001 Fi Interest Rate and Bond Calculations

Fixed Income Fundamentals1. Interest Rate & Bond Calculations – Day 1

Financial Markets Education

Interest Rates


1

Onn [email protected]

Kai-Hing [email protected]


Financial Markets Education provides instruction on all aspects of banking and finance for UBS employees and for our top clients.

Joe [email protected]

Walter [email protected]

Spencer [email protected]

Americas

Joe BoninStamford+ 1-203-719-6507 [email protected]

Europe APAC

2

Money Market Rates

♦ Investments of up to one year

♦ Retail

♦ Banks, Thrifts, Building Societies

♦ Brokerage “sweep” accounts

♦ Money Market Funds

♦ All of these take in cash deposits

♦ Pay interest

♦ Institutional Investors

♦ Money Centre Banks

♦ Financial Institutions

♦ Retail Investors

♦ All either have excess funds or need funds short term

♦ Money Market rates are usually add-on

– Interest earned is calculated based on the amount invested and added on to it

3

Interest Rate Terminology

♦ Interest Rate

♦ Future Value

♦ Present Value

♦ Interest Rates are determined by:

– time period

– currency

– credit quality

♦ Real Cash Flows:

– Money is deposited

– Interest is paid

– Based on the stated interest rate

4

Future Value / Present Value

1 year

rate = 5%100

105

1 year

rate = 5%95.24

100

How much needs to be deposited at 5% for 1 year so that the total amount will be 100?

Deposit 100 for 1 year at an interest rate of 5%

What will be the total amount after 1 year?

5

Spot Rate

♦ In the example 5.00% is called the one year spot rate

♦ One year deposit rate

♦ One year add-on rate

♦ One year zero rate

♦ They all mean the same thing!

6

Future Value Factor / Discount Factor

♦ Future Value of 1 = FVF

♦ Present Value of 1 = DF (discount factor) or PVF

♦ Depend on:

– Time

– Currency

– Credit

♦ e.g.

– Rate for a 1-month deposit of USD10,000 in a US commercial bank

– Rate for 6-month deposit of CHF50,000 in a publicly-traded Swiss money market fund

7

Simple Interest

♦ Used when the time period is at most one year

♦ No compounding (see later)

♦ “Simple” formulas

0.9524 DF

1.05 1 0.05 1 FVF :Example

DF

DF

time rate 1 FVF

1.051

time rate 11

FVF1

==

=×+=

=

=

×+=

×+

8

More Examples: Your turn

Time Rate FVF DF

1 6.25%

1/2 4.00%

1/4 9.00%

1 1.0855

1/2 1.0260

1/4 1.0150

1 0.9100

1/2 0.9434

1/4 0.9950

9

Compounding

♦ Rate applies to more than 1 period

♦ period could be:

– 1 year

– 6 months

– 3 months

♦ Examples:

– 2 year rate of 6%, compounded every 6 months

– 1 year rate of 8% compounded every 3 months

– 5 year rate of 4% compounded every year

10

Nominal and Effective Rates

♦ In the previous examples the rates were nominal rates

♦ A nominal rate of 8.00% compounded quarterly had a FVF of 1.08243

♦ So a deposit at such a rate would actually earn 8.243% in one year

♦ 8.243% is called the effective rate

♦ The same nominal rate could have a different effective rate depending on the compounding period

♦ If the 8.00% rate was compounded monthly what would the effective rate be?

11

Examples

Rate6.00%

Frequency241

Time215

FVF1.1255

PVF0.8885

8.00% 1.0824 0.92384.00% 1.2167 0.8219

A Rate, Time Period and a Compounding Frequency determine a FVF and a DF (PVF)

12

The Dreaded Formulae!

FVF1 PVF =

( ) frequency years

frequencyrate

1 FVF×

+=

( ) 1.1255 1 FVF :Example2 2

20.0600 =+=

×

0.8885 DF PVF1.1255

1 ===

13

More Examples

6 months 7%

Frequency = 1

FVF = 1.035

DF =

12 months 6%

Frequency = 4

FVF =

DF =

5 years 9%

Frequency = 2

FVF =

DF =

14

Suppose you have the FVF?

1.0824

1

You need to specify any two of these:

Rate

Time

Compounding Frequency

Then you can determine the third!

15

Most Common Problem

1 year 1.0824

Compounding Frequency Rate

1 8.24%

2 8.08%

4 8.00%

1

16

More Examples

1.19431

1Time = 2

Frequency = 4

Rate = _____

1Time = 1.5

Frequency = 2

Rate = _____1.09727

1Time = 10

Frequency = 2

Rate = _____

1.485947

17

Review: Discount Rates

♦ Used mostly for two types of securities:

– Treasury Bills

– Commercial Paper

♦ If you buy a 91 day Treasury bill with a “face value” of 10,000 you will receive USD10,000 in 91 days

♦ How much do you pay today?

18

Treasury Bill Discount Rate10,000

♦ Discount Rate = 4.60%

♦ time period = 91 days

♦ day basis = 360

9883.72 116.28 - 10,000 Price

116.28 0.0460 10,000 Discount 360

91

==

=××=

19

Homework Exercises

♦ Nominal Rate = 10% p.a.

♦ What is the effective rate if compounding is:

– Semi-annual _______

– Quarterly _______

– Monthly _______

– Daily (365) _______

♦ What is the DF and FVF for these simple rates:

♦ Rate Time DF FVF

♦ 6.35% 2 months ______ ______

♦ 9.20% 6 months ______ ______

♦ What are the DF and FVF for these compounded rates:

♦ Rate = 12% p.a. quarterly compounded; time = 2 years

♦ DF = ______ FVF = ______

♦ Rate = 4% p.a. monthly compounded; time = 18 months

♦ DF = ______ FVF = ______

20

Exercises♦ What annual compounded rate

has a 3 year FVF = 1.179257?

♦ What semi-annual compounded rate would have the same 3 year FVF I.e. 1.179257?

♦ What quarterly compounded rate would have a 9 month DF = 0.977833?

♦ What simple rate would have the same 9 month DF i.e. 0.977833?

♦ A 26-week T Bill is trading at a 5% discount. What is the price of USD1 million of this bill?

♦ What is its BEY?

