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158 views22 pagesTime Value of Money
Time value of money Financial Management
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yanagupta0904Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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/ 22
Time Value of Money
(Mathematics of Finance)
Time value of money means that . KM
imaney scceived tedeyietiere ee ‘of money is different in different time periods. The value
Coes wor Ti oe arian the vale of same amount receivable at some other time in future.
ee cay betes ta faire is less valuable than the same amount money received
ee en individual ora hast ek English proverb, ‘A bird in hand is worth two in the bush’. The
‘an rt receives money, the better it is. For example, if Mr. X is given an opti
ofreceiving €20,000 either now or one year i Say imrermeerserers
ing year later, he would definitely opt to receive today than after one
year. This is because the value of ruy i iecanrenand
ee aero ipee received today is definitely higher than the value of rupee
4 a - 1¢ Value of money is also referred to as ‘time preference for money”.
eee for Time Preference for Money : There are several reasons for preference for current
(i Risk : There is uncertainty about the receipt of money in future. The oth
insolvent or untraceable. e a
(ii) Preference for Present Consumption : Individuals, in general, prefer present consumption to
future consumption. The present needs to buy more food, clothes or consumer durables are considered
urgent as compared to future needs. The promise of a bowl of rice next week is of no value to a starving
man.
(iii) Inflation : Under inflationary conditions, the value of money, expressed in terms of its
purchasing power for purchasing goods and services declines.
(iv) Reinvestment Opportunities : Most of the individuals as well as the firms have a preference
for present money because of availability of opportunities for reinvestment. For example, if someone
has 71,000 today, he can deposit it in his bank account and ear interest, say six percent, At the end of
the year, the amount will accumulate to €1,060. Hence, if he has a choice between 1,000 today or after
cone year, it is the same as a choice between €1,060 next year or £1,000 next year, ‘Any rational person
is expected to prefer the larger amount.
Techniques of Time Value of Money : .
“The cash flows arising at different periods of time ean be made comparable by using any of the
following two techniques : |
(@ Compounding Technique 4
(6) Discounting or Present Value Technique e
i i i ique i Future Value
ding Technique : Compounding technique is used to caloulate the
of Te, In this technique interest eamed on an initial deposit (itil principal) becomes pat
of the principal at the end of the first compounding period. Thus the principal plus iain the i of
a period becomes the principal amount for calculating the for the next period. Compounding low : Future Value (EV: Ta
ac ofa sum may be estimated by the process of compounding, Under
compounding method interest ears interest.
In case of a single flow of a lump sum amount compounded annually the future vah
computed as under :
) also known as
we (FV) may bp
PV(L+nP
Future Value
Present Value
Rate of interest
Interest Factor for n period, To simplify the calculatig
© found out the value of (1 + 1° for Natious combinations of r and n. These Pre-calculateg
ted in Table A-1 at the end of the book,
ON 1.
Mr & deposits €50,000 at 6% pa. interest for five years compounded annually. How ‘much would
he get at the end of fifth year?
= PVdsnt
= 50,000 (1 + 0,6)5
= 50,000 (1.06)>
By searching Table A-1 for a Sombination of 6% and 5 years we get the figure of 1.338,
FV = 50,000 x 1.338
66,900
Non-annual
However, compounding may
ple, Compounding has been done annually,
for example it may be made every six monthe
the basic formula will remain the same, but the value of r and n
ing.
% paa. interest compounded half-yearly. What
Wr aitled soni
08 a
Pe ae
2 (ie. Half-yearly) 04
3 Years x 2 = 6 Periods
PY(L+nP
10,000 (1 + 0,04)6
By searching table A-I fora g
FV = 10,000 x 1,265
"y
‘ombination of 4% g
= 12,650 ited £50,000 is
oe a ae: a a fixed deposit account at 12% p.a. compounded quarterly for three years.
Ec aa rats ¢ (Delhi University B.Com. 2018)
SOLUTION:
r = 2 = 0.03
4 ie. quarterly»
n =3 years x4 = 12 Periods
FV= Pv(itnP
50,000 (1 + 0.03)!
For a combination of 0.03 (i.e. 3%) and 12 Periods table A-1 reveals 1.426
FV 50,000 x 1.426 = 71,300.
