time value of money

FINANCIAL PATHTHE PACIFIC INSTITUTE OF FINANCIAL STUDIES & RESEARCH

Time Value of Money

According to the time Value of the Money DECREASE but No. of Money INCREASE. Annuity is the term used to describe a series of periodic flows of equal amounts. Under the method of compounding, we find the Future Values (FV) of all the cash flows

at the end of the time horizon at a particular rate of interest. Under the method of discounting, we reckon the time value of money now i.e., at time

zero on the time line. So, we will be comparing the initial outflow with the sum of the present values (PV) of the future inflows at a given rate of interest.

There are 2 cases for calculation:

Lump sum Installments

The FIXED amount deposited The EQUALLY amount deposited orin ONE time. withdrawn in same period.

Lump sum

PV = A ×PVIF

Where:

PVIF= 1¿¿

A = Future Amount.

Installment (Annuity)

PV = A × PVIFA

Where:

PVIFA=¿¿

Here: k=rate of return , n =

maturity.

Rules & some more formulas:

1) If rate of interest is given in less than

1year (compounding period is semi

annually or quarterly) than

effective rate of interest can be

calculated by the following

formulae:-

FV = A × FVIF

Where:

FVIF=¿

A = Present Amount.

FV = A × FVIFA

Where:

FVIFA=¿¿

Effective rate of ∫ .=(1+ Km )

n

−1

m = compounding period.

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K= Nominal Rate of Interest.

(Effective rate of interest is always greater

then nominal rate of interest)

2) If installment given in monthly than

k = after using the above formulae:

If rate of interest is annually to

monthly than:

K = (1+k)1/12 – 1

If rate of interest is monthly to

annually than:

K= (1+k)12 – 1

K= Effective rate of Int.

3) If compounding period is monthly

than effective rate of interest:

= k / 12

4) If the word (beginning) given in

question than formulae multiplied

by: (1 + k)

5) If k is not given in question than:

K = L1+(L2−L1)×(H−B)(H−S)

Where:

L1 = Lower rate

L2 = Higher rate

H = Highest value

B = Base value

S = Smallest value

6) Present value of Perpetuity (Infinite

terms)

= A / K

Where:

A = Amount

K = Nominal rate of return

7) Capital recovery factor:

= 1

PVIFA

8) Sinking fund factor:

= 1

FVIFA

9) Rules of Doubling period:

Rules of 69 :

n = .35+ 69K

Rule of 72 :

n = 72K




Some Numerical Questions Based on Above Formulas

Q1. Mr.Sharma plans to save Rs.20,00,000 for his daughter’s education and marriage in fifteen years. The State Bank of India pays 8% interest compounded annually. If Mr.Sharma is planning to make equal annual contributions at the end of every year for a period of fifteen years, how much should he put in the bank annually? (a) Rs. 73,333.30 (b) Rs. 73,659.40 (c) Rs.1,66,666.70 (d) Rs.1,76,243.00 (e) Rs.2,38,596.50.

Q2. Suppose you plan to buy a Santro costing Rs.4,50,000 and go for a motor vehicleloan being offered by the ICICI bank. The rules of the bank require you to make a downpayment of Rs.45,000. The remaining amount would be financed by them and therepayment can be done over five years. The interest rate offered by the bank is 12%compounded monthly. If you accept this proposal, the equated monthly payment andthe effective annual interest rate respectively are (a) Rs. 6,750; 12.12% (b) Rs. 8,760; 12.68% (c) Rs. 9,008; 12.68% (d) Rs.10,008; 12.88% (e) Rs.10,567; 12.98%.

Q3. Mr. Ali takes a loan of Rs.1,00,000 from Fortune Finance Company Pvt. Ltd. As per the loan agreement, Mr. Ali has to pay off the loan in thirty-six monthly installments of Rs.5,000 each. The effective annual rate of interest that the finance company is charging Mr. Ali is

(a) 36.20% (b) 43.44% (c) 53.22% (d) 60.20% (e) 64.45%.

