Financial Management-Qand Solutions

8/10/2019 Financial Management-Qand Solutions

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TIME VALUE OF MONEY

TUTORIAL QUESTION AND

SOLUTION

Question 1 Find the following present and future values:

a. An initial 500 compounded for 1 year at 6 percent.

b. An initial 500 compounded for 2 years at 6 percent.

c. The present value of 500 due in 1 year at a discount rate of 6 percent.

d. The present value of 500 due in 2 years at a discount rate of 6 percent.SOLUTION

a. Given,

Present value (PV) = . 500

Interest rate (i) = 6%

FVn = PV(1 + i)n

FV1 = PV (1 + i)1

= 500 (1 + 0.06)1 = . 530

b. Present value (PV) = . 500


FVn = PV(1 + i)n

FV2 = PV (1 + i)2

= . 500 (1 + 0.06)2 = . 561.80

c. Answer:

= . 471.70

d. Future value (FV) = . 500


No. of periods (n) = 2

Present value (PV) = ?

PV =FVn

(1 + i)n=FV2

(1 + i)2

= 445

- 500

0

FV = ?

16%

- 500

0

FV = ?

26% 1

PV = ?

0

FV = 500

26% 1


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Question 2 Suppose John deposits 10,000 in a bank account that pays 10 percent interest annually.How much money will be in his account after 5 years?

SOLUTION Here, Present value (P) = 10,000,

Interest rate (k) = 10%

Number of years (n) = 5 years,

Future value (FV5) = ?

We have,

FV5 = PV (1 + k)n = 10,000 (1.10)5

= 10,000 1.6105

= 16,105.10

John will have 16,105.10 at the end of year 5 in his account.

Question 3 What is the present value of a security that promises to pay you 5,000 in 20 years?Assume that you can earn 7 percent if you were to invest in other securities of equal risk?

SOLUTION Here, Future value (FV) = 5,000

Number of years (n) = 20 years

Interest rate (k) = 7%

Present value (PV) = ?

We have,

PV =FV20

(1 + k)n= 5000/ (1.07)20

=5000/3.8697 = 1,292.09

Question 4 If you deposit money today into an account that pays 6.5 percent interest, how long will ittake for you to double your money?

SOLUTION Here, Interest rate (i) = 6.5%

Number of period (n) = ?

Present value (PV) = 1000 (assume)Future value (FV) = 2000

We have,

Present value (PV) =FV

(1 + i)n

or, 1000 = 2000 / (1+0.065)n

0 1 2 3 20

5,000PV = ?

7%

0 1 2 3 5

FV = ?10,000

10%

0 1 2 3 n = ?

FV = 2,000PV = 1000

6.5%


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or, (1 + 0.065)n = 2000/1000

or, (1.065)n= 2 .... (i)

Trying at n = 11We get,

If n = 11, the left hand side in above equation (i) is approximately equal to 2. Hencethe required no. of years to double the sum of money is 11 years.

Question 5 Your broker offers to sell a note for 13250 that will pay 2345.05 per year for 10 years. Ifyou buy the note, what rate of interest will you be earning? Calculate to the closestpercentage.

SOLUTION

Here,

Present value of annuity (PVA) = . 13,250

Periodic equal payment (PMT) = . 2345.05

No. of periods (n) = 10 years

Interest rate (i) = ?

Time Line

We have,

PVA = PMT PVIFA i n yrs.

or, . 13,250 = . 2345.05 PVIFAi% 10 yrsor, PVIFAi%, 10 yrs = 5.6502

From the PVIFA table, the value of 5.6502 in 10 years lies at 12%.

The required interest rate is 12%.

Question 6 Your parents are planning to retire in 18 years. They currently have 250,000, and theywould like to have 1,000,000 when they retire. What annual rate of interest would theyhave to earn on their 250,000 in order to reach their goal, assuming they save no moremoney?

SOLUTION Here, Future value (FV) = 1,000,000

Present value (PV) = 250,000

Time period (n) = 18 years

Interest rate (i) = ?

We have,

FV = PV (1 + i)n

or, 1,000,000 = 250,000 (1 + i)18

or, (1 + i)18 = 1,000,000/ 250,000

or, (1 + i)18 = 4

or, 1 + i = (4)1/18

or, i = 1.08 - 1 = 0.08 or 8%

The required rate of interest to reach the goal is 8%.

0 1 2 3 18

1,000,000 250,000

i = ?

PVA =

13250

0 101 2 3 4 5 6 7 8 9

2345.05 2345.05 2345.05 2345.05 2345.05 2345.05 2345.05 2345.05 2345.05 2345.05


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Question 7 What is the future value of a 5-year ordinary annuity that promises to pay you 300 each

year? The rate of interest is 7 percent.