Appendix

Continuously Compounded Interest

22

Compounding

♦ Nominal Rate of 8.00% per annum

♦ Compounding Frequency Effective Rate

♦ 2 8.1600%

♦ 4 8.2432%

♦ 12 ________%

♦ 365 ________%

23

The Limit

♦ As we increase the compounding frequency the effective rate increases

♦ But it “slows down”

nominal rate8.00% 1 2 4 12 365 1000 10000 limit

effective rate 8.00000% 8.16000% 8.24322% 8.29995% 8.32776% 8.32836% 8.32867% 8.32871%

frequency

♦ The limit is called “continuous” interest

♦ It is “easily” calculated

1etr trsimple

continuous −=× ×

24

Conversions Using Day Counts

♦ 6.00% actual/365 continuous rate

♦ Time period 120 days

♦ Is equivalent to what actual/365 simple rate?

0.0606r

0.01992r

0.019921e Interest

365120

3651200.06

=

=×

=−=×

25

And the other way

♦ 7.25% actual / 360 rate

♦ Time period 90 days

♦ Is equivalent to what actual/365 continuous rate?

0.0728r

0.017965)ln(1.01812r

1.018125e

0.0181251e

0.0181250.0725 Interest

36590

36090

36590

r

36590

r

=

==×

=

=−

=×=

×

×

The Yield Curves


1

But how do we build the curves?

♦ The yield curve is a set of interest rates consistent with market prices for liquidinstruments

♦ Which can then be used to price every position

of that currency and credit quality

Market

Prices

Implied

Curve

Curve

Building Tool

Pricing Tool

2

USD LIBOR 1998

♦ Our curve tool for USD LIBOR uses:

♦ Deposit rates

♦ Futures prices

♦ Swaps

3

Government Yield Curves

4

Types of Interest Rates

♦ We want to earn interest on an investment starting today, we need a ‘spot rate’

– Deposits

– Bills

– Commercial paper

– Strips

♦ We want to arrange today to invest at some time in the future, we want a ‘forward rate’

– Forwards

– Futures

♦ We want to receive a fixed (constant) interest payment periodically, we want a coupon rate or “par” rate

– Bonds

– Swaps

5

Spot Rate Example

♦ Short term (under one year)

♦ Deposit money at a bank for 9 months

♦ Bank quotes a rate of 6.00%

♦ Get back 4.5% more

♦ Longer term

♦ A 3-year zero coupon bond is trading for a price of 81.63

♦ Pay 81.63 today

♦ Receive 100 in 3 years

♦ Earn an interest rate of

♦ 7.00% is the 3-year annually-compounded spot rate

♦ Also called the 3-year zero rate

( ) 7.00%131

81.63100 =−

6

Spot Rates

Time period = t

Interest Rate = r

PV (price)

FV (redemption)

type rate on depending rt)-FV(1 or rt1

FV or

r)(1

FVtPrice

++=

Today or t0

7

Forward Deposit

♦ A client wants to deposit 10,000,000 for 3 months

♦ But not starting today

♦ Instead wants to do it in two months time

♦ A bank agrees today to take the deposit at an agreed rate of 5.25%

♦ Cash flows on the forward deposit:10,131,250

10,000,000

Today plus 5 months

Today plus 2 months

Today or t0

8

Alternatives to Forward Deposit

♦ Forward Deposits or loans are risk positions for the bank and the client

♦ If the client does not actually deposit the cash, the bank might have to pay a higher rate to fund itself

♦ If the bank defaults on the agreement, the customer might have to deposit elsewhere at a lower rate

♦ Usually institutional or corporate clients will “lock-in” rates by using Forward Rate Agreements (FRAs) or Futures

9

Forward Rate

♦ Bank agrees to “fix” a rate for a client on a 10,000,000 deposit

♦ Deposit will take place in 2 months

♦ Deposit will mature 3 months later

♦ Bank and the client agree to a rate of 5.25%

♦ 5.25% is the 2 x 5 forward rate

– Rate agreed today

– For a deposit or loan that begins in 2 months

– And terminates or matures in 5 months

♦ Bank has NOT agreed to take a deposit or make a loan

♦ Bank and the client have agreed if the deposit rate in 2 months is

– Less than 5.25%, the bank will pay the interest shortfall to the client

– More than 5.25%, the client will pay the excess interest to the bank

10

Example

♦ FRA (Forward Rate agreement)

♦ If the 3 month rate in 2 months is 5.00%

♦ FRA settlement is

♦ Up-front:

– When the start date of the deposit/loan occurs

♦ Discounted

– The interest variation from the agreed rate on the notional amount is calculated

– Then it is discounted by the observed rate for the period of the deposit or loan

– Instead of paying it out at the end of the period

– In reality daycount is not exactly ¼

– Contract can be tailored

♦ If the 3 month rate in 2 months is 5.50%

6172.8441

41

0.05001

0.002510,000,000=

×+

××

6165.2341

41

0.05501

0.002510,000,000=

×+

××

11

Exchange Traded Version of the FRA

♦ Short Term (3 month) Interest Rate Future

♦ All terms standardised

♦ Future = 94.75 corresponds to a rate of 100 – 94.75 = 5.25%

♦ One basis point (0.01%) is worth – 1,000,000 x 0.0001 x ¼ = 25 for EUR, USD

– 500,000 x 0.0001 x ¼ = 12.50 for GBP

♦ Futures are marked to market every day

♦ Buy 10 futures on EUR rate for 94.75 today

♦ Future closes at 95.00 Future closes at 94.50

♦ Receive EUR6250 tomorrow Pay EUR6250 tomorrow

♦ 10 Futures x 25 Basis Points x EUR25 per Basis Point

12

Par Yield / Par Coupon / Par Swap Rates

♦ Usually represented by bonds or by interest rate swaps

♦ Pay 100 today

♦ 5-year annual ‘Par rate’ of 6.00%

♦ Receive fixed coupon of 6 each year for 5 years

♦ In 5 years receive 100

♦ 100 is the principal amount

♦ 6.00% is the par coupon rate or par yield rate or par swap rate

♦ It’s the ‘fair’ or current rate

13

♦ What happens when 6% becomes ‘unfair’?

Cash Flows on a Par Bond

51

100

100

2 3 4

6 6666

14

♦ Rate is unknown but resets to what is fair/current

♦ We say that the rate is ‘floating’

♦ Is always worth 100…

Floating Rate Note (its always fair)

51

100

100

2 3 4

? ? ? ??