ILLUSTRATION 4.
Find the compounded value of %50,000 for 15 months at 8% compounded quarterly.
08 S
4 (ée. Quarterly) 0.02:
= 15 months _ «per
2 = ymonthe 75 Periods
PV(1+n)"
50,000 (1 + 0.02)5
= 50,000 x 1.104 = 55,200
Calculation of rate of interest :
ILLUSTRATION 5.
‘A company is required to invest €4,
inflow after 4 years is €5,43,600. What is
SOLUTION :
‘The amount of 74,00,000 is cash ‘outflow which m:
ina Bank Account that pays an unknown rate of interest
after 4 years.
(00,000 in a project with a life of 4 years. The projected cash
the company’s rate of retum on this project?
ay be treated as the principal amount deposited
put returns a compounded amount of 5,43,600
FV = py(itr
%5,43,600 = %4,00,000(1 +n)"
%5,43,600/4,00,000 dent
1,359 qty ,
In the compound value table A-1, value closest to the value of 1.359 in the 4 years row is found in
8% interest rate, Thus, the actual rate of interest on the pro} ny “
Calculation of Period +
ISTRATION 6. d
‘An amount of %20,000 is deposited in C—O
avis is €43,0
24 a
compounded yearly. The investor’s goal
realise the desired amount?
SOLUTIO! ‘ ‘
We have to find out the time period over which the amount of €20,000 will accumulate to 243,099
at 10% p.2- compounded annually.
FV = pviity®
43,000 = 20,000 (1 +-10)"
243,000/20,000 awl 10)"
2.15 = (1+-10)"
In the compound value table ‘A-I, value closest to the value of 2.15 in he 10% row
years row, Hence, it will take approximately 8 years to realise 743,000.
(ji) Future Value (FV) of a Series of Equal Cash Flows or ‘Annuity of Cash Flows : In many
instances a firm may be interested in the future value of a series of equal payments ox ieceipis Hal
every year for a number of years consecutively. For example, a firm deposits 710,000 each year at the
gnd of each year for the next 5 Year secevgnown as an annuity of deposit of €10,000 tesa
ffequal cash flows made at regular intervals either incoming or ene
to meet a future obligation, itis using annuity
ide a fixed sum each year
atthe end of each period the annuity is called regular annuity ot 2
eour atthe beginning of each period the annuity is called an annuity
an annuity is a stream ©:
when a company sets asi
When the cash flow occur
deferred annuity. If ‘the cash flow
due.
FV ofanannuity may be ascertained with the help of Table A-?2 (given at the end of the book). For
example, we want to find out FV of an annuity of €1,000 for 3 years at 8%. The relevant figure in Table
2 is 3.246, By multiplying this figure by 71,000 we get the FV of the annuity of £1,000 which is
3,246, The figure of 3.246 is also known as Compound Value of Annuity Factor (CVAF). Thus
‘Annuity Amount x CVAF (f, 2)
rate of interest
time period
n
4 p.a. interest compounded
WL TON 7.
'A 5 year annuity of %6,000 per year is deposited in a bank that pays 8
yearly. Find out the total amount available to the depositor at the end.
The FV of it
2 ieee may be calculated by using the following equation :
= Annuity Amount x CVAF (t, n)
Where, r x
i 8%
5
5.867 (As per Tab
le A-2
26,000 x 5.867 4
335,202
Hence, FY
A machine
for replaciy Costs %12,00,00 2
Placing the machine at Herts eftecttys life is estimated at 8
its effective life when its a Years ASighipeie alse
ip value is estimated at 40,000.
ie.
and CVAF (8%, 5)
)
. Calculate the amount which should
pa. compounded annually,
Effective cost of the machine = %12,00,000 - €40,000 = €11,60,000,
FV = Annuity Amount x CVAF (6%, 8 y)
€11,60,000 Annuity Amount x 9.897
Annuity Amount 11,60,000 + 9.897
bo Sty 1.207,
ILLUSTRATION 9,
A company has issued debentures of %60 lacs to be repaid after 7 years. How much should the
company invest in a sinking fund earning 12% pa. in order to be able to repay the debentures,
FV = Annuity Amount x CVAF (12%, 7 y)
%60,00,000 = Annuity Amount x 10.089
Annuity Amount = %60,00,000+ 10.089
= %5,94,707.
