Question-22 [May-2014] [M-5] AXY Ltd. is able to issue commercial paper of Rs.50,00,000

every 4 months at a rate of 12.5% p.a. The cost of placement of commercial paper issue is

Rs.2500 per issue. AXY Ltd. is required to maintain line of credit Rs.1,50,000 in bank balance. The applicable income tax rate for AXY Ltd. is 30%. What is the cost of funds (after taxes) to AXY Ltd. for commercial paper issue? The maturity of commercial paper is four months.

4.17 RELATIONSHIP BETWEEN BOND VALUE AND THE INTEREST RATE RISK

(a) As the interest rate varies, the bond value also changes.

There is a negative relationship between bond values and the market interest rates.

(b) This interest rate risk is higher in case of bonds of longer maturity as compared to the bonds of shorter maturity. The bond with a longer maturity is more exposed to variations in

interest rate. This argument also explains as to why the short term bonds have lower rate of interest than the long term bond.

[Nov-2005] [M-2.5] What is the interest rate risk, reinvestment risk and default risk & what

are the types of risk involved in investment in G Sec.

4.17.1 Bond Risk

Unsystematic risk – Internal Risk. It refers to default risk i.e, the issuer may default in payment of interest or principal or both.

Systematic risk – It is also known as interest risk. Interest risk refers to change in market

interest rate during the holding period. Systematic risk can be divided in two parts.

Interest Risk: Interest risk refers to change in market interest rate during the holding period.

Bond prices decrease with increase in Market interest and vice versa.

Reinvestment Risk: Change in market interest rate during the holding period affects the return

from the bond investment as the investor shall be reinvesting the interest income of the bond at the changed rate. The risk here is that the rate at which the interim cash flows are reinvested may fall thereby affecting the returns.

Default Risk: This type of risk in the context of a Government security is always zero. However, these securities suffer from a small variant of default risk i.e. maturity risk. Maturity risk is the risk associated with the likelihood of government issuing a new security in place of redeeming the

existing security. In case of Corporate Securities it is referred to as credit risk.

Two type of risk are involved in G-Sec (i) Interest Risk and (ii) Reinvestment Risk.


4.18 Bond Duration and Interest Rate Sensitivity [Q-23 to 29]

ICAI RTP ICSI ICWA

Nov-2005 M-8 + 8 Nov-2012 Dec-2007 M-6

May-2007 M-6 May-2014 3

May-2009 M-10

4.18.1 Bond Duration

Duration is the weighted average time to receive the PV of Bond. The weights are the present

value of the payment, using the bond’s YTM as the discount rate. (Duration is called as Macaulay Duration)

Two bond with similar maturity but different coupon rate and flow patterns will have different durations.

The duration of lower coupon bond is higher than the duration of higher coupon bond.

Maturity Period

[Non Zero Coupon Bond]

Bond Duration = Through Table OR

1+YTM - (1+YTM) + T(CR-YTM)

YTM CR[(1+YTM)T – 1] + YTM

Zero Coupon Bond Bond Duration = Maturity Period

Irredeemable Bond Bond Duration = (YTM+1)/YTM

4.18.2 Volatility of Bond

Volatility refers to the sensitivity of the bond price to change in the current interest rate.

The intensity of the price sensitivity depends on a bond’s maturity and the coupon rate of

interest.

The longer the maturity of a bond, the higher will be its sensitivity to the interest rate changes. Similarly, the price of a bond with low coupon rate will be more sensitive to interest rate changes.

The volatility or the interest rate sensitivity of a bond:

Volatility of bond = Duration (1+ YTM/n)

Where n is the no of interest payment in year.

Question-23

Face Value of the Bond = Rs.1000; Coupon Rate = 8.5%; Current market price = Rs.943.15

YTM = 10%. Maturity Period = 5 yrs.

(a) Calculate Bond duration and volatility of the Bond.

(b) Calculate price of bond if YTM is 11%

Question-23A

Face Value of the Bond = Rs.1000; Coupon Rate = 11.5%; Current market price = Rs.1056.85

YTM = 10%. Maturity Period = 5 yrs.

(a) Calculate Bond duration and volatility of the Bond.

(b) Calculate price of bond in YTM is 11%

Question-23B [Nov-2005] [M-8] [CS-Dec-2007] [M-6] [RTP-Nov-2012]



Question-24 [RTP-May-2014-3] [SP]

Mr A is planning for making investment in bonds of one of the 2 companies X Ltd and Y Ltd. The details of these bonds is as follows:

Company Face Value Coupon Rate Maturity Period

X Ltd Rs.10000 6% 5 Years

Y Ltd Rs.10000 4% 5 Years

The current market price of X Ltd’s bond is Rs.10796.80 and both bonds have same YTM. Since Mr A considers duration of bonds as the basis of decision making, you are required to calculate the duration of each bond and your decision.

Question-25 Duration of bond is 4.50 years. YTM =8% p.a. payable half yearly. Find the %

change in its price if the YTM declines from 8% to 7%.

Question-25A

4.18.3 Bond duration of Zero coupon Bond

Zero Coupon Bond Bond Duration = Maturity Period

Question-26 3 years maturity zero coupon bond is currently sold at Rs.816. Its maturity value is

Rs.1000. Find its duration. YTM is 7%

Imp Note

Bond duration of zero coupon bond is always equal to its maturity period.

4.18.4 Calculation of Bond value on the basis of Bond duration

Question-27 [May-2007] [M-6] [SP] Find the current market price of a bond having face value of Rs.1,00,000 redeemable after 6 year maturity with YTM at 16% payable annually and

duration 4.3202 years. Given 1.166 = 2.4364

4.18.5 Duration to Call

Question-28 [SP] 11%, ABC Ltd bond is a callable bond with a face value of Rs.1000. it has a remaining maturity of 5 yrs and the coupon is payable annually. The bond can be called by the

issuer after 3 yrs from now at a call price of Rs.1100. The bond is currently trading at a price of Rs.985. you are required to calculate the duration to call of 11%, ABC Ltd Bond.

4.19 Sale or purchase of bond on the basis of Bond Duration

There is inverse relationship between interest rate and bond price.

Bond Duration Volatility Change in Bond Price

Larger Larger Large

Smaller Smaller Small

Interest rate expected to be lower Interest rate expected to be higher

Bond price will increase Bond price will decrease

Bond of higher duration should be bought Bond of higher duration should be sold


Bond of lower duration should be sold Bond of lower duration should be bought

Bond Duration of Portfolio

Bond duration of portfolio = BD1*W1 + BD2*W2 + BD3*W3 + BD4*W4 + BD5*W5 + …………

Question-29 [Nov-2005] [M-8] The Investment portfolio bank is as follows:

Government Coupon Rate Purchase rate Duration

Bond (F.V. Rs.I00 per Bond) .(Years)

G.O.I. 2006 11.68 106.50 3.50

G.O.I. 2010 7.55 105.00 6.50

G.O.I.2015 7.38 105.00 7.50

G.O.I. 2022 8.35 110.00 8.75

G.O.I. 2032 7.95 101.00 13.00

Face value of total Investment is Rs.5 cores in each Government Bond. Calculate actual Investment in portfolio. What is a suitable action to churn out investment portfolio in the

following scenario?

(a) Interest rates are expected to be lower by 25 basis points.

(b) Interest rates are expected to be raised by 75 basis points.

Question-29A [SP] An inflow of Rs.1500000 is available and is to be invested in the following bond portfolio in the % specified.

Bond % of Money to be invested Duration of the bond (years)

1 10 10.35

2 22 4.25

3 19 7.50

4 7 9.50

5 17 12.67

6 6 5.82

7 11 8.50

8 8 6.71

(i) The face value of all the above bond is Rs.1000 and the YTM is 9%. Calculate the duration of

the bond portfolio.

(ii) What is a suitable action to churn out investment portfolio in the following scenario?

(a) Interest rates are expected to lower by 25 basis points

(b) Interest rates are expected to raise by 75 basis points

4.20 Convertible Bonds and Debenture [Q-30 to Q-32]

ICAI RTP ICSI ICWA

Nov-2008 M-8 Nov-2012 4

Nov-2009 M-4 May-2013 13

Nov-2010 M-5 May-2015 8

Nov-2011 M-6

May-2014 M-8

These are the bonds which have to be converted into specified number of equity shares of company issuing these bonds within a specified period.

In India most of the convertible bonds been issued on the basis of compulsory conversion i.e. the bonds are compulsorily convertible' number of specified number of shares; there is no discretion

of the bond holder. In USA European countries, convertible bonds are option convertible bonds i.e. conversion takes place if bond holder so desire.


An example of option convertible bond: Suppose a company issues 7% convertible bonds of $

each, maturity 7 years, redemption at par. The bond holder can get his bond converted into 4 equity shares after 2 years of issuance. Now whether the bond will be converted into equity shares or not, it is at the discretion of the bond holder. Suppose he gets the bonds converted into

shares, the company's liability towards principal and interest will extinguish. If he does not get the bond converted, he will be entitled to receive interest periodically and on maturity he will get the redemption amount of $100.

The convertibility option lowers the interest rate that the issuer would otherwise have to pay without this feature, and it appeals to investors who want current income, but would like to take

advantage of any growth in the issuer company.

Let us understand a few terms related to convertible debentures:

(a) Conversion ratio = The number of shares for which each bond can be converted

(b) Conversion price is the exercise price at which the investor converts his bond into equity

shares.

Conversion Price = Par value of Bond / Conversion Ratio

(c) Conversion parity price of Share or Market conversion price : It is the price of Share at which CMP of CB and Conversion value would be equal

Conversion parity price of Share = CMP of Convertible Debenture / Conversion Ratio

(d) Straight value of a Bond- This term is used with reference to optional convertible bonds.

The straight bond value is what the convertible bond would sell for if it could not be converted into equity shares. It is the price of an equivalent non-convertible bond.

Straight Value of Convertible Bond = CMP of Non Convertible Bond

(e) Downside risk: If the share price goes much below the conversion price, it is expected by

market forces that conversion option won't be exercised; the market price of the convertible bond will be equal to straight value of bond. This will result in loss for the investor. This loss is referred to as downside risk.

Downside risk or Premium over Investment Value (%)

= (CMP of CB – CMP of NCB)*100/CMP of NCB

Downside risk or Premium over Investment Value (Amt)

= (CMP of CB – CMP of NCB)

(f) Stock value of bond or Conversion Value of Bond

= CMP of share X Conversion ratio

(g) Conversion premium or Premium over conversion value: The extent by which the market value of a convertible security exceeds the conversion value.