SOLUTION Here, Future value of annuity (FVA) = ?

Payment (PMT) = 300

Number of period (n) = 5 years


We have,

FVA = PMT (1 + i)n - 1

i

= 300

(1 + 0.07)

5 - 10.07

= 300 5.7507 = 1,725.21

Question 8 What is the future value of a 5-year annuity due that promises to pay out 300 each year?Assume that all payments are reinvested at 7% a year, until year 5.

SOLUTION Here, Future value of annuity due (FVAdue) = ?

Payment (PMT) = 300

Number of period (n) = 5 yearsInterest rate (i) = 7%

We have,

FVAdue = PMT

(1 + i)

n - 1i (1 + i)

= 300

(1 + 0.07)

5 - 10.07 (1 + 0.07)

= 300 5.7507 1.07 = 1,845.97

Question 9

A company invests 4 million to clear a tract of land and to set out some young pinetrees. The trees will mature in 10 years, at which time the company plans to sell the forestat an expected price of 8 million. What is company's expected rate of return?

SOLUTION Here, Future value (FV) = 8,000,000

Present value (PV) = 4,000,000

Time period (n) = 10 years

Expected rate of return (i) = ?

First set up time line as follows:

0 1 2 3 5

300

FVA = ?

7%

300 300 300

0 1 2 3 5

FVA (due) = ?300

7%

300 300 300


5/12

We have,

FV = PV (1 + i)n

or, 8,000,000 = 4,000,000 (1 + i)10

or, (1 + i)10 =8,000,000/ 4,000,000

or, (1 + i)10 = 2

or, 1 + i = (2)1/10

i = 1.0718 - 1

= 0.0718 or 7.18%

Question 10 Rachel wants a refrigerator that costs 12000. She has arranged to borrow the totalpurchase price of refrigerator from a finance company at a simple interest rate equal to 12percent. The loan requires quarterly payments for a period of three years. If the first

payment is due three months after purchasing the refrigerator, what will be the amountof her quarterly payments on the loan?

SOLUTION

Given,

Present value of an annuity (PVA) = . 12,000

Simple interest rate per annum (i) = 12%

Loan requires quarterly payment i.e. m = 4

Number of years (n) = 3 year

Periodic equal payment (PMT) = ?

Time line:

We have,

PVA= PMT

1 -

1(1 + i/m)

n m

im

or, 12,000 = PMT

1 -

1(1 + 0.12/4)

3 4

0.124

or, 12,000 = PMT 9.9540

PMT = 12,000/ 9.9540

= 1,205.55

Question 11 You need to accumulate 10,000. To do so, you plan to make deposits of 1,250 per year,with the first payment being made a year from today, in a bank account which pays 12percent annual interest compounded annually. Your last deposit will be less than 1,250if less is needed to round out to 10,000. How many years will it take you to reach your 10,000 goal, and how large will the last deposit be?

SOLUTION Here,

Annual payment (PMT) = 1,250

Future value of annuity (FVAn) = 10,000

0 1 2 3 10

8 million 4 million

i = ?

-12000

0 12% 121 2 3 4

PMT PMT PMT PMT PMT


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Time to maturity (n) = ?

Last deposit = ?First, we determine the number of periods of the financial goal. This is calculated usingfuture value of annuity formula as follows:

We have,

FVAn = PMT FVIFAi, n 10,000 = 1,250 PViFA12, n

FVIFA12, n =10000

1250= 8

Looking FVIFA table the value 8 at 12 percent interest rate lies approximately in 6 years.Therefore the number of years to reach the financial goal is 6 years. Now we calculate thefuture value of 1,250 for 5 years at 12%, it is 7,941.06

FV = 1,250 FVIFA12, 5 = 1,250 6.3528 = 7,941

Compounding this value after 6 years and before the last payment is made, it is 7,941(1.12) = 8,893.92. Thus, we will have to make a payment of 10,000 - 8,893.92 = 1,106.08 at year 6, therefore it will take 6 years, and 1,106.08 must be paid in the lastinstallment.

Question 12 Your Company is planning to borrow 1,000,000 on a 5-year, 15 percent, annualpayments, and fully amortized term loan. What fraction of the payment made at the endof the second year will represent repayment of principal?