10.
A company offers to refund an amount of 1,88,075 at the end of 6 years for a deposit of 25,000
‘made annually. Find out the implicit rate of interest offered by the company.
SOLUTION:
Refund amount of 71,88,075 is the future value (FY) of annuity of 225,000 after 6 years at a
particular rate of interest. It can be presented as follows :
FV = Annuity Amount x CVAF (r, n)
31,88,075 25,000 x CVAF (r, 6 y)
CVAF (r, 6 y) 1,88,075 + 25,000
Find R23:
Now in Table A-2, the value of 7.523 corresponding to 6 years row is in 9% column. Hence, the
implicit rate of interest is 9%,
FY of an Annuity Due:
When the cash flows occur at the beginning of each period, the value of such annuity is called
annuity due and its future value (FV) can be calculated as under :
FV Adve = Annuity Amount x CVAF¢,n) x(1 +i)
ns a compound interest of
aad
jawoly es $20,000 x 5.867 x 1.08
%1,26,727
b) Discounting or Present Value Technique : 0 pean,
After ascertaining the FV of present money, the process of ascertaining the Me eiuean a
de discussed. This process is exactly opposite of compounding technique es
discounting technique. The discounting technique to find out the PV can be discu:
(i) Present Value ofa single future cash flow. 3
i Present Value a ofa series of equal future cash flows ee annuity.
(iii) Present Value (PV) ofa series of unequal future cash flows.
(iv) Present Value (PV) of a perpetuity.
(i) Present Value (PV) of a single future cash flow : For
future flow of money, we can use Table A-3 wherein present v
denoted as PVF(r, n).
determining the present value of a single
alues of a future sum is given and is
ILLUSTRATION 12. col
Mr. Arun is likely to receive €40,000 after 3 years. Ifthe rate of interest is 10% p.a. compounded
annually, what is its present value?
SOLUTION :
PV = FVXPVF(.n)
= %40,000x PVE\iox,3) |
= %40,000x.751
330,040.
ILLUSTRATION 13.
Assume that deposit sto be made at year zero into an account earning 8% compounded annually.
Itis desired to withdraw %2,00,000 three years from now and %3,00,000 seven years from now. What is
the size of the year zero deposit that will produce these future payments?
SOLUTION:
We have to calculate the PV of the two later withdrawals on the basis of Table A-3.
PV = FVXPVE (9)
%2,00,000 x PVF(8%, 3) + €3,00,000 PVF (8%, 7)
%2,00,000 x.794 + %3,00,000 x .583
%1,58,800 + &1,74,900
= %3,33,700,
Proof : The amount of %3,33,700 will compound to a value of €4,20,462 in three years (Using
Table ay aii, will be withdrawn then, leaving %2,20,462 which will compound to
approximately %3,00,000 in another four years, Her i i
Ea a nec y nee, an amount of €3,33,700 deposited today will
(ii) Present Value (PV) of a series of e
PY of a series of equal flows, the PVs o
calculated and then added, PVs of
Table (as per Table A-4 given at
qual future cash flows or annuity : In order to find out the
f different amounts accruing at different ti
equal flows may be calculated on the basis of, t
the end of the book), in a post office saving bank at an interest rate of
ie
Ae Xs depositing €25,000 annually for 5 year,
Find the present value of annuity.
‘Annuity Amount x PVAF(, 0)
‘Annuity Amount x PVAF(9%4,5)
25,000 x 3.89
297,250.
wna
present value of the annuity
Mr ¥ has to receive 50,000 per year for 8 years. Calculate the
1m interest on his investment at 10% pa.
assuming that he can ea
SOLUTION:
pv =. Annuity Amount x PVAF(,n)
‘Annuity Amount x PVAF(10%,8)
= ¥50,000 x 5.335 = %2,66,750
ILLUSTRATION 16.
‘A student is awarded a scholarship and two options are put before him : () to receive £11,000 now,
or (i) to receive €1,000 per month at the end of each of next 12 months. Which option should be chosen
if the rate of interest is 12% pa.