Q4. Excel Computers Ltd, is planning to accumulate Rs.10,00,000 in a sinking fund account torepay a debenture liability at the end of five years from now. The rate of interest paid by bankson their deposits is 8% per annum. How much should the company deposit in its sinking fundaccount every year? (a) Rs.1,45,566.67 (b) Rs.1,70,444.86 (c) Rs.2,00,000.00 (d) Rs.2,30,555.67 (e) Rs.2,70,444.86.




Q5. The fixed deposit scheme of Akshara Finance Corporation offers 12% p.a. interest for athree-year deposit. If the interest is compounded semi-annually, then the maturity value of adeposit of Rs.50,000 is approximately (a) Rs.59,559 (b) Rs.70,246 (c) Rs.70,926 (d) Rs.75,246 (e) Rs.82,936

.Q6. Novelty Industries is establishing a sinking fund to redeem Rs.5 crore bond issue, whichmatures in 15 years. How much do they have to put into the fund at the end of each year toaccumulate the Rs.5 crore, assuming the funds will earn a return of 7% compounded halfyearly? (a) Rs.17.76 lakh (b) Rs.18.88 lakh (c) Rs.19.71 lakh (d) Rs.20.09 lakh (e) Rs.21.13 lakh.

Q7. Hyderabad Grameena Bank (HGB) has various deposit schemes which offer the sameeffective rate of interest of 8 percent per annum compounded quarterly. Mr.Charan plans tosave Rs.150 at the beginning of every month with HGB in a monthly recurring deposit schemewhich has a maturity period of three years. What is the maturity value at the end of three years? (a) Rs.4,556 (b) Rs.5,224 (c) Rs.6,116 (d) Rs.6,656 (e) Rs.7,005.

Q8. Mr.Kumar borrowed an amount of Rs.5,50,000 from Margadarsi Finance Ltd. As per the loan agreement, he has to repay Rs.3 lakh at the end of 7th year, Rs.4 lakh at the end of 8th year, Rs.2lakh at the end of 9th year and Rs.2 lakh at the end of 10th year from now. In order to meet these payments, he wants to deposit money in a bank scheme at the end of every year for six years that offers an interest rate of 11% p.a. Thereafter he plans to retain the money in the bank account and withdraw required amounts as and when warranted. The interest rate is expected to remain 11% from 7th to 10th year. The approximate amount that Mr. Kumar should invest at the end of every year for a period of 6years is (a) Rs.1,02,090 (b) Rs.1,10,312 (c) Rs.1,21,293 (d) Rs.1,28,905 (e) Rs.1,89,389.




Q9. Mr. Sarathi deposits Rs.500 at the beginning of every quarter in the recurring deposit scheme of Majeera Bank for six years. If the bank offers an interest rate of 12 percent per annum compounded quarterly, the amount accumulated by the end of six years is (Round off your answer to the nearest integer)(a) Rs.15,546(b) Rs.17,729(c) Rs.18,765(d) Rs.19,254(e) Rs.20,187.

Q10. Mr.Aravind wants to invest in a pension fund according to which ten years from now he will start receiving a pension of Rs.2,000 every month. The payment will continue for 10 years. How much is the pension worth today if the interest rate is 11.50% compounded quarterly? (Round off your answer to the nearest integer)(a) Rs.25,238(b) Rs.29,237(c) Rs.34,568(d) Rs.39,876(e) Rs.45,997.

Q11. Mr. Kiran borrows Rs.1,00,000 at 8% compounded annually. Equal annual payments are to be made for six years. However, at the time of the fourth payment, the individual decides to pay off the entire loan. How much amount should Mr. Kiran pay at the end of the fourth year approximately?(a) Rs.45,609(b) Rs.50,207(c) Rs.56,578(d) Rs.60,208(e) Rs.65,428.

Q12. Bhagyanagar Finance Company Ltd. is offering a pension scheme for people who have just completed 40 years. According to the scheme the individuals who subscribe will have to deposit Rs.25,000 per year for 20 years. At the end of 20 years, every subscriber will receive a specific sum plus an annuity of Rs.1,00,000 for a period of 25 years. If the depositors wish to earn 11% rate of return, the specific amount to be paid by Bhagyanagar Finance Company Ltd. at the end of 20 years apart from the annuities is(a) Rs. 1,18,007(b) Rs. 7,62,875(c) Rs. 5,31,650(d) Rs. 9,39,433




(e) Rs.12,94,695.