Conversion Premium or Ratio of Conversion Premium (%) = (CMP of CB – Conversion Value)*100/ Conversion Value

Conversion Premium (Amt) = (CMP of CB – Conversion Value)

Conversion Premium per share = Conversion Premium/Conversion Ratio OR

Conversion Premium per share = Conversion Parity Price of Share - CMP of Share

(h) Favorable income differential = Interest Income p.a. – Dividend per Share*Conversion Ratio

Favorable income differential per Share = (Interest Income p.a. – Dividend per Share*Conversion Ratio)/Conversion Ratio

(i) Premium pay back period Financial Services or Break Even Period


= Conversion premium per share/Favorable income Differential Per share OR

= Conversion Premium/Favorable Income Differential

Question-30 M/s Reliance Industries have issued earlier a 11.5% convertible bond of face value Rs.1,000 maturing in 2007. After a period of 3 years on the option of the investor each of these

bonds can be converted into 50 equity shares of face value Rs.10 each. The investment value of similar non- convertible bond is Rs.870. The current market prices of the bond and the share are Rs.970 and Rs.18.50 respectively. The dividend per share for 2000-200 1 is Rs.2.12. You are

required to

(a) Conversion parity price of share

(b) Conversion Premium or Premium over conversion value

(c) Conversion premium per share

(d) Premium over investment value

(e) Favorable income differential

(f) Favorable income differential per share

(g) Break even period.

Question-30A [RTP-Nov-2012-4] [RTP-May-2013-13] [RTP-May-2015-8] [SP] The

following data is related to 8.5% Fully Convertible (into Equity shares) Debentures issued by JAC Ltd. at Rs. 1000.

Market Price of debenture Rs.900

Conversion Ratio 30

Straight value of Debenture 700

Market Price of Equity share on the date of Conversion 25

Expected Dividend Per share 1

You are required to calculate:

a) Conversion Value of Debenture

b) Market Conversion Price

c) Conversion Premium per share

d) Ratio of Conversion Premium

e) Premium over Straight Value of Debenture

f) Favorable income differential per share

g) Premium pay back period Financial Services

Question-30B [Nov-2010] [M-5]

Question-30C [Nov-2011] [M-6]

Question-30D [Nov-2008] [M-8]

Question-31 [May-2014] [M-8] GHI Ltd., AAA rated company has issued fully convertible bonds on the following terms, a year ago:

Face value of bond Rs. 1000

Coupon (interest rate) 8.5%

Time to Maturity (remaining) 3 year

Interest Payment Annual, at the end of the year

Principal Repayment At the end of bond maturity

Conversion ratio (number of shares per bond) 25

Current market of convertible bond Rs. 45

Market price of convertible bond Rs. 1175

AAA rated company can issue plain vanilla bonds without conversion option at an interest rates of

9.5%.


Required : Calculate as of today :

(i) Straight Value of bond.

(ii) Conversion Value of the bond.

(iii) Conversion Premium.

(iv) Percentage of downside risk.

(v) Conversion Party price.

T 1 2 3

PVIF of 9.5% 0.9132 0.8340 0.7617

Question-32 [Nov-2009] [M-4] Saran am Ltd. has issued convertible debentures with coupon

rate 12%. Each debenture has an option to convert to 20 equity shares at any time until the date of maturity. Debentures will be redeemed at Rs.100 on maturity of 5 years. An investor generally

requires a rate of return of 8% p.a. on a 5-year security. As an investor when will you exercise conversion for given market prices of the equity share of (a) Rs.4, (b) Rs.5 (c) Rs.6.

Cumulative PV factor for 8% for 5 years 3.993

Pv factor for 8% FOR YEAR 5 0.681

4.21 Redemption of old Bond and issue of new Bond [Q-33 to Q-34]

ICAI RTP ICSI ICWA

Nov-2001 Nov-2011

Nov-2005 M-10 Nov-2014

May-2009 M-06

May-2013 M-06

Calculation of PVCO if old bond is not redeemed

Particulars Cash Flow Tax Applicability Year PVCO

Interest on old Bond CO Intt p.a.*(1-TR) t1 to tn PVAF(DR%, n yrs)

RV of Old Bond CO RV tn PVF(DR%, nth yr)

Tax Saving on amortization of Issue cost on old Bond CI Amortisation p.a.*TR t1 to tn PVAF(6%,12 yrs)

Calculation of PVCO if old bond is redeemed with new bond

Particulars Cash Flow

Tax Applicability Year PVF/PVAF

Interest on new Bond CO Intt*(1-TR) t1 to tn PVAF(DR%, n yrs)

RV of new Bond CO RV tn PVF(DR%, nth yr)

Tax Saving on amortization of Issue cost on new Bond CI Amortisation p.a.*TR t1 to tn PVAF(DR%, n yrs)

Issue of new Bond CI (Issue Price-Issue Cost)*No of Bonds t0 1

Redemption of Old Bond including Premium CO Callable Value of old Bond t0 1

Tax saving on premium on old Bond CI Premium on old Bond*TR t1 PVF(DR%, 1 yrs)

Tax saving on writing off of unamortized cost of old

Bond

CI Unmortised Cost on old Bond*TR t1 PVF(DR%, 1 yrs)

Question-33 [May-2009] [M-6] [RTP-Nov-2011] ABC Ltd. has Rs.300 million, 12% bonds

outstanding with six years remaining to maturity. Since interest rates are falling. ABC Ltd. is contemplating of refunding these bonds with a Rs.300 million issue of 6 year bonds carrying a

coupon rate of 10%. Issue cost of the new bonds will be Rs.6 million and the call premium is 4%. Rs.9 million being the unamortized portion of issue cost of old bonds can be written off no sooner the old bonds are called off. Marginal tax rate of ABC Ltd. is 30%. You are required to analyse the bond refunding decision.

Question-33A [Nov-2001] A firm has a bond outstanding of Rs.3,00,00,000. The bond has 12

years remaining until maturity has a 12.5% coupon and is callable at Rs.1,050 per bond. It had flotation costs of Rs.4,20,000, which are being amortized at Rs.30,000 annually. The flotation


costs for a new issue will be Rs.9,00,000 and the current interest rate will be 10%. The after tax cost of the debt is 6%. Should the firm refund the outstanding debt? Show detailed working.

Consider corporate income tax at 50%.

Question-33B [May-2013] [M-6]

4.21.1 Overlapping Interest

Question-34 [Nov-2005] [M-10] [RTP-Nov-2014]

M/s Transindia Ltd is contemplating calling Rs.3 crores of 30 year’s Rs.1000 bond issued 5 years

ago with a coupon interest rate of 14%. The bonds have a call price of Rs.1140 and had initially collected proceeds of Rs.2.91 crores due to discount of Rs.30 per bond. The initial floating cost was Rs.360000. The company intends to sell Rs.3 crores of 12% coupon rate, 25 years bonds to

raise funds for retiring the old bonds. It proposes to sell the new bonds at their par value of Rs.1000. The estimated floatation cost is Rs.400000. The company is paying 40% tax and is after cost of debt is 8%. As the new bonds must first be sold and their proceeds, then used to retire old bonds, the company expects a two months period of overlapping interest during which

interest must be paid on both the old and new bonds. What is the feasibility of refunding bonds? [Ans: NPV = Rs.811980, Refunding of bonds is recommended]

4.22 Calculation of Forward Rate [Q-35 to Q-37]

ICAI ICSI ICWA

Nov-2007

Nov-2008 M-4

May-2010 M-8

This concept is based on Expectation Theory.

(a) Forward rate is the interest rate that we expect today to prevail in the market after

certain period.

(b) Forward rate for the first year is the rate of interest that we expect, in the beginning of

the first year to earn on our investment made in the beginning of the 1st year till the end of the first year. It is also referred as spot rate for 1st year.

(c) Forward rate for the Second year is the rate of interest that we expect today, to earn on

our investment made in the beginning of the 2nd year till the end of the 2nd year.

And so on.

(d) Calculation of Forward rate for the first year

(1 + YTM) = (1 + FR)

First year YTM and Forward rate is equal

(e) Calculation of forward rate for Second Year

(1+YTM)2 = (1+FR1)(1+FR2)

(f) Calculation of forward rate for third Year

(1+YTM)3 = (1+FR1)(1+FR2) (1+FR3)

Alternative method

B0 = Intt1/(1+FR1) + (Intt2 + RV)/(1+FR1)(1+FR2)

Question-35

What spot and forward rates are implied in the following bonds: (Face value Rs.100)

(a) Bond A Zero-Coupon; Maturity 1 year; Price Rs.93.46

(b) Bond B Coupon 5%; Maturity 2 years; Price Rs.98.13

(c) Bond C Coupon 9%; Maturity 3 years; Price Rs.104.62


Question-35A [May-2010] [M-8] [SP] Consider the following date for Government Securities:

Face value Interest (Rate%) Maturity (Years) Current price (Rs.)

1,00,000 0 1 91,000

1,00,000 10.5 2 99,000

1,00,000 11.0 3 99,500

1,00,000 11.5 4 99,900

Calculate the forward interest rates.

Question-35B [Nov-2007]

Question-36 [Nov-2008] [M-4] The following is the Yield structure of AAA rated debenture:

Period Yield (%)

3 month 8.5%

6 month 9.25

1 year 10.50

2 year 11.25

3 year and above 12.00

(a) Based on the expectation theory calculate the implicit one-year forward rates in year 2 and year 3.

(b) If the interest rate increases by 50 basis points, what will be the percentage change in the price of the bond having a maturity of 5 years? Assume that the bond is fairly priced at the moment at Rs.1,000.

Question-36A [SP] The following table represents the yield curve of a particular types of bonds issued by a company

Maturity period (Years) YTM %

1 10

2 11

3 12

What are the forward rates?

Question-37 [SP] The YTM of 1-year maturity zero coupon bond is 6% and that of 2-year maturity zero coupon bond is 7%. If the company issues a 2-year maturity 8% coupon bond of Rs. 1000 face value, what should be appropriate issue price?

4.23 Value of Bond at date Other than Due date of Interest

(a) First we will calculate ex interest value of bond on due date of Interest

(b) Value of Bond at date other than due date = Bond Value calculated above*(1+PIR)

Question-38 [Nov-2007] [M-6] MP Ltd. issued a new series of bonds on January 1, 2000. The bonds were sold at par (Rs.1,000), having a coupon rate 10% p.a. and mature on 31st December, 2015. Coupon payments are made semi-annually on June 30th and December 31st

each year. Assume that you purchased an outstanding MP Ltd. Bond on 1st March, 2008 when the going interest rate was 12%.

Required:

(a) What was the YTM of MP Ltd. Bonds as on January 1, 2000?

(b) What amount you should pay to complete the transaction? Of that amount how much should be accrued interest and how much would represent bonds basic value.


4.24 Contribution of Interest and Principal into changes in Price due to change in YTM

Question-39 [May-2009] [M-20]

Consider two bonds, one with 5 years to maturity and the other with 20 years to maturity. Both the bonds have a face value of Rs. 1,000 and coupon rate of 8% (with annual interest payments)

and both are selling at par. Assume that the yields of both the bonds fall to 6%, whether the price of bond will increase or decrease? What percentage of this increase/decrease comes from a change in the present value of bond's principal amount and what percentage of this increase/decrease comes from a change in the present value of bond's interest payments?

4.25 Calculation of actual return if Investor invests in different securities

First we have to calculate cash inflow and out flow from different securities and then discount it by two rate to get actual return like [IRR or YTM]

Question-40 Ram purchased at par a bond with a face value of Rs.1,000. The bond had five

years to maturity and a 10% coupon rate. The bond was called two years later for a price of Rs.1,200 after making its second annual interest payment. Ram then reinvested the proceeds in

a bond selling at its face value of Rs.1,000 with three years to maturity and a 7% coupon rate. What was Ram's actual YTM over the five year period?