SOLUTIO!
Option 1 : Since 211,000 is receivable now it is already expressed in present value and hence needs
no adjustment.
Option 2 : Since the rate of interest is 12% p.a. and the annuity consists of 12 periods, the rate of
interest per period may be expressed as 12%, On the basis of Table A~4, the present value may be
calculated as follows :
PV = Annuity Amount x PVAF(,n)
= Annuity Amount x PVAF(1%,12)
= %1,000x 11.255= 211,255
cent value in option 1, the student should
Since, the present value in option 2 is higher than the prest
choose option 2.
ILLUSTRATION 17.
(Present value of an Annuity Due or Annuity du
Mr. Zhas to receive €20,000 at the beginning o
of the annuity due assuming 10% p.a- interest.
SOLUTION ;
¢ at the beginning of each year)?
BA the present value
a
f each year for ues Calet
PV=
u
83,402. we Calculation of Annual Payments :
ILLUSTRATIO.
An investor deposits a sum of %6,00,000 in a bank account on which interest is aoe orate
How much amount can be withdrawn annually for a period of 12 years? OSes tag
SOLUTION:
S 9% p.a. It can
Deposit of %6,00,000 may be viewed as the PV of the future annuity of 12 years erp
be presented as follows :
PV = Annuity Amount x PVAF(,n)
%5,00,000 = Annuity Amount x 7.161
Annuity Amount = %6,00,000+ 7.161
= %83,787
Hence, the investor can withdraw an annuity of %69,823 for 12 years.
: y is to be repaid in five annual
ducational loan of %5,00,000 at 10% p.a. The loan is in fi
“ot eet cating from the end of the first year. Calculate the amount of equal annual installment.
TT a (Delhi University B.Com. 2018)
PV = Annuity Amount x PVAF (, n)
5,00,000 = Annuity Amount x PVAF (10%, 5 years)
Annuity Amount or Instalment Per year = 5,00,000 = PVAF (10%, 5 years)
_ 5,00,000
(By Searching Table A~4) aay ee Sa
Ten years from now, Mr. Naveen will start receiving a pension of €20,000 a year. The payment will
continue for sixteen years. How much is the pension worth now, if his interest rate is 12% p.a.
SOLUTION:
Step 1. Determine the PV of annuities by using Table A-4 :
PV = Annuity Amount x PVAF(, 9)
Annuity Amount x PVAF (1394 16)
= %20,000 x 6.974 = 21,39,480
- apne areas sis end of 10th year. Hence present yalue of €1,39,480 will be
PV= FVXPVF.n
oan x PVF 12%,9)
%1,39,480 x 361 = $50,352
LLUSTRATION 21,
A potential investor is interested in the
Purchase of a bond that has the followi
wing characteristics : ~~ The bond pays 9% per year on its fa
smarty the bondholder will receive inte eo
ud be paid for this bond ifthe invest
.000. The bond will mature in 15 years. At
est for 15 years plus €1,000 face value. What maximum price
for requires 12% rate of return?
SOLUTION:
‘Yearly interest receivable from the bond = 9% of £1,000 = %90, Maximum. purchase price for this
pond will be the PV of the future inflows discounted % requi Sais
receivable will be treated as annuity amount, eee 2
PV = Annuity Amount x PVAF,, n)
= 90XPVAF (2,15)
= %90x 6.811 = %612,99
Now, the present value of 21,000 (single payment) recei All be
ascertained as follows : (Use of Table A-3) : Payment) receivable at the end of 15 years will be
PV = FVXPVF@.n)
= FVXPVF(2%,15)
= %1,000x.183 = 7183
Thus, the maximum price will be = %612.99 + 183 = 8795.99,
JLLUSTRATION 22.
A firm purchases a plant for 40,00,000, It makes a down payment of 20% and borrows the
remainder at 12% interest rate, The loan is to be repaid in 7 equal annual instalments beginning from Sth
year. Calculate the amount of annual loan repayments.
SOLUTION :
Loan Amount = 80% of %40,00,000 = %32,00,000. Compound interest will be calculated over the
period of 11 years (i.e. 4 years + 7 years).