Q13. Mr.Pavan Kumar borrowed an amount of Rs.1,00,000 from a bank to be repaid in five equal annual installments. If the rate of interest is 14% p.a. on the reducing balances, the amount of principal amortized in the second installment is(a) Rs.12,040(b) Rs.14,122(c) Rs.17,249(d) Rs.23,131(e) Rs.29,128

Q14. Mr.Chalapati plans to buy a house for a price of Rs.25 lakh. Good People Bank offers 80% of the purchase price of the house as loan repayable in 10 years by equal quarterly installment of Rs.1,20,000. His friend agrees to provide the balance amount on condition that a lump sum amount of Rs.6 lakhs to be repaid to him after 1 year. What is the effective cost of borrowings to Mr.Chalapati under the above plan?(a) 15.35%(b) 17.42%(c) 20.24%(d) 22.43%(e) 25.00%

Q15. Mrs. Reddy is 60 years old and she is expecting that she will live for another twelve years. Her total savings are Rs.2,00,000, which she has deposited in a bank. She wants to spend her savings equally over these twelve years. If the interest earned on the deposit is 10% per annum, the withdrawal made at the end of every year from her deposit over the defined period is(a) Rs.19,458.363(b) Rs.20,119.063(c) Rs.25,258.363(d) Rs.29,352.663(e) Rs.39,352.063.

Q16. You are required to contribute Rs.2,000 per year in a pension plan for 10 years from the end of the current year. The plan will pay pension annually for a period of 20 years and the first payment will start after 16 years from now. If this plan is arranged through a savings bank that pays interest @ 7% per annum on the deposited funds, what is the size of the yearly pension?(a) Rs.2,578(b) Rs.2,945(c) Rs.3,456(d) Rs.3,570




(e) Rs.3,659.

Q17. Mr.Albert Rozario plans to purchase a flat in Goa after 2 years, whose present cost is Rs.10 lakhs. The cost of the flat is expected to increase by 15 % every year. Union Bank of India has agreed to finance upto 75% of the cost of flat whenever he buys the flat. He has recently deposited Rs.1 lakh with the bank at an interest rate of 10% p.a quarterly compounded which will become due for payment after 2 years. How much amount has Mr. Albert save every month to deposit under the cumulative deposit scheme of the bank so that the required amount of margin for the loan would be available after 2 years ? (Corrected to hundred)(a) Rs. 7,900(b) Rs. 8,200(c) Rs. 9,600(d) Rs.10,400(e) Rs.12,000

Q18. Ms.Aruna borrowed an amount of Rs.50,000 from a bank to be repaid in five equal annual installments. If the rate of interest is 12% p.a. on the reducing balances, the amount of principal amortized in the second installment is(a) Rs. 6,000 (b) Rs. 7,870(c) Rs. 8,814(d) Rs.13,870(e) Rs.20,000.

Q19. Everymans Bank pays interest at 10 percent p.a. compounded semi-annually. Janata Bank compounds interest on monthly basis. If Janata Bank wishes to pay the same effective rate of interest as that of Every mans Bank, the nominal rate of interest it should quote is(a) 9.80%(b) 10.00%(c) 10.25%(d) 10.75%(e) 11.00%

Q20. The IDBI deep discount bond offers investors Rs.2,00,000 after 25 years, for an initial investment of Rs.5,500. The implied interest rate on the bond, approximately is(a) 12.5%(b) 13.5%(c) 14.5%(d) 15.5%(e) 16.5%.




Q21. Chancellor Mills Ltd., availed a term loan of Rs.4 crore from Union Bank of India to finance their new project. As per the terms and conditions of the loan, the repayment will start after two years and the loan together with interest is to be liquidated in 10 equated half yearly installments. The loan will bear an interest rate of 13% p.a with quarterly compounding. The amount of each installment is(a) Rs.70,30,965(b) Rs.72,20,256(c) Rs.73,20,764(d) Rs.75,36,913(e) Rs.80,16,435

Q22. Mr. Srinivasan has against the pledge of jewellery borrowed Rs.10,000 from a local money lender . The said loan is to be paid back in 12 equated monthly installments of Rs.1,000 each commencing after one month. The effective interest paid by Srinivasan on the loan is(a) 15.2%(b) 20.6%(c) 36.5%(d) 41.3%(e) 52.6%.