15

A Fair Exchange

♦ These two are both worth 100 today

♦ The payments of 100 in year 5 are worth the same today so . . .

Floating Rate Note

Fixed Rate Bond

51

100

2 3 4

? ? ? ??

51

100

2 3 4

6 6666

16

These are worth the same

51 2 3 4

? ? ? ??

51 2 3 4

6 6666

17

♦ If you agree to pay 6 every year for 5 years and are paid the 1 year rate that is fair/current at each payment date

♦ This exchange has a present value of 0

♦ It is a fair trade

♦ Its called an interest rate swap

So this is worth 0 (Par)

51 2 3 4

? ? ? ??

6 6 6 6 6

18

Interest Rate Swap

♦ An interest rate swap is a tailored agreement between two counterparties:

♦ In this case, for five years

♦ One party agrees to pay a floating rate

♦ One party agrees to pay a rate fixed at the start

♦ Payments are netted

♦ The fixed rate on a swap worth 0 is called the ‘par swap rate’

19

Interest Rate Swaps

♦ Interest Rate Swaps are “fair value” agreements so long as the fixed rate is the current rate for its maturity

♦ In 1979 the World Bank and IBM did a landmark swap transaction

♦ Since that time interest rate swaps have become a commodity

♦ In a single currency interest rate swap there is no exchange of the principal amount

♦ So the notional size needs to be agreed as well

♦ Interest Rate Swaps are among the most frequently used derivatives in the financial world

♦ You can learn about their uses in the Interest Rate Swaps course

20

The Yield Curve

♦ The graph of the rate (y axis) for each point in time (x axis) is called the yield curve

♦ There are different yield curves for

– Spot rates

– Forward/futures rates

– Par rates

♦ There are different yield curves for different

– Currencies

– Credit qualities

– Quote conventions e.g. add-on, discount, compounded

♦ The yield curve is also called the “term structure” of interest rates

Curve Building


SECTION 1

Rate Arbitrage

2

Market Rates

3

Yield Curves = Term (time) Structure Of Rates

♦ Consider other currencies and different qualities

4

Types of interest rate

♦ Spot/zero rate

♦ Forward rate

♦ Par yield/coupon/swap rate

5


♦ Spot rates0

t1

0

t2

0

t3

6


♦ Forward rates0

t1

0

t2

0

t3

t1

t2

7


♦ Par bond yields0

t1

0

t2

0

t3

8

Where do we see these trading?

♦ Spot rate

– deposits

– t-bills

– CP

♦ Forward rate

– FRAs

– Eurodollar futures, Short sterling futures etc

♦ Par bond yield

– bonds

– swaps

9

Interest Rate relationships

♦ Spots, forwards and pars seem to be related - consider these cashflows...

0

0

0

0

0

t1

0

t2t1

000

t2

t1

100

105

99

106

100

66

100

+

=

10

Is there an opportunity?

♦ 3 mo rate = 5% p.a.

♦ 6 mo rate = 6.00% p.a.

♦ 3 x 6 forward = 6.00% p.a.

11

Spot and Forward

0

3M

0

0

1.0125

1

1.0125

1.02768751.0125

3M

6M

1.0125 x 1.015=1.0276875

6M3M

1

12

Spot Only

0

1.03

6M

1

13

No Arbitrage

♦ Arbitrage-free relationship means

– no profit at expiry

– Cashflows are the same at expiry

♦ What is the arbitrage-free 3x6 forward rate?

♦ Arbitrage-free rates are

– Fair?

– Correct?

– An academic idea?

14

No Arbitrage

♦ Spot and Forward trade:

♦ Spot only trade:

0

3M

0

0

1.0125

1

1.03

3M

6M

1.0125 x ????=1.03

6M

1

15

To be arbitrage-free what is the forward?

♦ For there to be no arbitrage, should be around 7%

♦ More precisely

– 1.01125 x ( 1 + r3x6 t) = 1.03

♦ which means that

– 1 + r3x6t = 1.03 / 1.0125

– 1 + r3x6t = 1.017284

– r3x6t = 0.017284

– As t= ¼, r3x6 = 0.017284 x 4 = 0.0691358

– or 6.91358% p.a.

♦ Note that 6M spot rate is almost the average of the forward rates covering the same time period

SECTION 2

Yield Curve Building

17

Forwards to spots

♦ Using this arbitrage relationship, we can relate spot and forward rates together

♦ Imagine we knew a series of forward rates and wanted to find the spot rates:

♦ Why would we want to do this?

18

Start with the forwards

Year: 1 2 3 4 5

Forward 0.0500 0.0625 0.0725 0.0800 0.0825

Spot

NB The forward rate given is for a 1 year investment ending at the given year

19

1 year forward vs 1 year spot

♦ Remember that our first forward starts at time 0, and so is the same as a spot rate

♦ 1 year spot:

♦ 0 x 1 forward:

♦ 1 year spot must also be 5%

0

t1

0

t1

20

Find the 2 year spot rate

♦ If we invest 1 for 1 year at the 1 year spot rate, we get

– 1.0500

♦ If we agree today to reinvest this at the 1 year forward 1 year rate, we get

– 1.05 x 1.0625 = 1.115625

♦ The arbitrage argument says that we must get the same FV if we invest for 2 years at the 2 year spot rate

♦ So, using the bond calculator

– PV = -1, FV = 1.115625, n = 2, PMT = 0

– Press the i button

– i =5.623%

♦ Repeat for the remaining spot rates

21

Forwards to spots

0 1 2 3 4 5Zero rate 5.0000%5.6232%6.1627%6.6191%6.9433%Forward rate 5.0000%6.2500%7.2500%8.0000%8.2500%

22

♦ Used to ‘present value’ cashflows

♦ One year discount factor:

♦ Two year discount factor:

0.952411 0.05)(1

1

r)(1

1 ==++

0.896422 0.0562)(1

1

r)(1

1 ==++

Discount factors

23

Discount factors

Note:

df1 = 1/ ( 1+ r0x1) df1 = 1/ ( 1+ r1)

df2 = df1 / (1 + r1x2) df2 = 1 / (1 + r2)2

df3 = df2 / (1 + r2x3) df3 = 1 / (1 + r3)3

etc.

