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TIME VALUE OF

MONEY : CHAPTER 2

PARVESH AGHI


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COMPOUND INTEREST

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Question 1

2000 is invested at annual rate of interest of10%. What is the amount after 2 years ifcompounding is done ?

(a) Annually ?

(b) Semiannually ?

(c) Monthly ?

(d) Daily ?

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SOLUTION

(a) Annually ?

FVn = P(1+i)n

FV2 = 2000(1.10)2

FV2 = 2000 X1.21

FV = 2420

(b) Semiannually ?

Fn = P(1 + i/m)n*m m= no of compounding per year

Fn = 2,000(1 + .10/2)2*2

= 2,000(1.05)4

FV = 2000 x 1.2155 = 2,431

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(c) Monthly ?Fn = P(1 + i/m)n*m

Fn = 2000(1 + .10/12)2*12

Fn = 2000(1.00833)24

Fn = 2000 X 1.22029 = 2440.58

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(c) Daily ?Fn = P(1 + i/m)n*m

Fn = 2000(1 + .1/365)2*365

Fn = 2000(1.00027)730

Fn = 2000 X 1.22135 = 2442.70

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Question 2

Determine the compound amount and compoundinterest on 1000 at 6% compounded semianually for 6 years .

Given that (1+i) n = 1.42576 for i = 3% and n=12

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Solution

Fn = P(1 + i/m)n*m

m= no of compounding peryear

Compound amount (CA ) = 1000 ( 1.03)12

=CA = 1000 X 1.42576 = 1425.76

Compound Interest = 1425.76-1000 = 425.76

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Question 3

What annual rate of interest compoundedannually doubles an investment in 7 years ?

Given that 21/7= 1.104090

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FVn = 2PFVn= P(1+i)

n

2P= P(1+i)7

2= (1+i)7

21/7= 1+i

1.104090= 1+ii= 1.104090-1 = .104090 = 10.41%

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Question 4

A person opened an account on April 2012 with a deposit of 800 .

The account paid 6% interest compounded annually quarterly . OnOctober 1 2012 , he closed the account and added enough additionalmoney to invest in a 6 month time deposit for 1000 earning 6%compounded monthly

(a) How much additional amount did the person invest on October 1?

(b) what was the maturity value of his Time deposit on April 1 , 2013 ?

(c) how much total interest was earned ?

Given that (1+i) is 1.03022500 for i= 11/2 % , n= 2 and is

1.03037751 for i= % and n= 6

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Question 5

Ramanuj has taken a 20 month car loan of 6,00,000 . The rate of interest is 12 percent perannum . What will be the amount of monthly loanamortization

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Solution

A=P / PVFAi N

A = 6,00,000 / PVFA1% 20

A = 6,00,000 / 18.0456

A = 33,249.1

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EFFECTIVE RATE OF INTEREST

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Question 6

A Ltd offers interest of 13% on its Public DepositsWhat is the effective rate of interest per annum ifcompounding is done

(a) Half yearly

(b) Quarterly

(c) Monthly

(d) Weekly ? Given that (1+i) is 1.1342 for i= 6.5% , n= 2 , 1.1365 for i= 3.25% and n= 4 ,

1.1380 for i = 1.083% , n= 12 , 1.1386 i = .25% n = 52

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PRESENT VALUE

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Question 7

What is the present value of 50,000 to bereceived after 10 years at 10 per centcompounded annually

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P = F X PVFni = 50,000 X .38554

= Rs 19, 277

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Question 8

An investor can make an investment in a realestate development and receive an expectedcash return of 45,000 after six years. Based ona careful study of other investment alternatives,

she believes that an 18 percent annual returncompounded quarterly is a reasonable return toearn on this investment. How much should shepay for it today?

PVFi,n = .3477 for i= 4.5% n = 24

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Solution

P/Y = 4, N = 6 4 = 24, I = 18, PMT = 0, FV =45,000 PV = 15,646.66.

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Question 9

Mr X has made real estate investment for 12,000 which he expects will have a maturityvalue equivalent to interest at 12% compoundedmonthly for 5 years . If most saving institutions

currently pay 8% compounded quarterly on a 5year term , what is the least amount for which MrX should sell his property ?

Given that (1+i)n = 1.81669670 for i = 1% and n=60 and that (1+i) -n = 0.67297133= 2% and n = 20

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It is a two part problem . First being determination of maturity value of

the investment of 12000 and then finding the present value of theobtained maturity value.