Question-41 Sen. acquired at par a bond for Rs.1,000 that offered a 15% coupon rate. At the

time of purchase the bond had four years to maturity. Assuming annual interest payments calculate Sen's actual yield-to-maturity if all the interest payments were reinvested in an

investment earning 18% per year. What would Sen's actual yield-to maturity be if all interest payments were spend immediately upon receipt?

4.26 Calculation of YTM if there is taxation

In case of taxation, we have to calculate cash inflow and cash outflow after tax and then discount it by two rate to get YTM

Question-42 [May-2004] [M-8] There is a 9% 5-year bond issue in the market. The issue

price is Rs.90 and the redemption price Rs.105. For an investor with marginal income tax rate of 30% and capital gains tax rate of 10% (assuming no indexation). What is the post-tax yield to

maturity?

CHAPTER-4 BOND VALUATION-SOLUTION 4.1

Solution-1

(a)

Face Value of the Bond = Rs.1000

Coupon Rate = 12%

Interest P.a. = Face Value of the Bond*Coupon Rate = Rs.1000*12% = Rs.120 p.a.

Maturity Period = 10 yrs

RV = Rs.1000 [Assuming]

Value of Bond for Ram

B0 at 12% = Intt x PVAF(10 years, 12%) + RV x PVF(10th years, 12%) = PVCI12% (Not Part of Ans in Exam)

B0 at 12% = 120*5.651 + 1000*0.322 = Rs.1000 = PVCI12% (Not Part of Ans in Exam)

Value of Bond for Shyam


B0 at 10% = 120*6.144 + 1000*0.386 = Rs.1123 = PVCI10% (Not Part of Ans in Exam)

Value of Bond for Mohan

B0 at 14% = PVCI14% = Intt x PVAF(10 years, 14%) + RV x PVF(10th years, 14%) = PVCI14% (Not Part of Ans in Exam)

B0 at 14% = PVCI14% = 120*5.217 + 1000*0.270 = Rs.896.04 = PVCI14% (Not Part of Ans in Exam)

(b) If Actual price of the bond = Rs.1050

Only Shyam will buy it but Ram and Mohan will not buy it.

(c) Actual return of Bond [YTM] for Ram, Shyam and Mohan if they buy it at Rs.1050

Cash Outflow for Ram, Shyam and Mohan = Rs.1050 [not Part of Ans in Exam]

Cash inflow for all of them = Rs.120 intt for 10 years and Rs.1000 at the end of 10th year. [not Part of Ans in Exam]

Since we have cash inflow and outflow, hence we can calculate IRR of above cash inflow and outflow, which is known as YTM of the Bond [not Part of Ans in Exam]

B0 at 12% = 120*5.651 + 1000*0.322 = Rs.1000

NPV at 12% = PVCI – PVCO

= B0 at 12% - Purchase price of Bond or current market price of bond

= Rs.1000 – Rs.1050 = - Rs.50

Since NPV is negative, then take Discount rate = 10%

B0 at 10% = 120*6.144 + 1000*0.386 = Rs.1123



= Rs.1123 – Rs.1050 = Rs.73

YTM or IRR or Actual Return of Bond = LR + [NPVLR]*Diff of rate/[NPVLR – NPVHR]

= 10% + (73*2) /73+50 = 11.18%

Solution-1A

(a)


Coupon Rate = 10%

Interest P.a. = Face Value of the Bond * Coupon Rate = Rs.5000*10% = Rs.500 p.a.



Value of Bond at 11% for an investor


B0 at 11% = 500*5.147 + 5000*0.434 = Rs.4743.50 = PVCI11% (Not Part of Ans in Exam)

(b) If Actual price of the bond = Rs.5000*97% = Rs.4850

Since B0 at 11% < Actual price of Bond, hence bond is overpriced. It should not be purchased.

(c) Actual return of Bond [YTM] if it is purchased at Rs.4850

Cash Outflow = Rs.4850 (Not Part of Ans in Exam)

Cash inflow = Rs.500 intt for 8 years and Rs.5000 at the end of 8th year. (Not Part of Ans in Exam)

Since we have cash inflow and outflow, hence we can calculate IRR of above cash inflow and outflow, which is known as YTM of the Bond (Not Part of Ans in Exam)

B0 at 11% 500*5.147 + 5000*0.434 = Rs.4743.50




= Rs.4743.50 – Rs.4850 = - Rs.106.50


B0 at 10% 500*5.334 + 5000*0.467 = Rs.5000



= Rs.5000 – Rs.4850 = Rs.150


= 10% + (150*1)/150+106.50 = 10.58%

Comment: Actual return from bond is 10.58% while required return is 11%, hence bond should not be purchased

Solution-1B

a)


Coupon Rate = 11%




Value of Bond at 13%

B0 at 13% = Intt x PVAF(3 years, 13%) + RV x PVF(3rd year, 13%) = PVCI13% (Not Part of Ans in Exam)

B0 at 13% = 11*2.361 + 100*0.693 = Rs.95.27 = PVCI13% (Not Part of Ans in Exam)

(b) If Actual price of the bond = Rs.97.60

Since B0 at 13% < Actual price of Bond, hence bond is overpriced. It should not be purchased.

(c) Actual return of Bond [YTM] if it is purchased at Rs.97.60

Cash Outflow = Rs.97.60 (Not Part of Ans in Exam)

Cash inflow = Rs.11 intt for 3 years and Rs.100 at the end of 3rd year. (Not Part of Ans in Exam)

B0 at 13% 11*2.361 + 100*0.693 = Rs.95.27



= Rs.95.27 – Rs.97.60 = - Rs.2.33


B0 at 11% = 11*2.444 + 100*0.731 = Rs.100



= Rs.100 – Rs.97.60 = Rs.2.40


= 11% + (2.40*2)/[2.40+2.33] = 12.01%

Comment: Actual return from bond is 12.01% while required return is 13%, hence bond should not be purchased

Solution-2


Coupon Rate = 7.5%

Interest P.a. = Face Value of the Bond * Coupon Rate = Rs.10000*7.5% = Rs.750 p.a.

Half yearly interest = Rs.750/2 = Rs.375


No of Period = 4 half year


Required rate = 6% p.a.

Required rate = 3% half year

Value of Bond at 3%

B0 at 3% = Half yearly Intt*PVAF(4 half year, 3%) + RV*PVF(4th half year, 3%)

B0 at 3% = 375*3.717 + 10000*0.888 = Rs.10273.88


Solution-2A


Coupon Rate = 10%


Half yearly interest = Rs.100/2 = Rs.50


No of Period = 6 half year



Required rate = 4.5% half year

Value of Bond at 4.5%

B0 at 4.5% = Half yearly Intt*PVAF(6 half year, 4.5%) + RV*PVF(6th half year, 4.5%)

B0 at 4.5% = 50*5.158 + 1000*0.768 = Rs.1025.90

Solution-3


Coupon Rate = varies


RV = at 5% premium = Rs.105



Years Cash inflow (Rs.) PVF @ 16% PV

1 9 0.862 7.758

2 9 0.743 6.687

3 9 0.641 5.769

4 9 0.552 4.968

5 10 0.476 4.76

6 10 0.410 4.10

7 10 0.354 3.54

8 10 0.305 3.05

9 14 0.263 3.682

10 14+105 =119 0.227 27.013

71.327

Thus the debentures should be priced at Rs. 71.327

Solution-3A

Calculation of PV

Years Cash out flow (Rs.) PVF @ 16% PV

1 8 0.862 6.90

2 8 0.743 5.95

3 12 0.641 7.69

4 12 0.552 6.63

5 15 0.476 7.14

6 15 0.410 6.16

7 15 0.354 5.31

7 105 0.354 37.15

82.93

Thus the debentures should be priced at Rs. 82.93

Solution-4


Coupon Rate = 10%

Interest p.a. = Rs.1000*10% = Rs.100



PVAF of Different required rate in different year

Method 1 Method 2


1 8% 1/(1.08) = 0.926 1/(1.08) = 0.926


2 9% 1/(1.08)(1.09) = 0.849 0.926/(1.09) = 0.849

3 11% 1/(1.08)(1.09)(1.11) = 0.765 0.849/(1.11) = 0.765

4 12% 1/(1.08)(1.09)(1.11)(1.12) = 0.683 0.765/(1.12) = 0.683

3.224 3.224

B0 = PVCI Intt*PVAF(4 years) + RV*PVF(4th year)

B0 = PVCI 100*3.224 + 1000*.683 = Rs.1005.40

(b) Actual return of Bond [YTM] if it is purchased at Rs.1005.40

Cash Outflow = Rs.1005.40 [Not part of Ans]

Cash inflow = Rs.100 intt for 4 years and Rs.1000 at the end of 4th year. [Not part of Ans]

B0 at 10% = PVCI10% Intt*PVAF(4 years, 10%) + RV*PVF(4th year, 10%)

B0 at 10% = PVCI10% 100*3.169 + 1000*0.683 = Rs.1000



= Rs.1000 – Rs.1005.40 = - Rs.5.40


B0 at 9% = PVCI9% 100*3.239 + 1000*0.708 = Rs.1031.90



= Rs.1031.90 – Rs.1005.40 = Rs.26.50


= 9% + (26.50*1)/[26.50+5.40] = 9.83%

Solution-4A


Coupon Rate = 6%

Interest p.a. = Rs.1000*6% = Rs.60




Method 1 Method 2


1 7% 1/(1.07) = 0.935 1/(1.07) = 0.935

2 8% 1/(1.08)(1.07) = 0.865 0.935/(1.08) = 0.865

3 9% 1/(1.07)(1.08)(1.09) = 0.794 0.865/(1.09) = 0.794

2.594 2.594

B0 = Intt*PVAF(3 years) + RV*PVF(3rd year)

B0 = 60*2.594 + 1000*.794 = Rs.949.64

Alternative Method

B0 = PVCI = Price of the bond = [60/(1.07) + 60/(1.07*1.08) + 1060/1.07*1.08*1.09)] = 949.53

Solution-5

(a)


Coupon Rate = 9%




Forward Interest Rate

1st Year = 12%

2nd Year = 11.25%

3rd Year = 10.75%



Method 1 Method 2


1 12% 1/(1.12) = 0.893 1/(1.12) = 0.893

2 11.25% 1/(1.12)(1.1125) = 0.803 0.893/(1.1125) = 0.803

3 10.75% 1/(1.07)(1.08)(1.09) = 0.725 0.803/(1.1075) = 0.725

2.421 2.421

B0 = Intt*PVAF(3 years) + RV*PVF(3rd year)

B0 = 90*2.421 + 1000*0.725 = Rs.942.89

Alternative Method

B0 = Price of the bond = [90/(1.12) + 90/(1.12*1.1125) + 1090/1.12*1.1125*1.1075)] = 942.89

(b)

Beta of Bond = 1.02 [It shows riskiness of Bond with respect to market]