0 1 2 3 4 5Zero rate 5.0000%5.6232%6.1627%6.6191%6.9433%Forward rate 5.0000%6.2500%7.2500%8.0000%8.2500%Discount Factor (df) 0.9524 0.8964 0.8358 0.7739 0.7149

24

Cumulative discount factors

♦ What is the value of 1 currency unit to be paid every year for 4 years?

♦ 0.9524+0.8964+0.8358+0.7739 = 3.4585

♦ we call this the 4-year cumulative discount factor (CDF4)

♦ this is useful for pricing annuities

♦ e.g. 800 every year for 4 years is worth 2,766.8 today

25

Cumulative discount factors

0 1 2 3 4 5Zero rate 5.0000%5.6232%6.1627%6.6191%6.9433%Forward rate 5.0000%6.2500%7.2500%8.0000%8.2500%Discount Factor (df) 0.9524 0.8964 0.8358 0.7739 0.7149Cumulative DF (cdf) 0.9524 1.8487 2.6845 3.4584 4.1732

26

Par yield/coupon/swap Rate

♦ We would like to find a bond coupon curve which is consistent with our spot and forward curves

♦ Consistent means arbitrage-free or same cashflows at expiry

♦ We will find the par yield /coupon/ swap rate curve

♦ We want to find the yields on the 1 through 5 year maturity bonds which will price at par in this environment

♦ Consider the 1-year annual par bond:

♦ What is the consistent or fair rate for c1?

0

1

100

100

c1

27

3-year par bond

0

321

100

100

c3 c3 c3

♦ 100 = c3 x 3-year cumulative discount factor + 100 x df3

♦ 100 = c3 x 2.6845 + 83.58

♦ c3 = (100 - 83.58 ) / 2.6845 = 6.1166 %

♦ In general:

n

nn cdf

df1coupon −=

28

Par yields

0 1 2 3 4 5Zero rate 5.0000%5.6232%6.1627%6.6191%6.9433%

Forward rate 5.0000%6.2500%7.2500%8.0000%8.2500%Discount Factor (df) 0.9524 0.8964 0.8358 0.7739 0.7149Cumulative DF (cdf) 0.9524 1.8487 2.6845 3.4584 4.1732

Swap / Coupon rate ( c) 5.0000%5.6061%6.1179%6.5390%6.8321%

29

Spot, Forward and Par Yield Curves

♦ Spot rates are like averages of forward rates

– spots are lower (in an upward-sloping environment)

– spots are higher (in an downward environment)

♦ Par rates give opportunity/cost for reinvestment

– higher forwards give a lower par yield than the spot rate

– lower forwards give a higher par yield than the spot rate

5-Year Yield Curve

4.50%

5.00%

5.50%

6.00%

6.50%

7.00%

7.50%

8.00%

8.50%

0 1 2 3 4 5

Year

Zero rateForward rateSwap / Coupon rate

30

Building Yield Curves

♦ We know that

– Zero rates

– Forward rates

– Par rates

♦ Are all related

♦ If we have a complete set of rates of one type i.e. for each point in time

♦ Then we can use that curve to build the others

31

US Treasury Strips

♦ A strip is the coupon from a US Treasury bond or note

♦ It “looks” exactly like a zero coupon bond

♦ You pay the price today

♦ No cash flow occurs until maturity

♦ Redeems at its face value

22-Dec-04 Price22-Dec-05 97.643922-Dec-06 94.646622-Dec-07 91.200722-Dec-08 88.023222-Dec-09 84.144222-Dec-10 80.177722-Dec-11 76.156222-Dec-12 72.379522-Dec-13 68.649722-Dec-14 65.0534

32

Zero Coupon Rates on 22/12/2004

♦ We can calculate the annual compounded zero coupon rate for each year for which we have the Price of a Treasury Strip

♦ The formula is

♦ Alternatively use a bond calculator

– n

– i

– PV

– PMT

– FV

1nprice

1 −

22-Dec-04 Price Zero rate22-Dec-05 97.6439 2.4129%22-Dec-06 94.6466 2.7892%22-Dec-07 91.2007 3.1179%22-Dec-08 88.0232 3.2406%22-Dec-09 84.1442 3.5131%22-Dec-10 80.1777 3.7507%22-Dec-11 76.1562 3.9679%22-Dec-12 72.3795 4.1233%22-Dec-13 68.6497 4.2681%22-Dec-14 65.0534 4.3934%

33

Zero Coupon Curve

Zero Coupon Rates

0.0000%

1.0000%

2.0000%

3.0000%

4.0000%

5.0000%

28-May-05

10-Oct-06

22-Feb-08

06-Jul-09

18-Nov-10

01-Apr-12

14-Aug-13

27-Dec-14

10-May-16time

Rat

e

34

Par Yields

♦ We can use this method to determine all the par yields for the same time periods as the zero coupon rates.

♦ The “formula” is:n

nn cdf

df1coupon

−=

df cdf par0.97644 0.97644 2.4129%0.94647 1.92291 2.7840%0.91201 2.83491 3.1039%0.88023 3.71514 3.2238%0.84144 4.55659 3.4798%0.80178 5.35836 3.6993%0.76156 6.11993 3.8961%0.72380 6.84372 4.0359%0.68650 7.53022 4.1633%0.65053 8.18075 4.2718%

35

Zero Yields and Par Yields

zero and par yields

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

5.00%

14-Jan-04 10-Oct-06 06-Jul-09 01-Apr-12 27-Dec-14 22-Sep-17time

yiel

d

36

Forward Rates

♦ From year 4 to 5:

–

♦ 1-year FVF for year 4 to 5:

–

♦ Forward rate from years 4 to 5:

–

♦ In general rate from year ‘n’ to year ‘m’:

–

4x545 r1

1dfdf+

×=

1r5

44x5 df

df −=

1rm

nnxm df

df −=

( )5

44x5 df

dfr1 =+

df Forward0.976439 2.4129%0.946466 3.1669%0.912007 3.7784%0.880232 3.6098%0.841442 4.6100%0.801777 4.9472%0.761562 5.2806%0.723795 5.2178%0.686497 5.4331%0.650534 5.5282%

37

Zero Yields, Par Yields and Forward Yields

Zero, Par and Forward Curves

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

5.00%

5.50%

6.00%

14-Jan-04

10-Oct-06

06-Jul-09

01-Apr-12

27-Dec-14

22-Sep-17

Time

Yiel

d Zero rateparForward

38

Bootstrapping

♦ We built the yield and forward curves using zero coupon bonds

♦ In markets other than the US Treasury it is not possible to do this

– The Strip market does not exist or

– The strips are not liquid enough to give a stable curve

♦ In that case we might want to simply start with the par curve and “back out”the discount factors i.e. the zero coupon rates

♦ This process is called bootstrapping.