Step 1. Calculate the total amount owed at the end of year 4 by compounding %32,00,000 for 4
years at 12% p.a.
FV= PV(i+1"
FV = %32,00,000 (1 +0.12)*
(By Searching Table A-1)
FV = %32,00,000 x 1.574
= %50,36,800.
Step 2. Now, the FV becomes the PV of 7 payments at 12% p.a. Hence, yearly payment will be
calculated by using annuity table A-4 :
PV = Annuity Amount x PVAF(,n)
= Annuity Amount x PVAF(12%,7)
%50,36,800 = Annuity Amount x 4.564
Annuity Amount= %50,36,800 + 4.564
= %11,03,593
(iii) Present Value (PV) of a series of unequal future cash flows : When the amounts receivable
in future are unequal, we can also calculate the present value of a series 0 n
present values of each individual payment as per Table A-3 and th
~ been explained in the following illustration mg Ncw
ILLUSTRATION 23. Hi anew
A firm can invest £60,000 in a project with a life of 5 years. The projected cash are a
follows :
sh Inflows (@)
eae oe 10,000
; 15,000
é 18,000
; 25,000
d 20,000
; 2
The cost of capital is 10%. Should the investment be made?
Since initial investment is %60,000 and present value of cash flows is 64,493, investment in
project can be made. ‘
This technique has been further discussed in Chapter relating to ‘Capital Budgeting’ in detail.
(#) Present Value (PV) of a Perpetuity : A perpetuity means an infinite series of equal cash flows
ccurring at regular time periods. For example, if a deposit of 71,000 is kept in a saving bank account
yielding 3.5% pa. interest for an indefinite period, then the annual interest of €35 will be called a
Perpetulty of interest. Present value of a perpetuity may be calculated with the help of the following
equation :
PV, = Annual Cash Flow/r
Here, PV, refers to present value of a perpetuity and r refers to rate of interest,
ILLUSTRATION 34, ’
Find out the present value of an investm ee mA, 08.060
indefinitely and he rate of interest is 13% pa, iS" 8 emPected to give a retum of %6,500 pa.
= Annual Cash Flow/r
= %6,500/.13
= 250,000 MONEY (MATHEMATICS OF FINANCE)
ILLUSTRA TION 25,
Mr. Vhad taken a freehold land for an annual rent of €6,000, Find out the present value of freehold
njoyable in perpetuity if the rate of interest i¢ 894 pa
= Annual Cash Flow/r
%6,000/.08
= 75,000
WLUSTRATION 36,
A finance company makes an offer to deposit a sum of %35,000 and then receive a return of 3,960
pa. perpetually. Should the fits be accepted ifthe rate ofinterext is 1286 p.a.? Will the decision change
if the rate of interest is 11% p.a.? (D.U. 2016, Non-Collegiate)
Of ara uasccebicd oaly IHthPViof tae pereruiy is Meet tie tad deposit of 235,000.
In case the rate of interest ig 12% :
PV, = Annual Cash Flow/r
= %3,960/.12 = 233,000
Since the PV of the perpetuity i. €33,000 is less than the initial deposit of %35,000, the offer need
not be accepted.
In case the rate of interest is 11% :
PV, = Annual Cash Flow/r
3,960/.11
= 736,000
Since the PV of the perpetuity i. €36,000 is more than te initial deposit of £35,000, the offer may
be accepted.
ILLUSTRATION 27,
Find out the present value ofa perpetuity of €1,000 starting in 3 year at an interest rate of 18% pa
SOLUTION:
Annual Cash Flow/t
21,000/.18
= %5,556
Since annual cash flow is to commence from 3rd year, the present value of 85,556 at the beginning
of first year must be calculated by using Table A-3 : os
PV = FVxPVF(r,n) ee
25,556 x.718 we
33,989, 1
ILLUSTRATION 28.
‘nat Company is selling a debenture which will provide
'ndefinite number of years. Should the debenture be purchased Ney
ted rate
710,500 and the required rate of return is 12 percent? What will be your answer if the requi
Teturn is 10 percent?