SOLUTION Here, Loan amount (PVA) = 1,000,000



First we determine the annual installment or payment (PMT)

We have,

PMT =PVA

PVIFAi n= 1,000,000/PVIFA15,5=1,000,000/ 3.3522 = 293,311.55

Preparation of Amortization Schedule,

Amortization schedule

Year Payments Interest Payment

of Principal

Ending Balance

1

2

298,311.5566

298,311.5566

150,000

127,753.2665

148,311.5566

170,558.2901

851,688.4434

681,130.1533

%principal in 2nd year =Principal payment in 2ndyear

Payments =57.17%

That is 57.17% of the payment in second year represents the principal.

Question 13 You are branch manager of town centre Natwest Bank, Manchester. A borrowerapproaches you for a term loan of 500,000. You agreed to give loan to be fullyamortized in a period of 5 year at 10 percent, annual payment. What will be the size of

0 1 2 3 n =?

Last deposit = ?

FVA = 10,000

12%

1,250 1,250 1,250


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8/12

Quarterly interest rate = 3%,

PMT = ?, PV = ?

We have,

PV =FV

(1 + i)n= 1,432.02/(1+0.03)35= 508.91

Now we calculate the payment (PMP)Here, n = 5 periods, i = 3%, PV = ?; FV = 508.91, FVA = 508.91PMT = ?

We have,

FVA = PMT

(1 + i)

n- 1i (1 + i)

or, 508.91 = PMT (1 + 0.03)5- 1

0.03 (1 + 0.03)

or, 508.91 = PMT 5.3091 1.03

PMT = 508.91/ 5.4684 = 93.06

Question 15 The prize in last week's Lottery was estimated to be worth 35 million. If you were luckyenough to win, then it will pay you 1.75 million per year over the next 20 years. Assumethat the first installment is received immediately.

a. If interest rates are 8 percent, what is the present value of the prize?b. If interest rates are 8 percent, what is the future value after 20 years?c. How would your answers change if the payments were received at the end of each

year?SOLUTION

Here, Payment (MPT) = 1.75 million

Number of periods (n) = 20 years,

a. Present value of annuity (PVA) = ? interest rate (i) = 8%

PVA = PMT

1 -

1(1 + i)n

i (1 + i) = 1.75

1 -

1(1 + 0.08)20

0.08 (1 + 0.08)

= 1.75 9.8181 1.08

= 18.56 million

b. Future value of annuity (FVA) = ?, Interest rate (i) = 8%

FVA = PMT

(1 + i)

n- 1i (1 + i)

= 1.75

(1 + 0.08)20- 10.08 (1 + 0.08)

= 1.75 45.7620 1.08

= 86.49 million

c. PVA and FVA assuming payments received at the end of year.

Present value of annuity (PVA) = ?, Interest rate (i) = 8%

We have,

PVA = PMT

1 -

1(1 + i)n

i

0 10

FV = 1,432.02

3%

PMT = ? PMT = ? PMT = ?

1

PMT = ?PMT = ?


9/12

= 1.75

1 -

1(1 + 0.08)20

0.08

= 1.75 9.8181

= 17.18 million

Future value of annuity (FV) = ?, Interest rate (i) 8%

FVA = PMT

(1 + i)

n- 1i = 1.75

(1 + 0.08)

20- 10.08

= 1.75 45.7620

= 80.08 million

Question 16 Ashley has 42,180.53 in brokerage account, and plans to contribute an additional 5,000

every year at an annual interest rate of 12 percent. If Ashley has to accumulate 250,000,how many years will it take for him to reach his goal?

SOLUTION Here, Present value (PV) = 42,180.53

Payment (PMT) = 5,000

Annual return (i) = 12%

Future value (FV) = 250,000

Number of years to reach goal (n) = ?

We have,

42,180.53 (1 + 0.12)n+ 5,000

(1 + 0.12)

n - 10.12 = 250,000

or, 5,061.66 (1.12)n + 5,000 (1.12)n - 5,000 = 30,000

or, 10,061.66 (1.12)n = 35,000

or, (1.12)n = 35,000/ 10,061.66

or, (1.12)n = 3.4786 ... (i)

In above eqn. (i) if we go for trying several values of 'n', the left hand side is exactlyequal to right hand side at n = 11.

The required no. of years to reach the goal is 11 years.

Question 17 Your client is 40 years old and wants to begin saving for retirement. You advise the clientto put 5,000 a year into the stock market. You estimate that the market's return will be,on average, 12 percent a year. Assume the investment will be made at the end of the year.

a. If the client follows your advice, how much money will she have by age 65?

b. How much will she have by age 70?

SOLUTION Here, Your client is 40 years old, Payment (PMT) = 5,000, Interest rate (i) = 12%

Investment will be made at the end of the yeara. Future value of annuity (FVA) at the age of 65?