39

Bootstrapping

♦ The yields on German Government Bonds on 22/12/2004 are shown here.

♦ Since the one year yield is 2.30405

– The one year zero rate is 2.30405

– The one year discount factor is 1/(1.0230405) = 0.977479

♦ That was easy

♦ What about the two year zero and the two year discount factor?

Date Yield22-Dec-05 2.3040%22-Dec-06 2.4708%22-Dec-07 2.6026%22-Dec-08 2.8200%22-Dec-09 2.9996%22-Dec-10 3.1761%22-Dec-11 3.3147%22-Dec-12 3.4398%22-Dec-13 3.5399%22-Dec-14 3.6336%

40

Two Year Rate

♦ The two year par yield is 2.4708

♦ So a 2 year bond with a coupon of 2.4708 is worth 100

♦ We could calculate the value by discounting the cash flows:

♦ We can use this to solve for the second discount factor:

2.4708102.4708

100

2df102.47080.977482.4708100 ×+×=

0.952319df

df

2

2 102.47080.977482.4708100

=

= ×−

41

Continuing the Process

♦ Every time we calculate a discount factor we can use it together with the next yield to calculate the next discount factor:

♦ This process is called “Bootstrapping” the curve

♦ General “formula”:

n

1nnn c1

)CDFc(1df+×− −=

Date Yield df cdf Zero22-Dec-05 2.3040% 0.977479 0.977479 2.3040%22-Dec-06 2.4708% 0.952319 1.929797 2.4729%22-Dec-07 2.6026% 0.925683 2.855481 2.6075%22-Dec-08 2.8200% 0.894257 3.749738 2.8334%22-Dec-09 2.9996% 0.861676 4.611414 3.0223%22-Dec-10 3.1761% 0.827262 5.438676 3.2110%22-Dec-11 3.3147% 0.793425 6.232101 3.3609%22-Dec-12 3.4398% 0.759503 6.991604 3.4984%22-Dec-13 3.5399% 0.726777 7.718381 3.6096%22-Dec-14 3.6336% 0.694316 8.412697 3.7156%

42

Homework Exercises

♦ UBS sells an FRA on GBP50 million to a client

– Rate = 6.50%

– Period is 6 months x 12 months

♦ When the FRA expires the 6 month rate turns out to be 6.10%

♦ Who pays on the FRA?

♦ How much is the settlement amount?

♦ Given these 1 year rates:

♦ 0 x1 1x2 2x3 3x4 4x5

♦ 8.00 7.50 7.25 7.10 6.90

♦ Find the discount factors for 1, 2, 3, 4 and 5 years

♦ Use these to find the 1, 2, 3, 4 and 5 year zero coupon rates

♦ Calculate the CDFs for 1, 2, 3, 4 and 5 years

♦ Use these and the DFs to find the 1, 2, 3, 4 and 5 year par yields

Fixed Income Fundamentals1. Interest Rate & Bond Calculations – Day 2


Bond Valuations and Yields


1

Bond Valuation Using a ‘Yield Curve’

♦ Year 1 2 3 4

♦ Spot 4% 5% 6% 7%

♦ DF

♦ CDF

♦ Value the 4-year annual 5% bond

♦ Guess the single rate to PV all cashflows at to get to the same value?

♦ Or would you rather solve quadratic polynomial, ?( ) ( ) ( )432 y1

105

y1

5

y1

5y1

5PV+

++

++

++

=

2

Bond Valuation Using a ‘Yield to Maturity’

♦ Year 1 2 3 4

♦ Ytm y% y% y% y%

♦ ‘DF’

♦ ‘CDF’


3

Yield to Maturity

♦ YTM is the one rate to discount all of the bond’s future cash flows to its price

♦ Its like a weighted average of spot rates

♦ YTM depends on rates and cashflows on the bond

♦ People sometimes say:

– “YTM is the expected return to the bond investor”

– “YTM is the assumed reinvestment rate of the bond’s coupons”

– “YTM is the equivalent rate on a deposit of the bond’s price for the time to maturity of the bond”

4

4 year 5% bond yielding 5%

5 5 5

105

1004.7619 4.5351 86.38384.3191

Futurevalues

Presentvalues

1 5 1/1.05 = 0.9524 4.76192 5 1/1.052 = 0.9070 4.53513 5 1/1.053 = 0.8638 4.31924 105 1/1.054 = 0.8227 86.3838

Total 100.0000 (par)

Year Cashflow DF Present value

Yield to Maturity - Example I

5


5 5 5

105

96.5349

4.7170 4.4500 83.16984.1981

Futurevalues

Presentvalues

1 5 1/1.06 = 0.9434 4.71702 5 1/1.062 = 0.8900 4.45003 5 1/1.063 = 0.8396 4.19814 105 1/1.064 = 0.7921 83.1698

Total 96.5349 (discount)


Yield to Maturity - Example II

6


5 5 5

105

103.6299

4.8077 4.6228 89.75444.4500

Futurevalues

Presentvalues

1 5 1/1.04 = 0.9615 4.80772 5 1/1.042 = 0.9246 4.62283 5 1/1.043 = 0.8890 4.44504 105 1/1.044 = 0.8548 89.7544

Total 103.6299 (premium)


Yield to Maturity - Example III

7

Price and Yield

8

Homework Exercises

♦ Annual Coupon = 7.00%

♦ Time to Maturity = 6 years

♦ YTM = 6.20%

♦ Price =

♦ Semi-annual Coupon = 6.00%

♦ Time to maturity = 5.50 years

♦ YTM = 7.25%

♦ Price =

♦ Annual Coupon = 10%

♦ Time to maturity = 5 years

♦ Price = 110

♦ YTM =

♦ Semi-annual coupon = 5.50%


♦ Price = 97.55

♦ YTM =

9

Exercises

♦ Annual coupon = 9.00%


♦ YTM = 8.00%

♦ Price =

♦ Assuming all coupons can be reinvested at 8.00%, calculate the total cash amount at maturity of the bond

♦ Calculate the return

♦ Semi-annual coupon = 12%


♦ YTM = 10%

♦ Price =



Bond Valuations and Yields


1

Bond Valuation Using a ‘Yield Curve’

♦ Year 1 2 3 4

♦ Spot 4% 5% 6% 7%

♦ DF

♦ CDF


♦ Guess the single rate to PV all cashflows at to get to the same value?