Q23. As per the ‘rule of 69’, if the amount deposited today doubles in four years and seven months, the effective interest rate per annum is(a) 15.1 percent(b) 15.5 percent(c) 15.8 percent(d) 16.0 percent(e) 16.3 percent

Answers of the Question

1. Answer: b




Reason: In order to accumulate Rs, 20,00,000, after 15 years, Mr Sharma has to deposit ‘A’ amount of rupees every year.Therefore, 20,00,000 = A * FVIFA(8% , 15yrs)

A= 20,00,000

FVIFA (8 % ,15 yrs)=20,00,000

27.152=Rs 73659.40

2. Answer: cReason: Amount of loan = Rs 4,50,000 – 45,000 = Rs 4,05,000.

Interest rate per month = 12%/12 = 1%Let the monthly payment be A.Therefore, 4,05,000 = A * PVIFA (1%, 60)

A = 4,05,000

PVIFA (1 % ,60 )= 4,05,000

44.96=Rs 9,008

Effective annual rate = (1+ 0.1212 )

12

−1=12.68 %

3. Answer :C Reason:- Pv = Rs 1,00,000 N = 36months Payment = Rs 5000 1,00,000 = 5000 × PVIFA(r%, 36)

Solving for r ( by interpolation method) For r = 3%, RHS = Rs 1,09,161.26 For r = 4%, RHS = Rs 94,541.41 By interpolation:

R = 3 %+ 1,09,161.26−1,00,0001,09,161.26−94,541.41

=3.62 %

Effective rate of interest = ( 1 + 0.0362 )12 – 1 = 53.22%.

4. Answer BReason : FV = Annuity amount × FVIFA (r%, 8)

Annuity amount = FV

FVIFA (8 % , 5 )=10,00,000

5.867=Rs .1,70,444 .86

Therefore, the company has to deposit Rs. 1,70,444.86 in the sinking fund account.

5. Answer C

Reason: Effective interest rate = (1+ 0.122 )

2

−1=12.36 %




Maturity value of the deposit is: = A × FVIF50,000 (1.1236)3 = Rs.70,926.

6. Answer : (c)Reason: First find out the effective annual rate (EAR)EAR = re = (1 + r/m)m – 1Where,r = nominal rate of interest, i.e., 7%m = number of times compounding done (Semi Annually) = 2

re = (1+ 0.072 )

2

−1=0.071225=7.123 %

Now, let the sinking fund payment be A.Rs, 5,00,000 = A × FVIFA(7.123%,15)

A = 5,00,00,000

25.368=Rs .19,70,987.07=Rs .19,70,987 .

Therefore Rs.19.71 lakh has to be paid into the sinking fund at the end of each year.

7. Answer (c)Reason: Effective annually rate of interest :Effective rate of Int. = (1+0.08/4)4 - 1 = (1.02)4

= .0824 or 8.24%Effective monthly rate of interest R=¿ = ¿ = .6623%This is a case of annuity due because the cash flows are occurring at the beginning of every month.Hence maturity value = Rs.150×FVIFA(r%,n)×(1+r) = Rs.150×FVIFA(0.6623%,36)

(1+0.006623)= 150×40.504×1.006623 = Rs.6,115.84 = Rs.6,116(approx)

8. Answer bReason: At the end of 6years, the future value of the amount that Mr.Kumar deposits each year for a period of 6 years, should be equal to the present value of the payments that he has to make under the loan agreement in the 7th, 8th, 9th, and 10th years.The discounted value of the payments to be made at the end of 7 th, 8th, 9th & 10th years, as at the end of 6th year = 3×PVIF(11%, 1)+ 4×PVIF(11%, 2)+ 2×PVIF(11%, 3)+2×PVIF(11%, 4)= 8.729 lakh.Let ‘X’ be the amount to be invested at the end of every period till 6 yearsFuture value of the deposits made at the end of every year, for a period of 6 years = X. FVIFA(11%,6years)

X. FVIFA(11%,6years) = Rs. 8.729 lakh




X = 8.729lakh

FVIFA (11 % ,6 years )=8.729 lakh

7.913=Rs , 1,10,312

Hence Mr. Kumar has to deposit Rs.1,10,312 at the end of every period for 6 years so as to be able to make the payments made under the loan agreement.