Number of periods (n) = 65 - 40 = 25 years

0 1 2 3 n

2,50,00042,180.53

12%

5000 5000 5000


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FVA = PMT

(1 + i)

n- 1i = 5,000

(1 + 0.12)

25- 10.12

= 5,000 133.3338 = 666,669b. Future value of annuity (FVA) at the age of 70?

Number of periods (n) = 70 - 40 = 30 years

FVA = PMT

(1 + i)

n- 1i = , 5,000

(1 + 0.12)

25- 10.12

= 5,000 241.3327

= 1,206,66

Question 18 Jason has inherited 25,000 and wishes to purchase an annuity that will provide him witha steady income over the next 12 years. He has heard that the local savings and loanassociation is currently paying 6 percent compound interest on an annual basis. If he wereto deposit his funds, what year-end equal pound amount (to the nearest pound) would hebe able to withdraw annually such that he would have a zero balance after his lastwithdrawal 12 years from now?

SOLUTION Here,

Present value of annuity (PVA) = 25,000



Equal annual withdraw (PMT) = ?

PVA = PMT

1 -

1(1 + i)n

i

or, 25,000 = PMT

1 -

1(1 + 0.06)12

0.06 or, 25,000 = PMT 8.3838

PMT = 2,981.9414

Question 19 You need to have 50,000 at the end of 10 years. To accumulate this sum, you havedecided to save a certain amount at the end of each of the next 10 years and deposit it inthe bank. The bank pays 8 percent interest compounded annually for long term deposits.How much will you have to save each year (to the nearest Pound)?

SOLUTION

FVA = PMT

(1 + i)

n- 1i

or, 50,000 = PMT

(1 + 0.08)10- 1

0.08

or, 50,000 = PTM 14.4866

PMT = 50,000/14.4866

= 3,451.46

Question 20 Louise wishes to borrow 10,000 for three years. A group of individuals agrees to lendher this amount if she contracts to pay them 16,000 at the end of the three years. What isthe implicit compound annual interest rate you receive (to the nearest whole percent)?

SOLUTION Here, Present value (PV) = 10,000


11/12

Number of year (n) = 3 years

Future value (FV) = 16,000

End payment, interest rate (i) = ?We have,

FV = PV (1 + i)n

or, 16,000 = 10,000 (1 + i)3

or, 1.6 = (1 + i)3

or, (1.6)1/3 - 1 = 1

or, i = 0.1695or 16.95%

Question 21 Calculate the present value of the following cash flow stream. Assume that the stated rateof interest is 14 percent per annum discounted semiannually.

SOLUTION If stated annual rate is 14 percent, discounted semiannually, first we calculate the effectiveannual rate as follows:

Effective interest rate (EAR) = (1 + 0.14/2)2 - 1 = 14.49%

Now present value of given cash flow stream discounted at 14.49 percent effective annualrate is calculated as follows:

Year Cash flow 14.49% PVIF PV

0

1

2

3

1,000

1,600

1,500

850

1.0000

0.8734

0.7629

0.6663

1,000.00

1,397.44

1,144.35

566.36

Total present value 4,108.15

Question 22 National Lottery has offered you the choice of the following alternative payments.

Alternative 1: 10,000 one year from now

Alternative 2: 20,000 five years from now.

a. Which should you choose if the discount rate is 0 percent? 20 percent?

b. What rate makes the options equally attractive?

SOLUTION

a. Calculation of present value if discount rate is 0 percentAlternative 1:

PV =FV

(1+i)n =10000(1+0)1 = 10,000

Alternative 2:

PV =FV

(1+i)n =20000(1+0)5 = 20,000

If discount rate is 0 percent, Alternative 2 is preferable because of higher presentvalue.

Calculation of present value if discount rate is 20 percent

1000

2 3

1600

End of year

Cash flow

1

1500 850

0


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Alternative 1:

PV =FV

(1+i)n =

10000

(1+0.20)1 = 8,333.33

Alternative 2:

PV =FV

(1+i)n =20000

(1+0.20)5 = 8,037.55

If discount rate is 20 percent, Alternative 1 is preferable because of higher presentvalue.

b. Calculation of rate of interest (i) at which both the options are equally likely

PV of Alternative 1 = PV of Alternative 2

10000(1+i)1 =

20000(1+i)5

(1+i)4= 2

i = (2)1/41 = 0.1892 or 18.92%

That is, if the discount rate is 18.92 percent both the alternative would produce equalpresent value so that both are equally likely.

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Financial Management-Qand Solutions