SOLUTION:
py, = Annual pee
Where PV, = Present value of a perpetuity
T = rate of interest
Ifris 12%
1,200
yy, = ~1200 _ 210,000
PV, 12 710,
Ifris 10%
PV, = uae 12,000
At present the bond is quoted at €10,500.
Hence, ‘
( ifris 12%, debenture should not be purchased as the current market price of €10,500
is more than current worth of 710,000. F ’
(@ ifr is 10%, debenture should be purchased as the current market price of €10,500 is
below its current worth of 12,000.
OBJECTIVE TYPE QUESTIONS
State whether each of the following statements is True or False :
1. Money has no time value.
2. One of the reasons for providing time value to money is that individuals prefer future
consumption to current consumption.
3. Understanding the time value of money essentially involves the understanding of the concept of
compounding and discounting.
|. Interest factor helps in incorporating time value to money.
. The discounting technique helps in finding out present value of a future sum.
Cash flows occurring at different points of time can be compared in absolute terms,
Rate of interest and time Period, both are needed to ascertain present or future value of money.
. is techniques of compounding and discounting are identical,
: i a pee Pointing ial ae he future value of present money whereas discounting deals with
ne eee ee lave 48 used to find out the future value of present money.
T. Anannuite ene’ of unequal payments,
annuity is an infinite series of cash flows,
12. The number of cash flows in a perpetuity ig
Ans. True :3, 4, 5,7,9, 19 eee
’ False 1, 2,6, 8, 10, 11
SI AAe
1. “A Rupee of today is not qual to the a rr i 018)
to the Pee of tomorrow,” p, ‘om, 21
-” Explain, (D.U. B.Com, 2018) for money.” What are the reasons for such a
»
“Cash flows of different y
they be made comparable,
4, Explain the mechanism of calc
5, Howis future value of series of
6, Explain the discounting techni
7. What is the formula for calculati
ars in absoh
lute terms are uncomparable.”? Give reasons and how can
(D.U. B.Com, 2017 External)
‘ulating the future value of present cash flows,
2
z
2
z
is
a
2
E
2
S
&
e
g
3
2
s
&
9, Write short notes on :
(A) Present value of a perpetuity
(ii) Annuity of cash flows
(ii) Present value of a future sum
10. Why is consideration of time important in financial decision making? How can time be adjusted?
(D.U. B.Com.2018)
(OU. B.Com. 2018)
PRACTICAL QUESTIONS
(Q 1. to Q. 28 are strictly in the serial order of Illustrations)
Q. 1. (@) Mr. A deposits 26,000 for 3 years at 9% p.a. compounded annually. How much would he
get at the end of 3 years?
[Ans. 27,770]
(6) Mr. B deposits 8,000 for 6 years at 8.5% p.a. compounded annually. How much would he get
at the end of 6 years?
[Ans. 713,056]
OV
Hint : Compounded value as per table A-1 for 8% is 1.587 and for 9% is 1.677. Hence for 8.5% will be >
(1.587 + 1.677) = 1.632.
Q. 2. An investor deposits 25,000 for a period of 4 years at 10% p.a. interest compounded
half-yearly. How much will he get at the end of 4 years?
[Ans, 36,925]
Q.3. An investor deposits 10,000 for a period of 3 years at 6%p.a. interest compounded quarterly.
What will be the total amount after three years?
[Ans. $11,975]
Q. 4. Find the compo
[Ans. 714,274]
i Fi initi 0,000. The
is offered a contract on which it has to incur an initial outlay of re a
Q. 5. A company is ofr flow of €7,36,800 after 3 years. What is the company’s rate of return
contract would provide a cash inflow . ge
on this contract? i
[Ans. 7% Approx.]
unded value of €12,000 for 21 months at 10% compounded quarterly. (MATHEMATICS op,
Q 6. An amount of %25,000 is deposited ae ee yielding 12% pa,
Compounded annually. The investor’s goal is oe ed?
compounded interest before the desired amount i
[Ans. 9 years approx.]
ars must the Praga
; ith 6p. ;
Q.7.Mr. X is depositing £4,000 in a recurit pak soe which pays 9% p.a. compound inten
How much amount Mr. X will get at the end of 10 ¥
[Ans. 223,940]
it ive life is estimated at 12 years. If the scrap
i SI d its effective life is est
8. es See eae of profit at the ea of oe year to accumulate at compot
000, what shou a
ee Daa $0 that a new machine can be purchased after 12 y
[Ans. 730,156]
0,00,000 bond issue which matures in 19
i ishing a sinking fund to redeem %40,00,
Q.9.Z i pesos fa ea into the sinking fund at the ead of each year to be able to
Ee fend sssuming the sinking fund is compounded at 8% annually?
redee!