9. Answer (b)Reason: Quarterly rate of interest is 12% ÷ 4 = 3.0 percent

Tenure of the scheme = 6 years × 4 = 24 quarters.The maturity value of this recurring deposit plan will be =

Rs.500×FVIFA(3%,24) (1+0.03) = 500 x 34.426 x 1.03 = Rs.17,729.39 (approximately).

10. Answer (e) Reason: First we need to find the annual interest rate. It is given that the interest rate is compounded quarterly, therefore the effective annual rate (EAR) of interest is found out as follows: EAR = re = (1 + r/m)m – 1 Monthly rate of return = ¿ = 0.0095 ×100 = 0.95% Total number of months = 120 The value of the pension annuity today can be found out in two steps. The present value of the annuity in the 9th year as the payment starts in the tenth year. i.e., Annuity value x PVIFA(0.95%, 120) = 2,000 x 71.53 = Rs.1,43,050 The present value of the figure obtained Rs.1,43,050 x PVIF(0.95%,120) = 1,43,050 x 0.322 = Rs.45,997

11. Answer dReason: the equated annual payment (EAP) which is computed comprises of both the interest and the principle amount. In this case, we need to first compute the EMI and then compute the break up of interest and principle amount. This will give us the amount of principle to be repaid at the end of each year. Thus while making the fourth payment, the EMI plus the principle amount remaining has to be paid.1,00,000 = EAP × PVIFA(8%,6yrs)

EAP = 1,00,000 ÷ PVIFA(8%,6yrs) = Rs.21,630.98Now let us see the break up of the EAP.Year Principle (1) Interest

component(2)= (1)×0.08

Principle component(3) = EAP – (2)

Principle remaining(4)

= (1) – (3)1 1,00,000 8,000.00 13,630.98 86,369.022 86,369.02 6,909.52 14,721.46 71,647.563 71,647.56 5,731.80 15,899.18 55,748.384 55,748.38 4,459.87 17,171.11 38,577.275 38,577.27 3,086.18 18,544.80 20,032.47*6 20,032.47 1,602.60 20028.38* -




*Actually these figures should be the same. But due to rounding off there is a minor deviation. Now the amount to be paid at the end of fourth year:= EMI + the principle amount remaining at the end of fourth year.= Rs.21,630.98 + Rs.38,577.27= Rs.60,208.25

12. Answer (b) According to the given information subscribers will deposit Rs.25,000 for 20 years and after 20 years scheme will pay Rs.1,00,000 at the end of every year for 25 years plus Rs. x at the end of 20 years from now.The discount rate is 11%.Therefore the data can be fit into an equation as25,000 × FVIFA(11%,20yrs) = X + 1,00,000 x PVIFA(11%,25yrs)

25,000 × 64.203 = X + 1,00,000 x 8.42216,05,075 = X + 8,42,200X= 16,05,075 – 8,42,000 = Rs.7,62,875

13. Answer (c) Reason: Let the equal annual installment be A.

1,00,000 = A x PVIFA(14%,5yrs)

Therefore A = 1,00,000 ÷ PVIFA(14%,5yrs) = Rs.29,131Every installment comprises an interest component and a principal component.The interest component in the first installment = 0.14 ×1,00,000 = Rs.14,000.Hence the amount of principal amortized by the first installment= 29,131 – 14,000 = Rs.15,131Interest component of the second installment = (1,00,000 - 15,131) × .14 = Rs.11,882