Fair Value of Bond = Rs.942.89

Expected Market Price of Bond = Rs.942.89*1.02 = Rs.961.75

Solution-6

Market Price of Debenture [Cum Interest] = Rs.103 as on 31/12/1989


Coupon Rate = 8%


Interest is payable annually hence intt of Rs.8 is included in above price

Market Price of Debenture [Ex Interest] = Cum Interest price of Debenture – Intt Due = Rs.103 – Rs.8 = Rs.95



App Return [First DR] = [Interest p.a. + (RV – NP)/N]/[(RV+NP)/2] = [8 + (100-95)/3]/(95+100)/2 = 9.91%

(b) Actual return of Bond [YTM] if it is purchased at Rs.95

Cash Outflow = Rs.95 [Not part of Ans]

Cash inflow = Rs.8 intt for 3 years and Rs.100 at the end of 3rd year. [Not part of Ans]

B0 at 10% = Intt x PVAF(3 years, 10%) + RV x PVF(3rd year, 10%)

B0 at 10% = 8*2.486 + 100*0.751 = Rs.94.98



= Rs.94.98 – Rs.95 = - Rs.0.02

YTM = 10%

Solution-7


Coupon Rate = 15%



RV = Rs.1000

T B rate =9%

Applicable discount rate

For AAA rating bond=9+3=12%

For AA rating bond=12% + 2% = 14%


B0 at 14% = PVCI14% = Intt x PVAF(5 years, 14%) + RV x PVF(5TH year, 14%)

B0 at 14% = PVCI14% = 150*3.432 + 1000*0.519 = Rs.1033.80

If Actual price of the bond = Rs.1025.86

Since B0 at 14% > Actual price of Bond, hence bond is underpriced. It should be purchased.


(c) Actual return of Bond [YTM] if it is purchased at Rs.1025.86

Cash Outflow = Rs.1025.86 [Not part of answer]

Cash inflow = Rs.150 intt for 5 years and Rs.1000 at the end of 5th year. [Not part of answer]

B0 at 14% = PVCI14% = 150*3.432 + 1000*0.519 = Rs.1033.80



= Rs.1033.80 – Rs.1025.86 = Rs.7.94

Since NPV is positive, then take Discount rate = 15%

B0 at 15% = PVCI14% = 150*3.353 + 1000*0.497 = Rs.999.95



= Rs.999.95 – Rs.1025.86 = -Rs.25.91


= 14% + (7.94*1)/[7.94+25.91] = 14.234%

Current Yield = Intt amt/Current market price of Bond = 150/1025.86 = 0.1462 = 14.62%

Solution-8

Face value of bond =500

Interest rate =9%

Annual redemption from 7th to 10 year=500/4=125

Year Principal

outstanding

Interest @9% Principal

repayment

Total PVF =13% Present value

1 500 45 - 45 p.a

3.998

2 500 45 - 45 p.a 179.91

3 500 45 - 45 p.a

4 500 45 - 45 p.a

5 500 45 - 45 p.a

6 500 45 - 45 p.a

7 500 45 125 170 .425 72.25

8 375 33.75 125 158.75 .376 59.69

9 250 22.50 125 147.50 .333 49.12

10 125 11.25 125 136.25 .295 40.19

Intrinsic value 401.16

B0 = Rs.401.16

Current Yield = Next income x 100 = 45 x 100 = 11.21%

Current price 401.16

Solution-8A

Face value of bond =1000

Annual repayment = Face value/life =1000/8=Rs.125

Year Principal outstanding

Interest @10% Principal repayment

Total PVF@8% Present value

1 1000 100 125 225 .926 208.35

2 875 87.50 125 212.50 .857 182.11

3 750 75 125 200 .794 158.50

4 625 62.50 125 187.50 .735 137.81

5 500 50 125 175 .681 119.18

6 375 37.50 125 162.50 .630 102.38

7 250 25 125 150 .583 87.45

8 125 12.5 125 137.5 .540 74.25

Intrinsic value or fair value of bond 1070.33

Solution-9

Face value of the bond = Rs.100

Purchase price of bond = 95% of face value = Rs.95

Coupon Rate = 10%


Tax rate of MIL = 40% [Not relevant for investor]


Half Yr O/s Amt of Bond Intt Redemption Amt Cash Flow PVF=5% PV PVF=6% PV

0 (95.00) 1.000 (95.00) 1.000 (95.00)

1 100.00 5.00 - 5.00 0.952 4.76 0.943 4.72

2 100.00 5.00 - 5.00 0.907 4.54 0.890 4.45

3 100.00 5.00 - 5.00 0.864 4.32 0.840 4.20

4 100.00 5.00 - 5.00 0.823 4.12 0.792 3.96

5 100.00 5.00 - 5.00 0.784 3.92 0.747 3.74

6 100.00 5.00 - 5.00 0.746 3.73 0.705 3.53

7 100.00 5.00 - 5.00 0.711 3.56 0.665 3.33

8 100.00 5.00 - 5.00 0.677 3.39 0.627 3.14

9 100.00 5.00 - 5.00 0.645 3.23 0.592 2.96

10 100.00 5.00 50.00 55.00 0.614 33.77 0.558 30.69

11 50.00 2.50 - 2.50 0.585 1.46 0.527 1.32

12 50.00 2.50 50.00 52.50 0.557 29.24 0.497 26.09

NPV 5.02 (2.89)


= 5% + (5.02*1)/[5.02+2.89] = 5.63%

Yearly YTM = 5.63%*2 = 11.26%

Solution-10 Holding Period: It is considered to be a period of one year. The bond will provide capital appreciation and also interest over a year.

Purchase Price = B0 = Rs.800

Sale Price = B1 = Rs.1000

Interest received = Rs.150

Holding Period return = [(B1 – B0) + Interest] x 100/B0 = [(1000-800)+150]*100/800 = 43.75%

Return due to capital gain = (B1 – B0)*100/B0 = (1000-800)*100/800 = 25%

Current yield = Interest p.a./B0 = 150/800 = 0.1875 = 18.75%

Solution-11


Coupon Rate = 15%



RV = Rs.125

Required rate of return = 18%

Value of Bond at 18% as on 01/04/2003

B0 at 18% = Intt x PVAF(5 years, 18%) + RV x PVF(5TH year, 18%)

B0 at 18% = 15*3.127 + 125*0.437 = Rs.101.53

Value of Bond as on 01/04/2004


B1 at 18% = 15*2.690 + 125*0.516 = Rs.104.85



B2 at 18% = 15*2.174 + 125*0.609 = Rs.108.73


B3 at 18% = Intt x PVAF(2 years, 18%) + RV x PVF(2nd year, 18%)

B3 at 18% = 15*1.565 + 125*0.718 = Rs.113.225


B4 at 18% = Intt x PVF(1 year, 18%) + RV x PVF(1st year, 18%)

B4 at 18% = 15*0.847 + 125*0.847 = Rs.118.58



B5 at 18% = Intt x PVF(0 year, 18%) + RV x PVF(0 year, 18%)

B5 at 18% = 15*1 + 125*1 = Rs.140

Solution-11A The bond was issued at start of 2002

Face Value of bond =1000

Half Yearly Interest =1000 x 10% x 6/12 = Rs.50

Current Yield =9% p.a i.e., 4.5% per half year

The Current date 2003 end assumed that the interest of 2003 has been paid, now remaining period = 6 half years

The intrinsic value of bond at 4.5% half yearly discount rate =

Value of Bond as on 31/12/2003 at 4.5% half year

B0 at 4.5% = Intt x PVAF(6 half year, 4.5%) + RV x PVF(6th half year, 4.5%)

B0 at 4.5% = 50*5.158 + 1000*0.768 = Rs.1025.90

Solution-12


Coupon Rate = 10%



RV = Rs.1000 [Assume]

Purchase price of Bond = Rs.1032.40


Cash inflow = Rs.100 intt for 4 years and Rs.1000 at the end of 4th year. [Not part of Ans]

Value of Bond at 9%


B0 at 9% = 100*3.239 + 1000*0.708 = Rs.1031.90



= Rs.1031.90 – Rs.1032.40 = - Rs.0.54


B0 at 8% = 100*3.312 + 1000*0.735 = Rs.1066.20



= Rs.1066.20 – Rs.1032.40 = Rs.33.80


= 8% + (33.80*1)/[33.80+0.50] = 8.98%

Calculation of YTC


Coupon Rate = 10%


Call Period = 2 yrs

RV = Rs.1100

Purchase price of Bond = Rs.1032.40


Cash inflow = Rs.100 intt for 2 years and Rs.1100 at the end of 2nd year. [Not part of Ans]


B0 at 12% = Intt x PVAF(2 years, 12%) + RV x PVF(2nd year, 12%)

B0 at 12% = 100*1.690 + 1100*0.797 = Rs.1045.70



= Rs.1045.70 – Rs.1032.40 = Rs.13.30


B0 at 13% = 100*1.668 + 1100*0.783 = Rs.1028.10




= Rs.1028.10 – Rs.1032.40 = - Rs.4.30


= 12% + (13.30*1)/[13.30+4.30] = 12.75%

Solution-12A

Calculation of YTC


Coupon Rate = 10%


Call Period = 3 yrs

RV = Rs.1050

Purchase price of Bond = Rs.950


Cash inflow = Rs.100 intt for 3 years and Rs.1050 at the end of 3rd year. [Not part of Ans]

Average return per year per Bond (till call): [(400) / (3)] =133.33

Approximate annual yield = [133.33 /950] ×100 =14.04%



B0 at 14% = 100*2.321 + 1050*0.675 = Rs.940.85



= Rs.940.85 – Rs.950 = - Rs.9.15


B0 at 13% = 100*2.361 + 1050*0.693 = Rs.963.75



= Rs.963.75 – Rs.950 = Rs.13.75


= 13% + (13.75*1)/[13.75+9.15] = 13.6%

Solution-13

FV of Bond = Rs.100

Coupon Rate = 10%

Interest p.a. = Rs.100*10% = Rs.10 for indefinite period

RV = 0

(a) Value of irredeemable Bond at 12% [Constant Cash inflows for indefinite period]

B0 at 12% = PVCI12% = Interest p.a./Required Return = Rs.10/0.12 = Rs.83.33

(b) YTM of irredeemable Bond

Maturity Period = Indefinite Period

RV = Rs.0


Cash inflow = Rs.10 for indefinite period [Not part of Ans]

Actual Return = Constant Cash inflow/B0 = 10/95 = 0.1053 = 10.53%

Solution-13A

Face value = Rs.1000

Annual interest = Rs.120 (12%)

YTM = 15%

Time = Infinite

Value of Debenture = Annual Interest/YTM

= 120/.15 = Rs.800

Market price of Debenture = Rs.700

This bond should be purchased.


Solution-14

Face Value = Rs.100 [Assume]

Coupon Rate = 10%

Interest P.a. = Face Value * Coupon Rate = Rs.100*10% = Rs.10 p.a.