[Ans. %2,10,781]
10. a) Aninvestment company offers to pay €3,04,825 at the end of 15 years to investors who
fe annuzlly €5,000, What interest rate is implicit in the offer?
[Ans. 18%]
value jg
und rate
() A company offers to pay you %2,03,040 at the end of 10 years if you deposit $10,000 annually.
‘What interest rate do you eam on the deposit?
[Ans. 15%]
i it i d
Q. LL. An investor deposits %8,000 at the beginning of each year in a bank earning compoun
interest of 9% p.a, Find out how much amount he will get at the end of 7 years?
[Ans. €80,224]
12, Find out the present value of %2,00,000 to be required after 4 years, if the rate of interes is
6%.
[Ans. 1,58,400]
{0.13.4 certain sum is to be deposited in a bank account earning 9% compounded annually. It is
desired to withdraw %50,000 five years from noy
should be deposited today
w and %80,000 eight years from now. What amount
yy that will result in the d
[Ans. 272,660]
lesired withdrawals?
6 yee ME P's depositing €5,000 per Year in a bank. Calculate the present value of the annuity for
Years assuming that he ears interest @ 12% p.a,
[Ans %20,555]
Q. 15. Mr. Q has to recel E
assuming than he can earn ee per year for 5 years. Calculate the Present value of the annuity
ret 299,895] “ton his investment at 8% pa. eS
. Mr, X woe
month atthe end of ene ne tions +
Pa,
h of next 6 @) receive
1 Yawanao.é,
months,
(©) receive 50,000 per
the rate ees 12%
tS atl
r £2,90,000 at present or
Which option should be chosen, ift
tt
Se Q. 17. Mr. R has to receive 710,000 at the beginning of each year f PCACuIbA thes preset
lue of the annuity due assuming 9% p.a. interest. iB year for 8 years. Calculat pres
[Ans. 60,332]
Q. 18. (a) Mr. Vivek deposits €4,00,000 on his retirement in a bank which pays 10% p.a. interest,
fow much he can withdraw annually for a period of 10 years?
[Ans. 65,094]
(0) Determine the amount of equal payment to be made for a loan of €10,00,000 taken for a period
of 15 years @ 12% p.a. interest,
[Ans. 71,46,821]
Q. 19. Determine the amount of equal payment to be made for a loan of €20,00,000 taken for a
period of 20 years @ 10% p.a. interest.
[Ans. &2,34,907]
Q.20. A 12 payment annuity of €20,000 will begin 8 years hence (the first payment occurs at the
end of 8 years). What is the present value of this annuity if the interest rate is 14% p.a?
[Ans. £45,280]
Q.21. Mr. Xis interested in buying a bond of the face value of €500 yielding 8% per year interest
‘on the face value. The bond will mature after 12 years. At maturity, the bondholder will receive interest
for 12 years plus the face value of the bond i.e. €500, What is the maximum price that should be paid
for this bond if the investor requires a 10% rate of return?
[Ans, 432]
Q.22. A firm borrows %60,00,000 at 9% p.a. The loan is to be repaid in 10 equal yearly instalments
beginning with 4th year. Calculate the amount of the required loan repayments.
[Ans. %12,10,658]
Q. 23. A firm can invest €10,000 in a project with a life of 4 years. The projected cash inflows are
as follows : r
Year Cash Inflows (2)
1 4,000
2 5,000
3 3,000
4 2,000
The cost of capital is 12%. Should the investment be made?
[Ans. Total present value of cash flows 10,965. Investment can be made.]