♦ Or would you rather solve quadratic polynomial, ?( ) ( ) ( )432 y1

105

y1

5

y1

5y1

5PV+

++

++

++

=

2

Bond Valuation Using a ‘Yield to Maturity’

♦ Year 1 2 3 4

♦ Ytm y% y% y% y%

♦ ‘DF’

♦ ‘CDF’


3

Yield to Maturity

♦ YTM is the one rate to discount all of the bond’s future cash flows to its price

♦ Its like a weighted average of spot rates

♦ YTM depends on rates and cashflows on the bond

♦ People sometimes say:

– “YTM is the expected return to the bond investor”

– “YTM is the assumed reinvestment rate of the bond’s coupons”

– “YTM is the equivalent rate on a deposit of the bond’s price for the time to maturity of the bond”

4


5 5 5

105

1004.7619 4.5351 86.38384.3191

Futurevalues

Presentvalues

1 5 1/1.05 = 0.9524 4.76192 5 1/1.052 = 0.9070 4.53513 5 1/1.053 = 0.8638 4.31924 105 1/1.054 = 0.8227 86.3838

Total 100.0000 (par)


Yield to Maturity - Example I

5


5 5 5

105

96.5349

4.7170 4.4500 83.16984.1981

Futurevalues

Presentvalues

1 5 1/1.06 = 0.9434 4.71702 5 1/1.062 = 0.8900 4.45003 5 1/1.063 = 0.8396 4.19814 105 1/1.064 = 0.7921 83.1698

Total 96.5349 (discount)


Yield to Maturity - Example II

6


5 5 5

105

103.6299

4.8077 4.6228 89.75444.4500

Futurevalues

Presentvalues

1 5 1/1.04 = 0.9615 4.80772 5 1/1.042 = 0.9246 4.62283 5 1/1.043 = 0.8890 4.44504 105 1/1.044 = 0.8548 89.7544

Total 103.6299 (premium)


Yield to Maturity - Example III

7

Price and Yield

8

Homework Exercises

♦ Annual Coupon = 7.00%


♦ YTM = 6.20%

♦ Price =

♦ Semi-annual Coupon = 6.00%

♦ Time to maturity = 5.50 years

♦ YTM = 7.25%

♦ Price =

♦ Annual Coupon = 10%


♦ Price = 110

♦ YTM =

♦ Semi-annual coupon = 5.50%


♦ Price = 97.55

♦ YTM =

9

Exercises

♦ Annual coupon = 9.00%


♦ YTM = 8.00%

♦ Price =



♦ Semi-annual coupon = 12%


♦ YTM = 10%

♦ Price =



Bond Futures


SECTION 1

Futures Contracts

2

Bond Futures

♦ Are an exchange traded contract

♦ Allow investors to gain exposure to bond yields

♦ Allow hedgers to reduce their exposure to bond yields

♦ Like all futures contracts they are marked to market and can be offset before expiry

♦ The futures months are March, June, September and December (H, M, U, Z are the symbols) with a separate contract for each expiry

3

Example

♦ The September 2005 Treasury Note Futures contract was priced at 108 – 20 on 30 March 2005

♦ Quotation is in 32nds so this means a decimal price of 108.625

♦ The underlying to the contract is a

– US Treasury Note

– 10 years to maturity on the first day of the futures month

– Semi-annual coupon of 6%

– Face amount of USD100,000

♦ Buying the future is “like” agreeing today to buy this note in September

♦ (So the price of the future is not the “cash” price of the note)

4

Deliverable Bonds

♦ The nominal underlying (6% coupon, 10 years to maturity) is an “ideal”

♦ In reality the person who is short the future chooses which of a list of deliverable bonds to deliver

♦ Criteria:

– Must be US Treasury Note

– At least 6 ½ years but not more than 10 years remaining to maturity at the first date of the futures month

♦ Payment to the short is:

– Futures Price x Conversion Factor for Delivered Bond

– Conversion Factor = price of the bond at 6% ytm on the first date of the futures month

5

Example

♦ A Bond Fund manager buys the September 10 year future at a price of 108-20

♦ In September the manager is still long the contract

♦ On 7 September a person who is short decides to deliver the 4% US Treasury note maturing on 17 February 2014

♦ The manager is “selected” to take delivery

♦ CBOT notifies the manager that delivery will occur on the next business day (8 September)

♦ The conversion factor for the note is 0.8713 (see Appendix or CBOT web site)

♦ Payment to the short is: 0.8713 x 108.625 = 94.64 (plus accrued interest)

6

Futures Contracts

♦ Bond Futures are among the most successful of all futures contracts

♦ There are futures on 5 year and 10 year notes in many markets:

– Germany

– UK

– US

– Japan

♦ In the US market there are also futures on 2 year and 30 year bonds but the most popular and liquid contracts are the 10 year futures

SECTION 2

Appendix: Conversion Factors

8

Calculation of the Conversion Factor

♦ For US Bond and Note futures the CBOT uses this method for calculating the conversion factor:

♦ Determine the amount of time left to maturity of the bond or note from the first day of the futures month

♦ Round this number DOWN to the nearest 3 months

♦ Calculate what the price of the bond or note would be if it had this amount of time left to maturity and was priced to yield 6%

9

Example

♦ 4% Treasury Note maturing on 17/02/2014

♦ Conversion factor relative to the September 2005 Future

♦ Time to maturity from 1/09/2005 to 17/02/2014 is 8.25 years rounded down to the nearest quarter year (actual time is 8 years 5 months and 17 days)