A . Maturity Value of the Investment

F = P ( 1+i)n

P = 12,000

i = 12%/12 = 1%

n = 12X 5 = 60

F = 12000 X ( 1.01)60

= 12000 X 1.8167

= 21,800.40

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B . Present value of the obtained maturityvalue

P = F X PVF i,n

P = 21,800 i= 8%/4 = 2% n = 5 x 4 = 20 PVF 2%,20 =

0.67297

P = 21,800.40 X .67297 = 14,671.02

X should not sell the property for less than 14,671.02

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Question 10

Suppose you have the opportunity to make an investment in a real

estate venture that expects to pay investors 750 at the end of eachmonth for the next eight years. You believe that a reasonable returnon your investment should be 17 percent compounded monthly.

Required :

a) How much should you pay for the investment? b) What will be the total sum of cash you will receive over the next

eight years?

c) Why is there such a large difference between (a) and (b)?

Given PVFAi,n = 52.2972 where i = 1.4167% n = 96

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Solution

(a)P/Y = 12, N = 8 12 = 96, I = 17, PMT = 750,FV = 0 39,222.96

(b) This can be solved by setting I = 0, PV = 0,and computing FV = 72,000.

(c ) The difference between the answers in parts(a) and (b) represents the foregone interest thatresults from receiving the payments in the future,

rather than today.

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Question 11

ABC is evaluating an investment that will providethe following returns at the end of each of thefollowing years:

year 1, 12,500; year 2, 10,000; year 3, 7,500;

year 4, 5,000; year 5, 2,500; year 6, 0; and year7, 12,500.

ABC believes that he should earn an annual rate

of 9 percent on this investment. How muchshould he pay for this investment?

Given Present value factors at 9% for years 1-7: 0.917, 0.842, 0.772,0.708 0.650, 0.596, 0.547

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Solution

This can be solved using the irregular cash flow worksheet:

CF0 = 0

C01 = 12,500

C02 = 10,000

C03 = 7,500 C04 = 5,000

C05 = 2,500

C06 = 0

C07 = 12,500

Set I = 9 and solve for NPV = 37,681

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ANNUITY

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Question 12

A person is required to pay four equal annualpayments of 4000 each in his Deposit

account that pays 10% interest per year .

Find out the future value of annuity at the end

of 4 years

Given : CVFA i,n = 4.641 where i = 10% n = 4

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Solution

F = A { (1+ i)n-1 }

i

F = A X CVFAn,i

F4= 4000 ( 1.10)4-1 )/ .10

= 4000 X ( .14641-1 /.10)

= 4000 X 4.641

= 18564

CVFA 4,10% = 4.641

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Question 13

200 is invested at the end of eachmonth in an account paying interest 6%per year compounding monthly . What is

the amount of this annuity after 10th

payment ?

Given that CVFA .005% ,10= 10.22

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Question 14

A investor wants to accumulate 43,746 at the end of four years from now

How much should deposited each year

at an interest rate of 6% so that it growsto 43,746 at the end of fourth year ?

Given : CVFAi.n where i = 6% n= 4

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Question 15

Y bought a TV costing 13,000 by making adown payment of 3000 and agreeing to makeequal annual payment for 4 years . How muchwould be each payment if the interest on unpaid

amount be 14% compounded annually ?Given PVFA i,n = 2.914 where i=14% n= 4

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Solution

Y bought a TV costing 13,000 by making a down payment of 3000 and

agreeing to make equal annual payment for 4 years . How much would beeach payment if the interest on unpaid amount be 14% compounded annually?

A = P/ PVFAi,n

A = 10,000 / PVFA i,n

= 10,000 /2.914

= 3431.71

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Perpetuity

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Question 16

Ramesh wants to retire and receive 3000 amonth . He wants to pass this monthly paymentto future generations after his death . He canearn an interest of 8% compounded annually .

How much will he need to set aside to achievehis perpetuity goal ?

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Solution

A = 3000

P = A

i

i = .08/12 = .006667

P = 3000

.00667

= Rs 4,49,775

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Question 17

When Sunil retires he wants to have apension of Rs 60,000 per annum to growat 5% per annum if the insurance

company can get a return of 11% after

meeting its expenses , what lump sumamount should Sunil pay now to

Insurance company.