Maturity Period = Indefinite Period

Current Market price of Bond = Rs.110

YTM of irredeemable Bond = Intt/Bo = 10/110 = .0909 = 9.09%

If yield go up by 1% i.e. 10.09

then market price would be = Annual Interest/YTM = 10/.1009 = Rs.99.108

Solution-15

Coupon Rate = 0% [No interest is paid every year but it is accumulated into bond value]


RV = Rs.25000


B0 at 15% = Intt x PVAF(10 years, 15%) + RV x PVF(10th year, 15%)

B0 at 15% = 0*5.019 + 25000*0.247 = Rs.6175

Solution-15A



RV = Rs.25000



B0 at 16% = 0*4.833 + 25000*0.227 = Rs.5675

Solution-16


RV = Rs.1000

Issue Price = Rs.260 [Purchase Price]

Cash Outflow = Rs.260 [Not Part of Ans]

Cash inflow = Rs.0 intt for 10 years and Rs.1000 at the end of 10th year. [Not Part of Ans]

Present value of Rs.1,000 to be received after 10 years = Rs.260 [Not Part of Ans]

PV of Re.1 to be received after 10 years =0.26. Consulting the PVF table, we find that the rate of interest in this case is in the range of 14% to 15%.



B0 at 14% = 0*5.217 + 1000*0.270 = Rs.270



= Rs.270 – Rs.260 = Rs.10


B0 at 15% = 0*5.019 + 1000*0.247 = Rs.247



= Rs.247 – Rs.260 = - Rs.13


= 14% + (10*1)/[10+13] = 14.434%

Solution-16A

(i)

Maturity Period = 1 yr

RV = Rs.1000

Issue Price = Rs.952.38 [Purchase Price]

Cash Outflow = Rs.952.38 [Not Part of Ans]

Cash inflow = Rs.0 intt for 1 year and Rs.1000 at the end of 1st year. [Not Part of Ans]

Present value of Rs.1,000 to be received after 1 year =Rs.952.38 [Not Part of Ans]


PV of Re.1 to be received after 1 years =0.95238 Consulting the PVF table, we find that the rate of interest in this case is in the range of 5%

Hence, YTM = 5%

(ii)


RV = Rs.1000



Cash inflow = Rs.0 intt for 2 years and Rs.1000 at the end of 2nd year. [Not Part of Ans]

Present value of Rs.1,000 to be received after 2 year =Rs.890 [Not Part of Ans]

PV of Re.1 to be received after 2 years =0.890 Consulting the PVF table, we find that the rate of interest in this case is in the range of 6%

Hence, YTM = 6%

Alternative,

1000*(1+r)2 = 890

(1+r)2 = 1.1235

1+r = 1.0599

r = .0599 = 5.99%

(iii)


RV = Rs.1000



Cash inflow = Rs.0 intt for 3 years and Rs.1000 at the end of 3rd year. [Not Part of Ans]

Present value of Rs.1,000 to be received after 3rd years =Rs.816.30 [Not Part of Ans]

PV of Re.1 to be received after 3rd years =0.81630. Consulting the PVF table, we find that the rate of interest in this case is in the range of 7%.

Hence, YTM = 7%

Solution-17



RV = Rs.1000

YTM = 10%

Value of Bond at 10% at t0


B0 at 10% = 0*6.144 + 1000*0.386 = Rs.386


B1 at 10% = RV x PVF(9th year, 10%)

B1 at 10% = 1000*0.424 = Rs.424


B2 at 10% = RV x PVF(8th year, 10%)

B2 at 10% = 1000*0.467 = Rs.467


B9 at 10% = RV x PVF(1st year, 10%)

B9 at 10% = 1000*0.909 = Rs.909


B10 at 10% = RV x PVF(0 yr, 10%)

B10 at 10% = 1000*1 = Rs.1000

Interest for the period

1st year B1 – B0 = 424 – 386 = Rs.38


2nd year B2 – B1 = 467 – 424 = Rs.43

10th Year B10 – B9 = 1000 – 909 = Rs.91

Solution-18

RV of Commercial Paper = FV of commercial Paper = Rs.10 Crs

Maturity Period = 12+30+31+17 = 90 Days

Interest rate = 7.25% p.a.

Interest rate for 90 days = 7.25*90/365 = 1.79%

Since interest is included in RV of CP for 90 days, hence

Issue price = RV/1+0.0179 = 10/1.0179 = 9.82 crs

Solution-18A

(i) Assume that 1 year = 365 days

RV of Commercial Paper = FV of commercial Paper = Rs.10 Crs

Maturity Period = 15+28+31+17 = 91 Days

Interest rate p.a. = 12.04%

Interest rate for 91 days = 0.1204 x 91/365 = 0.03 = 3%

Since interest is included in RV of CP for 91 days, hence

Issue price = RV x PVF(91 days, IR%)

Issue Price = 10 Cr/1.03 = Rs.9.708 Cr

Issue price = Rs.9.708 Crs

Solution-18B


No of days in maturity = 30/01/2004 to 01/11/2003 = 91 days

Annual interest = 8%


Required amt on 30.01.2004 = Rs.80 lakhs [For understanding, it is face value of Bond]

Amt to be deposited = Required Amt*PVF(91 days, IR%)

Amt to be deposited = 80 lacs/1.02 = Rs.78.43 lacs

Solution-18C


No of days in maturity = 91 days

Annual interest = 3.5%

Interest rate for 91 days = 0.035 x 91/365 = 0.0087 = 0.87%

Required amt on 30.01.2004 = Rs.80 lakhs [For understanding, it is face value of Bond]

Amt to be deposited = Required Amt*PVF(91 days, IR%)

Amt to be deposited = 80 lacs/1.02 = Rs.78.43 lacs

Solution-19

No of days in maturity = 90 days

1 year = 360 days

Interest p.a. = 8%


RV = Rs.100

Current Price = RV*PVF(90 days, IR%) = 100/1.02 = Rs.98.039

Current price of money market instrument = Rs.98.039

Equivalent yield or Effective interest rate p.a. = = (1 + PIR)4 – 1 = (1.02)360/90 – 1 = 1.0824 – 1 = 0.0824 = 8.24%

As per suggested answer

Equivalent yield = 2*12/3 = 8% [Assuming compounding annually]

Solution-19A


Interest p.a. = 6%

Interest rate for 45 days = 0.06 x 45/360 = 0.0075

RV = Rs.100

Because interest of 45 days is accumulated in RV


Current price of Instrument = RV*PVF(45 days, IR%)

Current Price = 100/1.0075 = Rs.99.26

(ii) Bond equivalent yield = 0.0075x360/45 = 6%

(iii) Effective annual return is calculated as compounded interest i.e. interest compounding every period of interest.

Hence Effective Interest = (1+PIR)No of Periods in a year – 1 = (1.0075)360/45 – 1 = 6.16%

Solution-19B


Interest rate for 91 days = 0.06 x 91/364 = 0.015 = 1.5%

RV = Rs.100

PV of Redemption value [Because interest of 91 days is accumulated in RV] = RV x PVF(91 days, IR%)

PV of Redemption price = 100/1.015 = Rs.98.52

Issue price = Rs.98.52

(ii) Effective annual return is calculated as compounded interest i.e. interest compounding every period of interest.

Hence Effective Interest = (1+PIR)No of Periods in a year – 1 = (1.015)364/91 – 1 = 1.0613 – 1 = .0613 = 6.13%

Solution-20

Deposit Amt = Rs.100

Maturity Amt = Rs.108

Period = 90 days

Interest of 90 days is accumulated in maturity amt, hence

Interest amt for 90 days = Maturity Amt – Deposit Amt = 108 – 100 = Rs.8

Interest rate for 3 months = Interest amt/Deposit Amt = Rs.8/100 = 8%

Assuming compounding quarterly

Effective Annual interest = (1 + Quarterly Interest)4 – 1 = (1.08)4 – 1 = 1.36 – 1 = 0.36 = 36%

Solution-20A

(a)

Redemption Value = Rs.1000

CMP of Treasury Bill = Rs.973

Maturity Period = 3 months

Since interest is included in RV of Treasury Bill for 3 months, hence

Interest amt for 3 months = RV – CMP of TB = 1000-973 = Rs.27

Periodic Interest rate for 3 months = Interest Amt for 3 months/CMP = 27/973 = 0.02775 = 2.775%

Assuming compounding quarterly

Effective Annual interest = (1 + Quarterly Interest)4 – 1 = (1.02775)4 – 1 = 1.1157 – 1 = .1157 = 11.57%

(b)

Coupon Rate = 10% payable half yearly

Half yearly interest = 5%

Assuming Compounding half yearly

Effective Annual interest = (1 + Half Yearly Interest)2 – 1 = (1.05)2 – 1 = .1025 = 10.25%

Solution-20B


CMP or Issue Price = Rs.9940

Maturity Period = 91 days

Interest for 91 days = RV – CMP = 10000-9940 = Rs.60

Periodic Interest rate for 91 days = Interest Amt for 91 days/CMP = 60/9940 = 0.0060 = 0.6%

Assuming compounding annually

Effective Annual interest = Period interest rate * 365/Period = 0.6*365/91 = 2.40%

Discount Rate for Govt Investment = Period interest rate * 360/Period = 0.6*360/91 = 2.37%

Solution-21


(a)


CMP or Issue Price of CP = Rs.97350


Since interest is included in RV of CP for 3 months, hence

Interest amt for 3 months = RV – CMP = 100000-97350 = Rs.2650




Cost of funds to the company

Effective interest p.a. 10.89%

Brokerage (0.125*4) 0.50%

Rating charge 0.50%

Stamp duty (0.125%*4) 0.50

Cost of funds p.a. 12.39%

Solution-21A

(a)


CMP or Issue Price = Rs.98000


Since interest is included in RV of CP for 4 months, hence

Interest amt for 4 months = RV – CMP = 100000-98000 = Rs.2000




Cost of funds to the company

Effective interest 6.12%

Brokerage 0.1% [Assuming Annually] 0.10%

Rating charge = 0.6% [Assuming Annually] 0.60%

Stamp duty = 0.15% [Assuming Annually] 0.15%

Cost of funds 6.97%

Note: In the question it has not been clearly mentioned whether issue expenses pertain to a year or 4 months. Although above solution is based on the assumption that this expense pertain to year.

Alternative Solution

Cost of Funds to the company [assuming expenses for issue is given for 4months basis]

Effective Interest rate = 6.12%

Brokerage = 0.10*12/4 = 0.30%

Rating Charges = 0.60*12/4 = 1.80%

Stamp Duty = 0.15*12/4 = 0.45%

Cost of Funds = 8.67%

Cost of funds in amount

Assuming issue exp is based on Rs.98000

Cost of Fund = 98000*8.67% = Rs.8496.00

(b) If Cost of Issue/ Fund is given in amount then

Cost of Issue in % = Cost of Fund in Amt/Net amount utilized for Operation

Solution-22

Issue Price = Rs.5000000


Interest Rate = 12.5% p.a.