14. Answer (d) Cost of the house = Rs.25,00,000 and the amount of loan = Rs.25,00,000 ×.8 = Rs.20,00,000Amount borrowed from his friend = Rs.5,00,000Amount to be repaid to the friend after 1 year = Rs.6,00,000Monthly installment to bank = Rs.1,20,000Hence, 25,00,000 = 120,000 PVIFA(k%,40) + 6,00,000 PVIF(k%,4)

RHS, K at 5%, = 1,20,000×17.1591 + 6,00,000×0.8227= 20,59,092 + 4,93,620 =Rs.25,52,712K at 6% ,RHS = 1,20,000 x 15.0463 + 6,00,000×0.7921= 18,05,556 + 4,75,260 = Rs.22,80,816

By interpolation; K=5 %+(6 %−5 % ) × 25,52,712−25,00,00025,52,712−22,80,816

=5.19 %

Effective annual interest rate = (1+ 0.0519)4 – 1 = 22.43%

15. Answer (d) Equated annual withdrawal × PVIFA(10%,12yrs) = Rs.2,00,000




Equated annual withdrawal × 6.8137 = 2,00,000Equated annual withdrawal = 2,00,000 ÷ 6.8137 = Rs29,352.663

16. Answer (e) E Future value of the 10-payments annuity = 2000×FVIFA(7%,10yrs) = 2000×13.816 = Rs.27632. The amount of Rs.27632 is available immediately after the last payment. Thefuture value of this at the end of the 15th yearFV = 27632× FVIF(7%,5yrs) = 27632× 1.403 = Rs.38768The annuity amount of the retirement annuity = Rs.38768A×PVIFA(7%,20yrs) =38768 A= 38768 ÷ 10.594 =Rs.3659

17. Answer (a)

Effective interest rate = (1+ km )

m

−1

Effective interest rate is =(1+ 0.104 )

4

−1=10.38 %

Effective monthly rate = (1 + 0.1038)1/12 = 0.826%Expected cost of the flat after 2 years = 10 (1 + 0.15)2 = Rs.13,22,500Bank loan = 13,22,500×0.75 = Rs.9,91,875Margin required = 13,22,500 – 9,91,875 = Rs.3,30,625The maturity amount of two year fixed deposit = 1,00,000×FVIF(10%,2yrs)= Rs.1,21,840The amount to be received under cumulative deposit = 3,30,625 – 1,21,840 = Rs.2,08,785.The monthly amount to be saved : 2,08,785 = A×PVIFA(.826%,24) = Rs.7901.33.

18. Answer (c) Let the equal annual installment be A.50,000 = A x PVIFA(12%,5)

Therefore A =50,0003.6048

= Rs.13,870

Every installment comprises an interest component and a principal component.The interest component in the first installment = 0.12×50,000 = Rs.6,000.Hence the amount of principal amortized by the first installment= 13,870 – 6,000 = Rs.7,870Interest component of the second installment = (50,000- 7,870)×.12 = Rs.5,056The amount of principal amortized in the second installment=13,870 – 5,056 = Rs.8,814.

19. Answer (a) Effective rate of interest of Everyman Bank = 10.25%Nominal rate of interest of Janata Bank = 9.80

20. Answer (d)




The future value of a single cash flow is given as FV = PV(1+ k)n

2,00,000 = 5,500×(1+ k)25 By interpolationRHS K= 15%LHS K= 16%k = 15.45% or 15.50%.

21. Answer (b) Effective Annual rate = 13.65%The loan amount after the moratorium period= 4,00,00,000 = Rs.5,16,63,106.Effective half-yearly rate = (1 + 0.1365)1/2-1 = 6.6%5,16,63,106 = Rs.72,20,256

22 Answer (d) PV = A × PVIFA(k%,12yrs)

10,000 = 1000 × PVIFA(k%,12yrs)

Let K be At 2% ; Rs.10,575.3At 3%; Rs.9962

By interpolation; 2 %+ (3 %−2% ) × 10,575.3−1000010,575.3−9962

=2.925 %

Effective rate per annum = (1 + 0.02925) 12 -1 = 41.33%

23. Answer (e) According to the rule of 69,Doubling period (in years) = 0.35 + 4.(7÷12) = 0.35 + (69÷k)Solving the above equation we get k = 16.3%.



time value of money

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