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Solution

P= 60,000/11%-5%

P= 60,000/6% = 10,00,000

39

=

AP

i g

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Question 18

You own an oil pipe line which will generate a 2 million cash return over the coming year . Thepipelines operating costs are negligible , and it isexpected to last for a very long time .

Unfortunately, the volume of oil shipped isdeclining , and cash flows are expected to declineby 4% per year. The discount rate is 10%

what is the present value of the pipelines cashflow if its cash flows are assumed to last forever ?

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Solution

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Question 19

You plan to retire 33 years from now. You expect that you will live 27

years after retiring.

You want to have enough money upon reaching retirement age towithdraw 12,00,000 from the account at the beginning of each year

you expect to live, and yet still have 50,00,000 left in the account at

the time of your expected death (60 years from now).

You plan to accumulate the retirement fund by making equal annualdeposits at the end of each year for the next 33 years. You expect that

you will be able to earn 12% per year on your deposits. However, you

only expect to earn 6% per year on your investment after you retire

since you will choose to place the money in less risky investments.

What equal annual deposits must you make each year to reach yourretirement goal?

Given : CVFA i,n where i= 12% n =33 , PVFA i,n where i = 6% n = 27

PVF i,n where i=6% n = 27

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Solution

You must solve this problem in two steps.First, calculate the PV at the time of

retirement of the amount needed to give

you the annuity and remaining sum wanted.Second, calculate the payment necessary

each year over the period from now until

retirement to generate the goal.

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Annuity = 12,00,000

I = 6%

N = 27

PV = A X PVFA 6%,27 X (1.06)

PV = 12,00,000 X 13.21053X1.06

PV = 1,68,04,392

TOTAL = 1,68,03,794+10,36,850= 1,78,40,644

A = F / CFVA 12%,33

A = 1,78,40,644/ 342.4294 = 52,100

FV = 50,00,000

I = 6%

N= 27

PV = FX PVF 6%,27

PV = 50,00,000 X . 0.20737 PV = 10,36,850

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Question 20

How much amount is required to be investedevery year so as to accumulate 3,00,000 at theend of 10 years if the interest is compoundedannually at 10%

Given CVFA in = 15.9374 where i is 10% and n=10

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Solution

How much amount is required to be investedevery year so as to accumulate 3,00,000 at theend of 10 years if the interest is compoundedannually at 10%

FV = 3,00,000

A = P / CVFA 10 , 10%

= 3,00,000 /15.9374

18,823.62

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Question 21


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Question 21

XYZ company is creating a sinking fund toredeem its preference capital of 10 lakhsissued on April 6 ,2012 and maturing on April 5,2023 . The first annual payment will be made on

April 6 2012 . The company will make equalannual payments and expects that the fund willearn 12 percent per year . How much will be theamount of sinking fund payment ?

CVFA i% n = 24.1331 where i = 12% n= 12

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Solution


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Solution

A = P / CVFA12,12%

X 1.12

= 10,00,000/ 24.1331 x 1.12

= 10,00,000 / 27,029072

= 36 , 997.35

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Question 22


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Question 22

A doctor is planning to buy an X-ray machine forhis hospital . He has two options . He can eitherpurchase it by making a cash payment of 5lakhs or 6,15,000 are to be paid in six equal

annual instalments . Which option do you suggestto the doctor assuming the rate of return is 12% ?

Present value of annuity of Re 1 at 12 percentrate of discount for six years is 4.111

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A doctor is planning to buy an X-ray machine for his hospital . He has two

options . He can either purchase it by making a cash payment of 5 lakhs or 6,15,000 are to be paid in six equal annual instalments . Which option doyou suggest to the doctor assuming the rate of return is 12% ? Present valueof annuity of Re 1 at 12 percent rate of discount for six years is 4.111

Annual Instalment = 6,15,000 / 6 = 1,02,500

I = 12% n= 6

P= A X PVFAi,n

P = 1,02,500 X 4.111 = 4,21,377.50 The doctor should buy x ray machine on instalment basis as PV is less than

the Cash price

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Question 23


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Question 23

Consider that an investor has an opportunity of

receiving following amounts at the end every year forfive years

Year 1= 1000

year 2= 1500

Year 3 = 800

Year4 = 1100

Year 5 = 400

Find the present value of this stream of cash flows if

the investors required rate of return is 8%

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Discounted Cash FlowSystem AYear Cash Flow (Rs) Discount Factor(8%)Present Value (Rs)

(CF x DF)

1 1000 .926 926

2 1500 0.857 1285.5

3 800 0.794 635.2

4 1100 0.735 808.5

5 400 0.681 272.4

Total 3927.6

Question 24


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Question 24

A company offers a fixed deposit schemewhereby 10,000 matures to 12625 after 2years , on a halfyearly compounding basis .