♦ We used excel as shown below (note that 8.25 years from 1/09/05 is 1/12/13)

settle 01-Sep-05maturity 01-Dec-13coupon 0.04ytm 0.06freq 2face 100price 87.12676

Bond Repo and Carry Costs


SECTION 1

Repo

2

The Repo Market

♦ Bond Traders / Market-Makers / Dealers

♦ Have a need to finance bonds they own – borrow cash

♦ Need to access bonds they have sold short – borrow bonds

♦ They do both of these in the RepoMarket

♦ Investors / Government Agencies / Pension Funds / Insurance Companies

♦ Want to earn interest on unneeded cash – lend cash

♦ Sometimes want to raise cash for investment purposes – lend bonds

♦ They do both of these in the Repomarket

3

Repo market

♦ Money market: Short term loans

♦ Collateral

♦ US Treasury bonds

♦ Sovereigns

♦ Highly rated corporates

4

Examples

♦ UBS buys £10 million of a UKT from a client

♦ Still has the position at the end of the day

♦ Needs to be funded

♦ Lend the bond in the repo market

♦ Take in the cash price

♦ Tomorrow if we sell the bond

♦ Repay the loan

♦ Get the bond back

♦ A hedge fund wants to sell $50 million of UST bond

♦ Sells the bond in the market

♦ Borrows the bond in the repo market

♦ Gives up the cash received for the bond

♦ Delivers the bond to the buyer

♦ When it wants to close the trade:

– Buys the bond in the market

– Returns it in the repo market

– Is paid cash plus interest

– Pays the seller of the bond

5

Repo transaction

♦ Repo: Repurchase agreement

♦ Today: UBS buys a UK Treasury bond from a customer

– UBS borrows purchase price from Salomon Brothersgiving the bond as collateral

♦ In a few days, UBS sells the bond to another customer

– UBS pays the original purchase price plus interestto Salomon

– UBS receives the bond back from Salomon and deliversit to the customer, receiving new full price

6

Repo

♦ UBS has done a repo transaction

♦ Repo – borrow money giving a bond as collateral

♦ Salomon Brothers has done a reverse-repo transaction

♦ Reverse-repo – lend money, taking a bond as collateral

7

Repo Rates

♦ General Collateral

– general level of repo rates for all bonds of a given issuer

– e.g. all Gilts

♦ Special

– different repo rate for a particular issue

8

Example expanded

♦ UK Treasury 5¾’s 7th Dec 09

♦ Settlement date : 1 Mar 2004

♦ Cash Price: 102.70

♦ Accrued Interest: 1.335

♦ Invoice Price: 104.035

♦ UBS

– buys £10,000,000 face value

– borrows £10,403,538 from Salomon at 5.6%, depositing bond as collateral

– (Repo rate in the UK is actual/365)

9

Salomon UBS Customer1

Bond

Invoiceprice

Bond

Invoiceprice

Start:

Repo Transaction

Repo transaction Outright purchase

10

Salomon UBS Customer2

Repay loan

Bond

New invoice price

Bond

Suppose UBS sells the bond a few days later:

Settlement date: 8 March 2004

New cash price: 102.00

New accrued interest: 1.445

Invoice price: 103.445

UBS sells £10,000,000 face value,

Receives (on sale) £10,344,535

Pays (on repo close out) £10,403,538 x ( 1 + 5.6% x 7/365)

= £10,414,711

Repo Example

11

Total P/L Receives £10,344,535

Repays £10,414,711

Net £ (70,176)

This can be decomposed into:

Price change: ( 102.00 – 102.70 ) x 100,000 = (70,000)

Coupon earned: 5.75/2 x 7/183 x 100,000 = 10,997

Interest paid: 10,403,538 x 0.056 x 7/365 = (11,173)

Total £ (70,176)

Analysis of Transaction

SECTION 2

Carry Cost

13

Financing positions

♦ When we put on bond positions we care about carry cost:

♦ How much is paid / earned to be long a bond?

♦ In this case

– Earn coupon:

– £10,997 per £10m face for 7 days

– 0.01571 per 100 face per day

– Pay repo:

– £11,173 per £10m face for 7 days

– 0.01596 per 100 face per day

♦ In this case Net Carry = - 0.00025 per 100 face per day

♦ Negative carry means there is a net cost to hold the bond

14

The Punchline

♦ A bond forward is just like a bond future

♦ Forwards are priced using a ‘cost of carry’ principle

♦ In the bond example before, what is the fair 7-day forward price?

Bond Risk


1

Bond Price and Yield Curve

♦ Bond prices are determined by the yield curve

♦ Bond price risk stems from yield curve changes

♦ In this section, we seek to quantify this risk

♦ Risk measures are used to

⎯ hedge

⎯ implement views

♦ When the yield or interest rates go up, the price of a bond drops

♦ When the yield or interest rates drop, the price of a bond increases

2

What is Risk?

♦ Risk is exposure to change

♦ Which is riskier, a USD bond issued by– US Treasury?

– UBS?

♦ Which is riskier?– 2 year US Treasury note

– 30 year US Treasury bond

♦ We will focus on market/price risk– how does the value (price) of a bond change as interest rates change?

3

Bond Risk

4

Single Cash flow - Zero Coupon Bonds

♦ We first consider zero-coupon bonds

♦ How does the value of a single cashflow change?

♦ Interest rates = 10%

♦ Zero coupon bond values:

♦ What if rates go to 10.1%?

Maturity Value today1 90.9092 82.6455 62.092

10 38.55430 5.731

5

♦ If rates go up to 10.1%, the values go down

♦ Values before and after, and changes:

♦ % change seems to be proportional to the time to maturity

10% 10.1%Maturity Value today New value Change % Change

1 90.909 90.827 -0.083 -0.09%2 82.645 82.495 -0.150 -0.18%5 62.092 61.811 -0.281 -0.45%

10 38.554 38.206 -0.349 -0.90%30 5.731 5.577 -0.154 -2.69%

Single Cashflow

6

Single Cashflow

♦ % price change is proportional to maturity

♦ In the example, % price change = – 0.0009 x maturity

♦ Where does the number 0.0009 come from? It turns out that

♦ % price change =

♦ Price change =

yield 1change yield

0.1010.10100.100.0009

+=

+−=

[ ] maturity yield 1

change yield ×−+

[ ] price maturity yield 1

change yield ××−+

7

Weighted Average Maturity - continued

♦ You have this portfolio:

– $500 of a 1-year zero coupon bond

– $500 of a 5-year zero coupon bond

♦ The price risk is the same as $xxxx of a x-year zero coupon bond

8

Weighted Average Maturity - Example

♦ What if you have:

– $500 of 1 year

– $200 of 3 year

– $300 of 5 year

♦ In this case

♦ The price risk is the same as:

– $1000 of 2.6 year ZCB

♦ Instead of thinking of the portfolio as 3 bonds of different maturities, we think of it as $1000 invested in 2.6 year zero coupon bonds

( ) ( ) ( ) 2.6 5 3 1 1000300

1000200

1000500 =×+×+×

9

Risk on a Bond

♦ A bond is like a portfolio of cashflows

♦ We know how to measure the risk on a single cashflow - it is proportional to the maturity of the cashflow

♦ We can take a portfolio of cashflows and find a single cashflow that is equivalent to the portfolio in terms of price risk

10

Bond Price

♦ 4 year bond with 8% coupon

♦ Yield = 7%

♦ What is the price?

4 n

100 FV

8 PMT

7 i

PV = 103.39

♦ But how is the price made up?

11

Bond as a Portfolio

♦ Bond value broken down:

♦ The bond is like

– 7.48 of the 1 year cashflow




♦ So what is the maturity of the single cashflow that is equivalent?

Year Cashflow Value1 8 7.482 8 6.993 8 6.534 108 82.39

103.39

12

Bond Duration

♦ Bond is like a portfolio where

– 7.48 / 103.39 = 7.2% is invested in the 1 year ZCB




♦ This is equivalent to 103.39 invested in the 3.58 year zero-coupon bond:

Year Cashflow Value Proportion Proportion x maturity1 8 7.48 7.2% 0.0722 8 6.99 6.8% 0.1353 8 6.53 6.3% 0.1894 108 82.39 79.7% 3.188

103.39 1.00 3.58

13

Macauley Duration

♦ 3.58 years is the maturity of the zero-coupon bond that has the same sensitivity to interest rate changes as this coupon bond

♦ It is called the bond’s Macauley Duration

♦ A 3.58-year zero coupon bond has risk defined by:

– Price change = – [ yield change / ( 1 + yield) ] x 3.58 x price

♦ Our coupon bond has the same sensitivity, so:

– Price change = – [ yield change / ( 1 + yield) ] x 3.58 x price

♦ So, for a coupon bond:

– Price change = – [ yield change / ( 1 + yield) ] x Duration x price

= – [ Duration / ( 1 + yield) ] x price x yield change

14

Price Value of a Basis Point (PVBP)

♦ Price change =

♦ For our bond, modified duration = 3.58 / 1.07 = 3.35

♦ For a 1 b.p. change in yield, the change in price is

♦ (– 3.35 x 103.39 x 0.01%) = -0.0346

♦ This is called the Price Value of a Basis Point (PVBP)

♦ If the yield goes from 7% to 7.01%, we expect the price to go down by 3.46 cents on a $100 face. Let’s check …

modified duration

[ ] change yield price yield 1

duration ××−+

15

Testing PVBP

♦ Using PVBP, we can predict changes in price from changes in yield:

♦ How do these compare with actual bond prices (using bond calculator)

♦ PVBP only seems to work for small changes

down 100 down 1 current up 1 up 100106.85 103.42 103.39 103.35 99.92353

down 100 down 1 current up 1 up 100Predicted 106.85 103.42 103.39 103.35 99.92Actual 106.93 103.42 103.39 103.35 100

16

Gamma

4 4.5 5 5.5 6 6.5 6.99 7 7.01 7.5 8 8.5 9 9.5 10Predicted 113.78 112.05 110.31 108.58 106.85 105.12 103.42 103.39 103.35 101.66 99.92 98.19 96.46 94.73 93.00Actual 114.52 112.56 110.64 108.76 106.93 105.14 103.42 103.39 103.35 101.67 100 98.36 96.76 95.19 93.66

90.00

95.00

100.00

105.00

110.00

115.00

4 5 6 7 8 9 10

Yield

Pric

e

17

Gamma

♦ PVBP (or bond delta) depends on the yield too

♦ As yields increase, the PVBP decreases because the price of the bond and its duration decrease

♦ Gamma is the measure of how the PVBP changes when yields change

18

Another View of Duration

♦ A bond has two risks

♦ These risks move inversely

♦ The point in time where they cancel each other out is called the Duration

Price risk

Reinvestment risk

change in price due to change in yield

change in coupon reinvestment income due to change in yield

♦ What does this mean?

19

4 year, 5% coupon bond yielding 7%

Example

1 5 4.6729 0.05012 5 4.3672 0.04683 5 4.0815 0.04384 105 80.1040 0.8592

Total 93.2256 1.0000

Year Cashflow Present value % of price

3.712156 0.85924 0.04383 0.04682 0.05011 D =×+×+×+×=

The Macauley duration

20

Example - continued

♦ Suppose immediately after we purchase the bond, the yield either rises to 8% or falls to 6%

—New price at 8% = 90.06 - Did we lose money?

—New price at 6% = 96.53 - Did we make money?

♦ What if we hold the bond for several years?

♦ If we hold the bond for 3.71 years, the bond will return 7%

HoldingPeriod

Return(YTM=6%)

Return(YTM=7%)

Return(YTM=8%)

1 year 9.76% 7% 4.34%

2 years 7.87% 7% 6.15%

3 years 7.24% 7% 6.76%

4 years 6.93% 7% 7.07%

21

♦ Also can be understood as the point in time where thepast and future cashflows are balanced (as in a see-saw)

– 10% coupon, 4-year bond

– 8% yield to maturity

♦ Macauley Duration is 3.504 years

9.3 8.6 7.97.4

1 2 3 4

74

Macauley’s Duration

22

Factors affecting Duration

♦ Duration is impacted by all things that determine the price of a bond

time to maturity

coupon

yield

frequency of coupon payments

23

Zero Coupon Bond

24

Increasing coupon

25

Increasing yield

26

Increasing time

27

Increasing coupon frequency

28

Exercise: Risk Measures of German Bond

29

Exercise: Risk Measures of German Bond

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001 Fi Interest Rate and Bond Calculations

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