Interest amt of 4 months = 5000000*12.5%*4/12 = Rs.208333

Cost of Placement per issue = Rs.2500

Total Cost of Issue = Rs.208333+2500 = Rs.210833 [Assuming incurred at the beginning]


Tax rate = 30%

Cost of Issue [NOT] = Rs.210833*0.7 = Rs.147583

Fund remain blocked in bank Account = Rs.150000

Net amount that can be utilized for operation = Rs.5000000-210833-150000 = Rs.4639167

Cost of Fund = Cost of Issue/Net Amount of Utilisation = 147583/4639167 = 0.0318 = 3.18% for 4 months

Cost of Fund p.a. = 3.18*12/4 = 9.54%

Solution-23

Face value of the Bond = Rs.1000

Coupon Rate = 8.5%

Interest Amt = Rs.85


RV = Rs.1000

Bond Duration

Year Cash Inflow PVF=10% PV Weight Weight * Year

1 85.00 0.909 77.27 0.082 0.082

2 85.00 0.826 70.21 0.074 0.149

3 85.00 0.751 63.84 0.068 0.203

4 85.00 0.683 58.06 0.062 0.246

5 1,085.00 0.621 673.79 0.714 3.572

PV of Bond 943.15 1.000 4.252

Bond Duration = 4.252 yrs

Volatility of Bond = Bond Duration/[1+YTM/n]

Volatility of Bond = 4.252/1.1 = 3.865%

Volatility of bond says that if YTM increase by 1% then price of bond will decrease by 3.865% and vice versa

(b)

Volatility of Bond = 3.865%

Current market price of Bond = Rs.943.15

Current YTM = 10%

If YTM increase by 1% then price of bond will decrease by 3.865%

Hence price of bond at 11% YTM = Current maket price of Bond * [1-Volatility of Bond*(Increase in YTM)]

= Rs.943.15(1-0.0386*1) = Rs.943.15*0.9614 = Rs.906.7444

Cross verification of price of Bond at YTM = 11% [Not part of ans in Exam]

Year Cash Inflow PVF=11% PV

1 85.00 0.901 76.59

2 85.00 0.812 69.02

3 85.00 0.731 62.14

4 85.00 0.659 56.02

5 1,085.00 0.593 643.41

PV of Bond 907.16

Solution-23A

Face value of the Bond = Rs.1000

Coupon Rate = 11.5%

Interest Amt = Rs.115


RV = Rs.1000

Bond Duration


1 115.00 0.909 104.54 0.099 0.099

2 115.00 0.826 94.99 0.090 0.180

3 115.00 0.751 86.37 0.082 0.245

4 115.00 0.683 78.55 0.074 0.297

5 1,115.00 0.621 692.42 0.655 3.276

PV of Bond 1,056.85 1.000 4.097





Volatility of bond says that if YTM increase by 1% then price of bond will decrease by 3.724% and vice versa

(b)

Volatility of Bond = 3.724%

Current market price of Bond = Rs.1056.85

Current YTM = 10%

If YTM increase by 1% then price of bond will decrease by 3.724%

Hence price of bond at 11% YTM = Current maket price of Bond * [1-Volatility of Bond*(Increase in YTM)]

= Rs.1056.85*(1-.03724*1) = Rs.1056.85*0.96276 = Rs.1017.493

Cross verification of price of Bond at YTM = 11% [Not part of Ans in Exam]

Year Cash Inflow PVF=10% PV

1 115.00 0.901 103.62

2 115.00 0.812 93.38

3 115.00 0.731 84.07

4 115.00 0.659 75.79

5 1,115.00 0.593 661.20

PV of Bond 1,018.04

Solution-24

For calculation of duration of bond, we need YTM

WN-Calculation of YTM of X Ltd’s Bond

CMP of Bond = Rs.10796.80

FV = Rs.10000

CR = 6%

Interest p.a. = Rs.10000*6% = Rs.600 p.a.

Maturity Period = 5 years

B0 at 6% = Rs.10000 [Since FV and RV is equal]

NPV at 6% = B0 at 6% - CMP of bond

= Rs.10000 – Rs.10796.80 = - Rs.796.80


B0 at 4% = 600*4.453 + 10000*0.822 = Rs.10891.80

NPV at 12% = Rs.10891.80 – Rs.10796.80 = Rs.95


= 4% + (95*2)/(95+796.80) = 4.20%

Calculation of Bond duration of X Ltd’s Bond at YTM = 4.20%


1 600 0.960 576.00 0.053 0.053

2 600 0.921 552.60 0.051 0.102

3 600 0.884 530.40 0.049 0.147

4 600 0.848 508.80 0.047 0.189

5 10,600.00 0.814 8,628.40 0.799 3.996

PV of Bond 10796.20 1.000 4.488

Bond Duration of X Ltd’s Bond= 4.488 yrs

Calculation of Bond duration of Y Ltd’s Bond at YTM = 4.20%

Coupon Rate = 4%


1 400 0.960 384.00 0.039 0.039

2 400 0.921 368.40 0.037 0.074

3 400 0.884 353.60 0.036 0.107

4 400 0.848 339.20 0.034 0.137

5 10,400.00 0.814 8,465.60 0.854 4.271

PV of Bond 9,910.80 1.000 4.628

Bond Duration of Y Ltd’s Bond= 4.628 yrs


Decision: Since the duration of Bond of X Ltd is lower, hence it should be preferred.

Solution-25


YTM = 8% p.a.



Volatility of bond says that if YTM increase by 1% then price of bond will decrease by 4.326% and vice versa.

There is inverse relationship between Bond price and YTM, if YTM decreases by 1%, then price of the Bond will increase by 4.326%

Solution-26

Period Cash Inflow PVF = 7% PV of cash inflows

Proportion of PV Period x Proportion

1 0 0.935 0 0 0

2 0 0.873 0 0 0

3 1000 0.816 816 1 3

Total 816 1 3

Duration = 3

Solution-27

Let annual interest = C

Year Cash Flow PVF = 16% PV of Cash Inflow PV of Cash Inflow * Year

1 C 0.862 C×0.862 0.862c

2 C 0.743 C×0.743 1.486c

3 C 0.641 C×0.641 1.923c

4 C 0.552 C×0.552 2.208c

5 C 0.476 C×0.476 2.308c

6 C 0.410 C×0.410 2.460c

6 100000 0.410 1,00,000×0.410 2,46,000

3.684C +41000 246000 +11.319c

Bond Duration = Sum of [PV of Cash Inflow*Year]/Current price of Bond

4.3203 = (246000 +11.319C) / (3.684c +41000)

15.915985 C +177132.3 =246000 +11.319 C

4.596985 C =68867.7

C =14981 say 15000

Coupon rate =15%

Interest p.a. = FV of Debenture*Coupon rate = Rs.100000*15% = Rs.15000

Current price of debenture = Intt*PVAF(6 yrs, 16%) + RV*PVF(6th yr, 16%) = 15000*3.685 +100000*0.410 =96275

Solution-28

Calculation of YTC


Coupon Rate = 11%


Call Period = 3 yrs

RV = Rs.1100




B0 at 13% = 110*2.361 + 1100*0.693 = Rs.1022.01



= Rs.1022.01 – Rs.985 = Rs.37.01


B0 at 15% = 110*2.284 + 1100*0.658 = Rs.975.04




= Rs.975.04 – Rs.985 = - Rs.9.96

YTC or IRR or Actual Return of Bond = LR + [NPVLR]*Diff of rate/[NPVLR – NPVHR]

= 13% + (37.01*2)/[37.01+9.96] = 14.575%

Duration to call for 11%, ABC Ltd Bond

Year Cash Inflow PVF=14.575% PV Weight Weight * Year

1 110.00 0.873 96.03 0.098 0.098

2 110.00 0.762 83.82 0.085 0.170

3 1,210.00 0.665 804.65 0.817 2.452

984.50 1.00 2.72


Solution-29 Calculation of Actual investment of Portfolio and Bond Duration of Portfolio

Security Purchase Price

Face of Security Purchased

Amt of Investment Weight Bond Duration

Bond of Duration of Portfolio

GOI 2006 106.50 5.00 532.50 0.20 3.500 0.707

GOI 2010 105.00 5.00 525.00 0.20 6.500 1.294

GOI 2015 105.00 5.00 525.00 0.20 7.500 1.493

GOI 2022 110.00 5.00 550.00 0.21 8.750 1.829

GOI 2032 101.00 5.00 505.00 0.19 13.000 2.483

Total Investment 2,637.50 1.00 Bond Duration 7.806

Suitable action to churn out investment portfolio in following scenario

To reduce risk and to maximize profit or minimize losses

(a) There is inverse relationship between Interest rate and price of Bond. Bond having high duration will have high volatility and vice versa. Hence if interest rates are expected to be lower by 25 basis points, in such case, we should increase the average duration of our portfolio by purchasing bond having high duration i.e. GOI 2032 and selling bond having low duration i.e. GOI 2006.

(b) There is inverse relationship between Interest rate and price of Bond. Bond having high duration will have high volatility and vice versa. Hence if interest rates are expected to be increased by 75 basis points, in such case, we should decrease the average duration of our portfolio by purchasing bond having low duration i.e. GOI 2010 and selling bond having having duration i.e. GOI 2032.

Solution-29A

(i) Calculation of Weighted average duration of portfolio

Bond % of Money to be invested Duration of the bond (years) % * Duration

1 10 10.35 1.035

2 22 4.25 0.935

3 19 7.50 1.425

4 7 9.50 0.665

5 17 12.67 2.1539

6 6 5.82 0.3492

7 11 8.50 0.935

8 8 6.71 0.5368

8.00

Bond duration of Portfolio = 8.00 yrs

(ii) Suitable action to churn out investment portfolio in following scenario

To reduce risk and to maximize profit or minimize losses.

(a) There is inverse relationship between Interest rate and price of Bond. Bond having high duration will have high volatility and vice versa. Hence if interest rates are expected to be lower by 25 basis points, in such case, we should increase the average duration of our portfolio by purchasing bond having high duration i.e. Bond 5 and selling bond having low duration i.e. Bond 4.

(b) There is inverse relationship between Interest rate and price of Bond. Bond having high duration will have high volatility and vice versa. Hence if interest rates are expected to be increased by 75 basis points, in such case, we should


decrease the average duration of our portfolio by purchasing bond having low duration i.e. Bond 6 and selling bond having duration i.e. Bond 5.