If the company wishes to amend the scheme by

compounding interest every quarter , what will bethe revised maturity value ?

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F = P (1+i) n

12625 = 10,000 (1+i)4

10,000 (1+i)4 = 12625

10,000 = 12625 / (1+i)4

10,000 = 12625 X PVF i,4

PVF i,4 = 10,000/12625

PVF i,4 = .79207 = I = 6% for half year

i= 12% for full year

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If the company wishes to amend the scheme bycom pounding interest every quarter the revisedmaturity

F = 10,000 (1+ 12%/4)4x2

= 10,000 (1.03)8

= 10,000 X 1.267 = 12,670

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Question 25


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Question 25

You are considering the purchase of aninvestment that is expected to generate cashflows of 15,000 per year for the next five years.

After that, cash flows are expected in increase at

the rate of 5 percent per year for the indefinitefuture.

Thus, in year 6 the cash flow will be 15,750,

etc. How much is this investment worth to youtoday if your required return is 15 percent?

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Question 26

You are valuing an investment that will pay you 26,000per year for the first 9 years, 34,000 per year for thenext 11 years, and 47,000 per year the following 14years (all payments are at the end of each year).

Another similar risk investment alternative is an accountwith a quoted annual interest rate of 9.00% with monthlycompounding of interest. What is the value in today'srupee terms of the set of cash flows you have beenoffered?

Given : ( 1.0075)12= 1.0938 , PVFAin = 5.904 where i=9.38% and n= 9 , PVFAi,n = 8.887 where i=9.38% andn= 20, PVFAi,n = 10.155 where i= 9.38% and n= 34

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Solution


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Solution

Since the payments occur annually, but the interest is

compounded monthly, we first must calculate the effectiveannual interest rate.

EIR = (1+i/m)nm-1

EIR = ( 1+ 9%/12)12

= ( 1.0075)12

-1 = 1.0938 -1 = .0938 =9.38%

PVAi,n = 5.904

PVAi,n = 8.887

PVAi,n = 10.155

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Present value of the cash in flows

Particulars Cash Flowper year

PVFA Amount

26,000 per year forthe first 9 years

26000 5.904 1,53,504

34,000 per year forthe next 11 years

34,000 8.887 -5.904=2.983

1,01,422

47,000 per year thefollowing 14 years

47,000 10.155-8.887 =

1.268

59596

Value in today's rupee terms of the set of cash flowsyou have been offered?

3,14,522

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Question 27


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Question 27

ABC Ltd is considering the purchase of anapartment project for 100,000. They estimatethat they will receive 15,000 at the end of eachyear for the next 10 years.

At the end of the 10th year, the apartment projectwill be worth nothing. If the company insists on a9 percent return compounded annually on itsinvestment, is this a good investment?

PVA n.i = 6.41766 where i= 9% and n= 10

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P/Y = 1, N = 10, I = 9, PMT = 15,000, FV = 0 PV = $96,264.87. Based on

the NPV rule, this is a poor investment becausethe present value of future cash

flows is less than the required investment of$100,000.

Alternatively, you could enter PV = $100,000

and solve for I = 8.14%. Because the IRR of thisinvestment is less than the 9% hurdle rate, Dallasshould not invest in this project

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Question 28


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Question 28

Amit is considering the purchase of a Plot. He

can buy the Plot today and expects the price torise to 15,00,000 at the end of 10 years. Hebelieves that he should earn an investment yield

of 10 percent annually on this investment. Theasking price for the plot is 7,00,000. Should hebuy it?

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P/Y = 1, N = 10, I = 10, PMT = 0, FV = 15,00,000PV = 5,78,314.93 . Because

the present value of this investment is less thanthe 7,00,000 asking price for the lot,

Amit should not buy it.

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Question 29


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Question 29

The quarterly returns on investment account of an

investor over the past year have been as follows:

First Quarter 5%

Second Quarter 2%

Third Quarter7%

Fourth Quarter1.5%

If Investor had 12,000 in the account at the beginning of

the year, what is the value in the account at the end of thefourth quarter?