Q.24. Asceriain the present value of a perpetuity which is expected to yi
and the rate of interest is 12% p.a. . ia
[Ans. %50,000]
a,
Q. 25, Find out the present value of freehold f
,600 and rate of interest is 9% p.a.
40,000] 2.16
Q. 26. An investment company makes an offer to deposit a sum of £50,000 and then receive a
of £4,200 p.a. perpetually. Should the offer be gecepted if the rate of interest is 8% p.a? Tetum
[Ans. Present value of perpetuity £52,500. The offer should be accepted.]
ty of £600 starting in4 years at an interest rato 12%
Q.27. Find out the present value ofa perpetuit
p.a.
[Ans. 73,560]
is selli ich will provi
Q.28. A company is selling a debenture whic annual int :
indefinite umber of 20S Should the debenture be purchased if it is being quoted in the market for
10,000 and the rate of interes ‘What will be your answer In the above case if the rate of
interest is 8 per cent? F (D.U. 2014, External)
‘Ans. (i) At 12% interest present value js £8,333. As the present value is Jess than market price of
710,000 debenture should not be purchased. :
(i) At8% interest present value is 212,500. AS the present value is more than the market price,
it should be purchased.
ide annual interest payment of 1,000 for
t is 12 per cent
ADDITIONAL QUESTIONS
Q. 29. Mr. X invests £10,000 at ihe end of each year at 10% rate of interest per year. State what
amount he will receive at the end of 4 years:
[Ans. €46,410]
Q.30. A company has
the company invest ina sinking fund earning 7%
[Ans. €7,95,893]
Q.31. Ascertain the present value of €5,00,000 receivabl
12%.
[Ans. &2,02,000]
Q.32, Find out the present value of an amount of 880,000 deposi in Axi iod
Re 000 deposited now in Axis Bank for a per
[Ans, 750,400]
Q. 33. An investor has to receive €50,001 5
2 é ,000 per year for 5 years. Calculate th t value of the
annuity assuming that he can eam interest on his inv Sere ae
TAns. €1,89,550] is investment at 10% p.a.
issued debentures of £200 lacs to be repaid after 15 years. How much should
in order to be able to repay the debentures?
Je after 8 years, if the rate of interest is
Q. 34, Find out th E
pa ut the present value ofan ordinary annuity of €12,000 per annum for 10 years at 1226
[Ans. 67,800]
Q. 35. What will b
Li%par 3¢ the present value of €10,000 drawn at the beginning of each year for 8 years @
[Ans. 757,121] i :
Q. 36. What is the
Present worth i
effective annual rate of i of operating expendi
hisetabeph spenditure of i
ae rest is 12% p.a? %20,000 per year for 8 years if the
tae! VALUE OF MONEY (MATHEMATICS OF FINANCE) 27
___DELHI UNIVERSITY EXAMINATION QUESTIONS
Q. 1. Vijay borrows from Kings Bank an amount of %10,00,000 @12% p.a. on 1-4-2012. As per
agreement, repayment including interest is to be made in five equal annual instalments with first
jnstalment falling due after three years, i.e., on 31-3-2015. What would be the amount of each
instalment? (B.Com. D.U. 2012, Regular)
FV = PV(i+rP
10,00,000 (1 + 0.12)?
(By Searching Table A-1)
%10,00,000 x 1.254
%12,54,400
Now, the FV (as calculated above) becomes the PV of the 5-payments. Hence, yearly payment will
be calculated by using Table A-4.
PV = Instalment amount x PVAF(,9)
12,54,400 = Instalment amount x PVAF (12%, 5)
12,54,400 = Instalment amount x 3.605
_ 12,54,400
Ce a TCO)
= 347,961
(Refer Ilustration 22)
Q.2. Mr. Ram wona prize in a game show. He is provided with the following three alternatives of
receiving his prize :
I: %1,10,000 right now.
II : Amannuity of 715,000 at the end of each year for 10 years.
I : A perpetuity of 210,000 for ever.
Which of the alternatives is most suitable for Mr. X if he has opportunity to earn 10% p.a.
(B. Com. D.U. 2012, External)
Option I
Itis expressed in terms of its present value only ~. no adjustment is required = %1,10,000
F iW. 7
agar ey
on oh