Solution-30

CMP of convertible bond = 970

No of equity shares on conversion = 50 shares

CMP of share =18.50

Convertible value of convertible bond =18.50 x 50 = 925

CMP of non-convertible bond or Straight value of Bond =870

Interest on convertible bond =11.5% of 1000 = 115

Dividend per share =2.12

(i)

Conversion parity price of Share = CMP of CB /Conversion Ratio = 970/50 Rs.19.40

(ii) Premium over conversion value

Conversion Premium or Premium over conversion value or Ratio of Conversion Premium

= (CMP of CB – Conversion Value)*100/ Conversion Value

= (970-925)*100/925 = 4.86%

Conversion Premium or Premium over conversion value (Amt) = (CMP of CB – Conversion Value) = 970-925 = Rs.45

(iii) Conversion premium per share

Conversion premium per Share = Conversion Premium/Conversion Ratio = 45/50 = Rs.0.90

Conversion Premium per share = Conversion Parity Price of Share - CMP of Share = = 19.40 – 18.50 = Rs.0.90

(iv) Premium over Investment

Downside risk or Premium over Investment Value (%) = (CMP of CB – CMP of NCB)*100/CMP of NCB

= (970-870)*100/870 = 11.49%

Downside risk or Premium over Investment Value (Amt) = (CMP of CB – CMP of NCB) = (970-870) = Rs.100

(v)

Annual dividend on 50 shares = Rs.2.12*5 = Rs.106

Annual interest on CB = Rs.115

Favorable income differential = Interest Income p.a. – Dividend per Share*Conversion Ratio = 115-106 = Rs.9

Favorable income differential per Share = (Interest Income p.a. – Dividend per Share*Conversion Ratio)/Conversion Ratio = (115-106)/50 = Rs.0.18

(vi)

Premium pay back period Financial Services or Break Even Period = Conversion Premium/Favorable Income Differential

= Rs.45/5 = 5 Years OR

= Conversion Premium per Share/Favorable Income differential per share = 0.9/0.18 = 5 years

Solution-30A

a) Conversion value of Debenture = CMP of Equity share x Conversion Ratio =25x30 = Rs.750

b) Market Conversion Price = CMP of CB /Conversion Ratio = 900/30 = Rs.30

c) Conversion Premium per share

Conversion Premium = (CMP of CB – Conversion Value) = 900-750 = Rs.150

Conversion Premium per share = Conversion Premium/Conversion Ratio = Rs.150/30 = Rs.5 OR

= Conversion Parity Price of Share – CMP of Share =30-25 = Rs.5

(d)

Conversion Premium or Premium over conversion value or Ratio of Conversion Premium

= (CMP of CB – Conversion Value)*100/ Conversion Value = (900-750)*100/750 = 20%

(e) Premium over Straight Value Debenture

Downside risk or Premium over Investment Value (%) = (CMP of CB – CMP of NCB)*100/CMP of NCB

= (900-700)*100/700 = 2 8.57%

Downside risk or Premium over Investment Value (Amt) = (CMP of CB – CMP of NCB) = (900-700) = Rs.200


(f)

Annual dividend on 30 shares = Rs.1*30 = Rs.30

Annual interest on CB = Rs.85

Favorable income differential = Interest Income p.a. – Dividend per Share*Conversion Ratio = 85-30 = Rs.55

Favorable income differential per Share = (Interest Income p.a. – Dividend per Share*Conversion Ratio)/Conversion Ratio = (85-30)/30 = Rs.1.833

(g)

Premium pay back period Financial Services or Break Even Period = Conversion Premium/Favorable Income Differential

= Rs.150/55 = 2.73 Years OR

= Conversion Premium per Share/Favorable Income differential per share = 5/1.833 = 2.73 years

Solution-31


Coupon Rate of CB = 8.5%

Interest P.a. = Face Value of the Bond*Coupon Rate = Rs.1000*8.5% = Rs.85 p.a.



Coupon rate of NCB = 9.5%


B0 at 9.5% = Intt x PVAF(3 years, 9.5%) + RV x PVF(3rd years, 9.5%)

B0 at 12% = 85*2.5089 + 1000*0.7617 = Rs.975

(i) Straight Value of Bond = Rs.975

CMP of convertible bond Rs.1175

Conversion Ratio 25

CMP of share Rs.45

CMP of non-convertible bond or Straight value of Bond Rs.975

Interest on convertible bond 8.5%

(ii)

Conversion value of Bond = CMP of share*Conversion Ratio = Rs.45*25 = Rs.1125

(ii) Conversion Premium or Premium over conversion value or Ratio of Conversion Premium

= (CMP of CB – Conversion Value)*100/ Conversion Value

= (1175-1125)*100/1125 = 4.44%

Conversion Premium or Premium over conversion value (Amt) = (CMP of CB – Conversion Value) = 1175-1125 = Rs.50

(ii) Conversion premium per share [As per Suggested]

Conversion premium per Share = Conversion Premium/Conversion Ratio = 50/25 = Rs.2

(iii) % of Downside risk or Premium over Investment Value (%) = (CMP of CB – CMP of NCB)*100/CMP of NCB

= (1175-975)*100/975 = 20.51%

(iv) Conversion parity price of Share = CMP of CB/ Conversion Ratio = 1175/25 = Rs.47

Solution-32

Face Value of the Debenture = Rs.100

Coupon Rate = 12%



RV = Rs.100

RR = 8%

Value of Debenture at 8% if it is not converted into share

B0 at 8% = Intt x PVAF(5 years, 8%) + RV x PVF(5th years, 8%)

B0 at 8% = 12*3.993 + 100*0.681 = Rs.116.01

Convertible value of debenture

CMP of Share Convertible Ratio Convertible Value of Debenture

Rs.4 20 Rs.80


Rs.5 20 Rs.100

Rs.6 20 Rs.120

Hence, unless the market price is Rs. 6 conversion should not be exercised.

Solution-33

Option 1 If Old Bond is not redeemed

Existing outstanding debts =300 M

Interest rate =12%

Interest P.a. = Rs.300*12% = Rs.36 m for 6 years

Remaining life =6 years

Unamortized issue cost = 9 M

Amortisation of Issue cost p.a. = 9/6 = 1.5m

Assume Unamortised cost is tax deductible

Tax rate = 30%

Cost of Capital = 10%*0.7 = 7%

PV of Cash out flow if old Bond is continuously used (Amt in m)

Year Particulars Amt PVF=7% PVCO

1 to 6 Interest on old Bond 36*0.7 4.766 120.10

6th RV of Old Bond 300 0.666 199.80

1 to 6 Tax Saving on amortization of Issue cost on old Bond -1.5*0.3 4.766 -2.145

317.75

Option 2 If Old Bond is redeemed and new Bond is issued

Callable Value of Old Bond = 4% Premium

Total Callable Value of Old Bond = 300*1.04 = Rs.312 m

Premium payable on old bond =Rs.12m

Floatation cost of new bond = Rs.6m

Amortisation cost of New Bond p.a. = Rs.6/6 = 1m p.a.

Coupon Rate of New Bond = 10%

New bond issue value = 300 m

Intt p.a. on new Bond = Rs.300*10% = Rs.30m

Net receipt on issue of new bond = 300 - 6(issue cost)] = 294 m


1 to 6 Interest on new Bond 30*0.7 4.766 100.08

6th RV of new Bond 300 0.666 199.80

1 to 6 Tax Saving on amortization of Issue cost on new Bond -1*0.3 4.766 -1.43

0 Issue of new Bond (Net Receipts) -294 1 -294

0 Redemption of Old Bond including Premium 312 1 312

1 Tax saving on premium on old Bond -12*0.3 0.935 -3.37

Tax saving on writing off of unamortized cost of old Bond -9*0.3 0.935 -2.58

310.50

Since PVCO under option 2 is less than option 1, hence Bond should be redeemed

Solution-33A


Existing outstanding debts = 300 lakh

Interest rate =12.5%

Interest P.a. = Rs.300*12.5% = Rs.37.50 lakhs for 12 years

Remaining life =12 years

Unamortized issue cost = 12*30000 =3.60 lakhs


Tax rate = 50%

PV of Cash out flow if old Bond is continuously used

Year Particulars Amt (Lacs) PVF=6% PVCO

1 to 12 Interest on old Bond 37.50*0.5 8.383 157.18

12th RV of Old Bond 300 0.497 149.10



305.03


Callable Value of Old Bond = 1050 per Bond

Total Callable Value of Old Bond = 1050*300/1000 = Rs.315 lacs

Premium payable on old bond =Rs.15 lacs

Floatation cost of new bond = Rs.9 lacs

Amortisation cost of New Bond p.a. = Rs.9/12 = 0.75 lacs p.a.


New bond issue value = 300 lacs

Intt p.a. on new Bond = Rs.300*10% = Rs.30 lacs

Net receipt on issue of new bond = 300 - 9(issue cost)] = 291 lacs



12th RV of new Bond 300 0.497 149.10

1 to 12 Tax Saving on amortization of Issue cost on new Bond -0.75*0.5 8.383 -3.14




1 Tax saving on writing off of unamortized cost of old Bond -3.60*0.5 0.943 -1.70

286.93


Solution-34


Existing outstanding debts = 300 lacs

Interest rate =14%

Interest P.a. = Rs.300*14% = Rs.42 lacs for 25 years

Remaining life = 25 years

Discount on old Bond = 9 lacs

Floatation cost on Old Bond = 3.6 lacs

Amortisation of Issue cost p.a. = 12.6/30 = 0.42 lacs

Unamortised Cost of Old Bond = 12.6 – 0.42*5 = 10.5 lacs


Tax rate = 40%

Cost of Capital = 8%

PV of Cash out flow if old Bond is continuously used

Year Particulars Amt (Lacs) PVF=8% PVCO

1 to 25 Interest on old Bond 42*0.6 10.674 268.98

25th RV of Old Bond 300 0.146 43.80


310.99


Callable Value of Old Bond = 1140 per Bond

Total Callable Value of Old Bond = 1140*300/1000 = Rs.342 lacs

Premium payable on old bond =Rs.42 lacs

Floatation cost of new bond = Rs.4 lacs

Amortisation cost of New Bond p.a. = Rs.4/25 = 0.16 lacs p.a.


New bond issue value = 300 lacs

Intt p.a. on new Bond = Rs.300*12% = Rs.36 lacs

Net receipt on issue of new bond = 300 - 4(issue cost)] = 296 lacs




25th RV of new Bond 300 0.146 43.80

1 to 25 Tax Saving on amortization of Issue cost on new Bond -0.16*0.4 10.674 -0.68




1 Tax saving on writing off of unamortized cost of old Bond -10.5*0.4 0.926 -3.89

0 Overlapping Interest of 2 months = 300*14%*2/12 = 7 7 1 7

0 Tax saving on Overlapping interest -7*0.4 0.926 -2.59

304.69


Solution-35

(a)

For Zero coupon Bond,

Bo = RV*PVF1

RV*1/(1+DR) = Bo

1+DR = RV/Bo

DR = 100/93.46 – 1 = 0.0699 = 7%

Spot rate = 7%

(b) For bond with coupon rate

FV = Rs.100

CR = 5%

Intt p.a. = Rs.5

RV = 100


Bo = Rs.98.13

Discount rate for first year = 7%

Bo = Intt1/1.07 + (Intt2+RV)/1.07*(1+FR2)

5/1.07 + 105/(1.07)(1+ FR2) = Rs.98.13

FR2 =5%

(c) For bond with coupon rate

FV = Rs.100

CR = 9%

Intt p.a. = Rs.9

RV = 100


Discount rate for first year = 7%

Discount rate for second year = 5%

Bo = Rs.104.62

Bo = Intt1/1.07 + Intt2/[1.07*1.05] + [Intt3 + RV]/[1.07*1.05*(1+FR3]

