Application Of mathematics in Finance- Time Value of Money

12.1 INTRODUCTION
The entire operation of our industrial civilization is based on the concept that money
belonging to one individualmay be used by others in return for periodic payments. Thus
money can be rented in the same way as real estate, machine to ols, vehicles or other
equipment. The only difference is that the rent or interest paid for the use of money is itself
money, whereas, for example, the rent for a fleet of trucks is not usually additional trucks.
Interest plays so important a part in business that many individuals and agencies are engaged
directly in the business of lending money, while others do so indirectly. In this chapter we
discuss selected topics in fnance that deal with the time value of money, such as investments,
loans and so on.

12.2 INTEREST
Certain terms are used so often in business transactions that it is better to define the
explicitly.
Capital is a stock of accumulated wealth-money or its equivalent.
Consideration is a fee.
Interest is consideration for the use of invested or loaned capital.
The principal is the capital originally invested.
The time is the number of years or fraction of ayear for which the principal is borrowed or
loaned.
The amount is the sum ofthe principal and the interest.
The rate ofinterest is the amount charged for the use of the principal for a given length of
time, usually on ayearly or per annum basis.
Rates ofinterest are usualy expressed as percentages :5% per annum; 10% per annum and so
on. However, when using rates ofinterest in calculations, we use the decimal equivalent :0.05
for 5%; 0.10 for 10% and so on.
There are two kinds of interest. If interest is computed on the original principal, it is called
simple interest. Thus,in transactions involving simple interest, the principal on which interest
is computed remains unchanged throughout the term ofthe loan and the interest becomes due
either at the end of the loan or at the end of stated intervals during the term of the loan,
However, when the principal is increased by interest at the end of each petiod and interest is
thuscomputed on a principal which grows periodically, we are dealing with compound
interest.
12.3 SIMPLE INTEREST
The simple interest on agiven principal at agiven rate for agiven time is the product ofthree
factors:the principal, the rate ofinterest (per annum) and the time (in years). Thus the formula
for simple interest expressed in symbols is :
where
I=Prt
I=Interest in rupees (or other units of money)
P= Principal in rupees (or other units ofmoney)
r= Rate ofinterest(in decimal form)
t =Time (in years)
If S is used to symbolize the amount, then the definition of amount leads to the formula:
S=P+I=P+Prt =
P(1+ rt)
These formulas can be used to caiculate the interest on a stated principal for agiven time at
agiven rate, and to find the amount.

Mathematics ofFinance
Eample
1.Findthe simpleinterest on S00 for
and
Now,
Example
6months?
Example2.Findthe time required for 2500 to yield 300in simple interest at 8%.
Solution.Fromthe formulaI= Prt, we obtain
I= (500)(0.04)(3) = 60
t=
S = P+I= 500+ 60 =560
I =Prt
I=300, P= 2500and r = 0.08.
300
2500 (0.08)
Pr
300
2001.5 (years).
3. At what interest rate will 3000 yield 120 in simple interest in
Solution. From the formula / =Prt, we obtain
SubstitutingI=120, P= 3000 and t =
P=
Pt
P=
1
2
120
(3000)
we have
Example 4. What principal will amount to 645 in1
= 0,08 or 8%
Solution. From the formula S' =
P(1 +r), we have
S
1+rt
Substituting S
= 645, r=0.05 and t= 1.5, we obtain
645
1+(0.05) (1.5)
EXERCISE 12.1
years at 5% simple interest ?
600
lFind the simple interest and amount of aprincipal of ? 600 for 2years at 5%.
12.3
A
person invests 1000 in asavings bank paying 6% simple interest. What is the balance or
amount ofthe savings account after 6 months ?
ndthe amount at 6% simple interest of 1200 due in 9 months.
r3years at 4%, andfindthe amount.
Solution, Here
eP= 500, r= 0.04 and t=3. Since

12.4 COMPOUND INTEREST
In transactions involving compound interest, the interest earned by an invested amount oe
money (or principal) isreinvested so that it too earns interest. That is, interest is converted (or
compounded) into principal and hence there is interest on interest".
Interest may be converted into principal annually, semiannually, quarterly, monthly or at any
otherregular periods of time. The frequency of conversion is anumber indicatinghow many
times interest is compounded in one year. The time between two successive conversions of
interest is called the conversion period. Thus, if interest is compounded quarterly, the
frequency ofconversion is 4and the conversion period is 3months. The total amount due at
the end of the last period is called the compound amount.The difference between the
compound amount and the original principal is called the compound interest.
Remark.It may be remarked that regardless of the frequency of conversion, the rate of
interest is usually expressed as an annual rate. When the conversion period is other than a
year, the rate per conversion period is found by dividing the stated annual rate by the number
ofconversion periods in ayear. Thus, ifthe quoted rate is 8?% compounded quarterly, the rate
per conversion period is 2% or 0.02. Further, when the frequency ofconversion is not stated,
interest will be understood to be converted annually. Thus, the expression "interest at 4%" or
"money worth 4%" will mean 4% converted annually.
12.5 FORMULA FORCOMPOUND AMOUNT
In this section we shall develop aformula for computing the amount, when interest is
compounded. Let
P= the originalprincipal
n =
the number of conversion periods
i = the interest rate (in decimal form) per conversion period
S =the amount at the end ofn periods