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Solution:


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Solution:

This is simply a series of percent changes, where the

percent change is different in each period. To solve forthe future value, you must multiply by (1 + r) for eachperiod, and the r changes each time. So first converteach percent (Big R) to the (1 + r) format: Divide by 100

and add 1. First Quarter (1 + r) = 1.05

Second Quarter (1 + r) = 1.02

Third Quarter (1 + r) = 0.93 Fourth Quarter (1 + r) = 0.985

FV = 12,000 x 1.05 x 1.02 x 0.93 x 0.985 11,773.07

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Question 30


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Question 30

What are the monthly payments on each of the

following Housing loans? Which loan is the bestoption for a homeowner who can afford paymentsof 875 per month? What is the total amount that

will be paid for each loan? Assume each Loan is 100,000.

Loan A: 30-year loan with a fixed interest rate of 8.5 percent

Loan B: 15-year loan with a fixed interest rate of 7.75 percent

Loan C: 20-year loan with a fixed interest rate of 8.125 percent

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Loan A. To determine the monthly payment for a 30-year loan with an 8.5-percent fixedinterest rate, clear your calculators memory, then set your calculator to 12 monthlypayments and end mode. Input the following to solve this equation:

PV =$100,000

N = 360 (Calculate the number of monthly periods by multiplying the length of the loan

by the number of months in a year: 30 * 12 = 360.)

I = 8.5/12 PMT = ?

Your monthly payment for this loan would be $768.91, and the total amount of all

payments would be $768.91 * 360, or $276,807.60.

The formula is: PV/((1-(1/(1+(I/P))^(N*P)))/(I/P))

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Loan B. For a 15-year loan at 7.75 percent

interest, follow the same steps explained above.This time, input the information listed below:

PV =$100,000

N = 15 * 12 = 180

I = 7.75

PMT = ? The monthly payment for this loan wouldbe $941.28, and the total amount of all paymentswould be $941.28 * 180, or $169,430.40.

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Loan C. For a 20-year loan at 8.125 percent

interest, the calculations are still the same. Inputthe

following in your financial calculator:

PV =$100,000 N = 20 * 12 = 240

I = 8.125

PMT = ?

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The monthly payment for this loan would be

$844.24, and the total amount of all paymentswould be $844.24 * 240, or $202,617.60.

Considering the mortgage payment the

homeowner can afford, the best financial option isLoan Cthe 20-year fixed-rate mortgage at8.125 percent interest. This loan would allow thehomeowner to pay off the home in 10 fewer years

than if he or she had the 30-year loan and to pay$74,190 less

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THANK YOU

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Question


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Question

You plan to retire 33 years from now. You expect that you will

live 27 years after retiring. You want to have enough money upon reaching retirement age

to withdraw 180,000 from the account at the beginning of each

year you expect to live, and yet still have 2,500,000 left in the

account at the time of your expected death (60 years from now).

You plan to accumulate the retirement fund by making equalannual deposits at the end of each year for the next 33 years.

You expect that you will be able to earn 12% per year on your

deposits. However, you only expect to earn 6% per year on your

investment after you retire since you will choose to place the

money in less risky investments.

What equal annual deposits must you make each year to reachyour retirement goal?


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27. You must solve this problem in two steps. First, calculate the PV at the time ofretirement of the amount needed to give you the annuity and remaining sum wanted.Second, calculate the payment necessary each year over the period from now untilretirement to generate the goal.

n = 27

i = 6

FV = 2500000

PMT = 180000

solve for PV (answer: = 3,038,989.79)

(make sure you are in begin mode)

n = 33

i = 12

FV = 3038989.79

solve for PMT (answer: = 8,874.79)

(make sure you are in end mode)

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XYZ Ltd is thinking of acquiring a new computer system

that will enhance productivity for five years to come. Thiscomputer system project essentially - requires an initialinvestment of 1 million today, - but yields in return thefollowing sequence of cash inflows in the future, as a

result from the enhanced productivity: Year 1: 100,000

Years 2, 3, 4 300,000

Year 5: 100,000Assess the computer system project at cost of capital of5% NPV = =951,662,

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q

You are considering the purchase of two different

insurance annuities. Annuity A will pay you16,000 at the beginning of each year for 8years. Annuity B will pay you 12,000 at the endof each year for 12 years. Assuming your moneyis worth 7%, and each costs you 75,000 today,which would you prefer? [102228 and 95312]

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