9/1.07 + 9/(1.07)(1.05) + 109/(1.07)(1.05)(1+r) = Rs.104.62

FR3 =10%

Solution-35A

(a) Consider one year security

For Zero coupon Bond,

Bo = RV*PVF1

RV*1/(1+DR) = Bo

1+DR = RV/Bo

DR = 100000/91000 – 1 = 1.099 – 1 = 9.9%

(b) Consider 2 year security of coupon rate

FV = Rs.100000


CR = 10.5%

Intt p.a. = Rs.10500

RV = 100000


Bo = Rs.99000

Discount rate for first year = 9.9%

Bo = Intt1/1.0928 + (Intt2+RV)/1.0928*(1+FR2)

10500/1.099 + 110500/(1.099)(1+ FR2) = Rs.99000

9554.14 + 100546.00/(1+ FR2) = 99000

(1 + FR2) = 100546/(99000-9554.14)

FR2 = 1.1240 -1 = .1240 = 12.40%

(c) Consider 3 year security of coupon rate

FV = Rs.100000

CR = 11%


RV = 100000


Bo = Rs.99500


Discount rate for second year = 12.4%

Bo = Intt1/1.099 + Intt2/1.099*1.124 + (Intt3 + RV)/1.099*1.124*(1+FR3)

11000/1.099 + 11000/(1.099*1.124) + 111000/1.099*1.124*(1+FR3) = Rs.99500

10009.10 + 8904.90 + 89858.46/(1+FR3) = 99500

(1 + FR3) = 89858.46/(99500-10009.10-8904.90)

FR3 = 1.115 - 1 = 0.115 = 11.50%

(d) Consider 4 year security of coupon rate

FV = Rs.100000

CR = 11.5%


RV = 100000


Bo = Rs.99900


Discount rate for second year = 12.4%

Discount rate for 3rd year = 11.5%

Bo = Intt1/1.099 + Intt2/(1.099*1.124) + Intt3/(1.099*1.124*1.115) + (Intt4 + RV)/(1.099*1.124*1.115)*(1+FR4)

11500/1.099 + 11500/(1.099*1.124) + 11500/(1.099*1.124*1.115)+111500/(1.099*1.124*1.115)*(1+FR4)= Rs.99900

10464.06 + 9309.66 + 8349.47 + 80953.57/(1+FR4) = 99900

(1 + FR4) = 80953.57/(99900-10464.06-9309.66-8349.47)

FR4 = 1.128 - 1 = 0.128 = 12.80%

Solution-36

(a) For 2 years YTM = 11.25

For first year Spot rate = 10.5% = FR1

Based on expectation theory

(1+YTM)2 = (1+FR1)(1+FR2)

(1.1125)2 = (1.105)(1+FR2)

FR2 = 12%

For 3 years YTM = 12%

For first year Sport rate = 10.5% = FR1

For Second year forward rate = 12%%


(1+YTM)3 = (1+FR1)(1+FR2) (1+FR3)

(1.12)3 = (1.105)(1.12) (1+FR3)


FR3 = 13.52%

(b) If fairly priced at Rs.1000 and rate of interest increases to 12.5% the percentage change will be as follows:

Price = 1000(1.12)5/(1.125)5 = 1762.34168/1.8020

= 977.99 or Rs.978

% change = 1000 – 978/1000 x 100 = 22/1000 x100 = 2.2%

Alternative Method


Coupon Rate = 12% = YTM [As bond is fairly priced]





B0 at 12.5% = Intt x PVAF(5 years, 12.5%) + RV x PVF(5th years, 12.5%)

B0 at 12.5% = 120*3.560 + 1000*0.555 = Rs.982.20

% change = (1000 – 982)*100/1000 = 18*100/1000 = 1.8%

Solution-36A

(a) For 2 years YTM = 11%

For first year Spot rate = 10% = FR1


(1+YTM)2 = (1+FR1)(1+FR2)

(1.11)2 = (1.1)(1+FR2)

FR2 = 1.2321/1.1 -1 = 1.12 -1 = 12%

For 3 years YTM = 12%

For first year Sport rate = 10% = FR1

For Second year forward rate = 12%%


(1+YTM)3 = (1+FR1)(1+FR2) (1+FR3)

(1.12)3 = (1.1)(1.12) (1+FR3)

FR3 = 1.4049/1.232 – 1 = 1.1403 – 1 = 14.03%

Solution-37

Appropriate price = 80/ (1.06) + 1080/ (1.07)2= 1018.79

Alternative way:

Suppose we invest Rs. 100 today, it will grow to Rs. 106 after 1 year.

Suppose we invest Rs. 100 today, it will grow to 100(1.07)2 after 2 years

Forward rate for year 1 = 6%

Forward rate for year 2 = [100(1.07)2/ 1.06] - 1 = 8.00996

Appropriate price = 80/ (1.06) + 1080/ {(1.06) (1.08009)} = 1018.79

Solution-38

(i)

Market value of Bond = Face value of Bond [RV = FV]

YTM = CR

Since the bonds were sold at par, hence YTM of Bond as on 1, Jan 2000 were 10%

(ii)


Coupon Rate = 10%


Half yearly Interest = Rs.50

Maturity Period from 01/01/2008 = 8 yrs


Value of Bond at 12% as on 01/01/2008

B0 at 12% = Intt x PVAF(16 Period, 6%) + RV x PVF(16th Period, 6%)

B0 at 12% = 50*10.105 + 1000*0.394 = Rs.899.25


Value of Bond as on 01/01/2008 = Rs.899.25

Value of Bond as on 01/03/2008 = 899.25*1.02 = 917.24

Payment for Complete transaction = 917.24

Interest accrued = 50*2/6 = 16.67

Bonds basic value = 917.24 – 16.67 = 900.57

Solution-39

(a) If the yield of the bond falls the price will always increase. This can be shown by following calculation.

5 year Bond


Coupon Rate = 8%




Value of Bond at 6%

B0 at 6% = Intt x PVAF(5 years, 6%) + RV x PVF(5th Year, 6%)

B0 at 6% = 80*4.212 + 1000*0.747 = Rs.1083.96

Increase in 5 year's bond price = Rs. 83.96

20 year Bond


Coupon Rate = 8%




Value of Bond at 6%

B0 at 6% = Intt x PVAF(20 years, 6%) + RV x PVF(20th Year, 6%)

B0 at 6% = 80*11.469 + 1000*0.312 = Rs.1229.52

So increase in bond price is Rs. 229.52

PRICE INCREASE DUE TO CHANGE IN PV OF PRINCIPAL

5 yrs. Bond

= RV*PVF(5th Year, 6%) - RV*PVF(5th Year, 8%)

= 1000*0.747 – 1000*0.681 = 747 - 681 = Rs.66.00

% Change in price due to change in PV of Principal = Rs.66/83.96 = 79%

20 yrs. Bond

= RV*PVF(20th Year, 6%) - RV*PVF(20th Year, 8%)

= 1000*0.312 – 1000*0.215 = 312 - 215 = Rs.97.00

% Change in price due to change in PV of Principal = Rs.97/229.52 = 42%

PRICE CHANGE DUE TO CHANGE IN PV OF INTEREST

5 yrs. Bond

= Rs.80 x (PVAF 6%, 5) - Rs.80 x (PVIAF 8%, 5)

= Rs.80 x 4.212 - Rs.80 x 3.993 = Rs.336.96 - Rs.319.44 = Rs.17.52

Change in price Rs.17.52/ Rs.83.96 x 100 = 20.86%

20 yrs. Bond

= Rs.80 x PVAF 6%, 20) - Rs.80 x (PVAF 8%,20)

= Rs.80 x 11.47 - Rs.80 x 9.82 = Rs.917.60 - Rs.785.60 = Rs.132

Change in price = Rs.132/ Rs.229.60 x 100 = 57.49%

Solution-40

Initial purchase of bond =Rs.1000

Annual interest for 2 years =Rs.100 each

Callable value at end of year 2 =1200


Again Investment in another bond Rs. 1000 (Assuming Bond cannot be issuedin part)

Annual Interest =7% of 1000=Rs.70

Redeemable value =1000

[Not part of answer in exam]

Cash flows

Year:

0 =(-)1000

1 100

2 100+1200-1000=300

3 70

4 70

5 70+1000=1070

Taking a discount rate 9% [slightly higher than the interest rate of 7% because inflow on 1 & 2 year is better i.e., 100 & 300 as compare to 70]

= [100*0.917 + 300*0.842 + 70*0.772 + 70*0.708 + 1070*0.650] – 1000

= [91.70+252.60+54.04+49.56+695.50] -1000 = 1143.40-1000 = 143.40

Taking a discount rate 15% , PV of inflows

= [100*0.870 + 300*0.756 + 70*0.658 + 70*0.572 + 1070*0.497] – 1000 = (-) 68.31

YTM or IRR or Actual Return from investment = LR + [NPVLR]*Diff of rate/[NPVLR – NPVHR]

= 9% + (143.40*6)/(143.30+68.31) = 13.06%

Solution-41 While calculating NPV, it is assumed that the intermediate cash inflows are reinvested at a rate equal to cost

of capital being uses as discount rate. Here the reinvestment is at 18%.

Face Value of bond =1000

Annual Interest =150

Life =4 years

Re investment of Interest =18%

Cash inflow at the end of 4th year = 150*(1.18)3 + 150*(1.18)2 + 150*(1.18)1 + 150*(1.18)0 + 1000

= 246.45 + 208.86 + 177 + 150 + 1000 =1782.31

Initial Outflows = 1000

Applying Discount rate 20%

PVCI at 20% = 1782.31/ (1.20)4 = 859.52

NPV =859.52-1000 = -140.48

Applying Discount rate 15%

PV of inflows =1782.31/(1.15)4 = 1019.48

NPV = 1019.48 -1000 = 19.48

YTM or IRR or Actual Return from investment = LR + [NPVLR]*Diff of rate/[NPVLR – NPVHR]

= 15% + (19.48*5)/(19.48.30+140.48) = 15.60%

Solution-42


Face value of the Bond = Rs.100 [Assuming]

Coupon rate = 9%

Interest p.a = Rs.100*9% = Rs.9 p.a.

RV = Rs.105


Cash Outflow = Rs.90

Cash inflow = Rs.9 intt for 5 years and Rs.105 at the end of 5th year before tax.

Income Tax rate = 30% and Capital gain tax rate = 10%

Interest after tax = Rs.9*0.7 = Rs.6.30 for 5 yrs

Capital Gain = RV – Purchase price = 105-90 = Rs.15

Capital gain tax = Rs.15*10% = Rs.1.50

RV [after capital gain tax] = Rs.105-1.50 = Rs.103.50

App annual return = (6.30 + [(103.50-90)/5])/[(90+103.50)/2] = (6.30+2.70)/96.75 = 9.30%

Calculation of Actual Return

First discount rate = 10% as calculated above.

B0 at 10% = 6.30*3.790 + 103.50*0.621 = Rs.88.15




= Rs.88.15 – Rs.90 = - Rs.1.85


B0 at 9% = = 6.30*3.889 + 103.50*0.650 = Rs.91.77



= Rs.91.77 – Rs.90 = Rs.1.77


= 9% + (1.77*1)/(1.77+1.85) = 9.49%

Date post:	25-Dec-2021
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CHAPTER-4 BOND VALUATION - Helping Hand Institute

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