Then
the amount at the end offirst conversion period =P+ Pi = P(|+i)
the amount at the end of second conversion period
= P(1+i)+P(l+ i)i= P(l+i)(1+i) = P(l +i)
the amount at the end ofthird conversion period
the amount at the end of the nth conversion period = P(1+ i)"
once the compound amountS of aprincipal Pat the end ofnconversion periods at the
interest rate of iper conversion period is given by
= P(1+i +P(l+ i i= P(1 + i
i=
S= P(1+i)"
This formula is usually referTed to as the compound interest formula. When using the
compound interest fomula, remember that
number of conversion periods per year
annual rate of interest
Eor example, if the annual rate of interest is 8% and the compounding is semiannually, then
there are 2converSion periods per year and
i= 0.08/2 = 0.04
S=P1+
If the values of iand n are given, the value of (1+i may be found directly from Table I.
The table tells us what one rupee will amount to for various values of iand n.
S= P
..(1)
Note 1. It may be noted that if a principal of P is invested for 't years at an annual interest
rate of r and interest is compounded mtimes a year, then the interest rate per period is rlm
and there are mt conversion periods throughout the term of the investment. Hence fusing
Formula (1)), the compound amnount Sat the end of tyears becomes
m/
mt
That is, the compound amount Sat the end of yearsat an annual interest ofr
Compounded mtimesa year isgiven by
...2)

12.6
using the following formula:
Note2. Once compound amount is obtained, compoundinterest may then be obtainedby
() Annually
CompoundInterest=S-P
Example S. Findthe compound amount of 2000for 4years at 6%converted:
(ii) Quarterly
(i) Semiannually
(iv) Monthly
Solution. (i) When the interest is convertedannualy, we have
P= 2000, i =0.06 and n =4
S = P(l+ i)" = 2000(1.06)*
= 2000(1.262476)
=2524.95
(ii) When the interest is convertedsemiannually, we have
i= 0.06/2 = 0.03 and n = 4(2)= 8
S =P(l +i)" =2000(1.03)8
= 2000 (1.266770)
= 2533.54
Mathematicsjor
(ii)When the interest is converted quarterly, we have
S =P(l +i)" =2000 (1.015)°
= 2000 (1.268985)
=2537.97
i= 0.06/4 = 0.015 and n =4(4) =16
(iv) When the interest is converted monthly, we have
S=P(l+ iy =2000 (1.005)
= 2000 (1.270489)
= 2540.97
BSS DUuales
(using Table I)
(using Table I)
i= 0.06/12 = 0.00S and n = 4(12) = 48
(using Table )
(using Table I)
6. Find the compound amouht and the compound interest of?700 invested for D
years at 8% compounded semiannually.
Solution. Here P= 700. With interest compounded semiannually, we have n= 15 (2) =30
and the periodic rate iis 0.08/2 =0.04. Substituting these values in Formula (1), we obtain

12.7 COMPOUND AMOUNT AT CHANGING RATES
(husfar we have assumed a constant rate of interest for the entire duration of an investment.
However, interest rates may change from time to time. Thus, a bank, which pay

Mathematics of Finance
g8% when adeposit is made, may, after a number of vears, raise the rate to 9% and later on
nerhaps reduce it to 7%.The final compound amount is the product of the original principal
12.11
andtwoor more factors of the form (1+i or e,each with its proper value for i and n or
rand t.This is best illustrated with the help of followingexamples.
Example 15. Aman made a deposit of 2500 in a savings account. The deposit was left to
accumulate at 6% compounded quarterlyfor the first 5 years and at 8% compounded
semiannually for the next 8 years. Find the compound amount at the end of 13 years.
|Delhi Univ. B.Com. (H) 2014
Solution. For the first 5 years, i= 0.06/4 = 0.015, n = 5(4) =
20. For the next 8 years,
i= 0.08/2 = 0.04, n =8(2) = 16.Hence the amount atthe end of 13 years is:
S= 2500 (L.015)20 (1.04)16
= 2500 (1.34685) (1.87298)
=6306.55 (app.)

NOMINAL 'ANDEFFECTIVE RATES OF INTEREST Tuey)
In
transactions involving compound interest, the stated annual rate of interest in called the
nominal rate of interest. Thus if an investment is made at 6% converted semiannually, the
nominal rate of interest on this investment is 6%. It may be noted that the actual interest
earnedon the given investment will be more than 6% per year. For example, ? 100 invested
at 6% converted semiannually amounts in one year to 100 (1.03)2 = 106.09. Thus the
orest actually earned on this principal of ?100is 6.09 which represents an annual return
of 6.09%. We say that the effective rate in this case is 6.09%. When the conversion period 1S
ayear, the effective rate is the same as the stated annual rate.
dRelationship Betweenthe Effective Rate and the NominalRate.Let r, denote the effective
rate correspondingto the nominal rater,converted mtimes a year. We shall use i exclusively
for rate per conversion period. Thus i=rm. At the rate i, the principal P amounts in one
vear to P(1+ i)". Since an efféctive rate is the actual rate compounded annually, therefore at
the effective rate r, the principalP amounts in one year to P(1+r,). Thus
5 P(l+r)= P(1+i)" Or
year
1+r, = (1+ i)"
).
Thus the effective rate, r, , equivalent to the nominal rate r converted m times a
is given by =(1+i)"-1=1 +
m
-1
r, = (1+ i)" -1
ana
Exantu
...(1)

torce o lnterest. The nomninal rate r compounded continuously and equivalent
ie.
effective rate r. is called the force of interest.
i.e.,
Example 17. What effective rate is equivalent to a nominal rate of 8% converted
nominal rate of39%
Solution. Using Formula (1), the effective rate r equivalent to
converted quarterly is given by
-(1-i-[1--(1.02) -1
= 1.0824 -1= 0.0824
m
Thus the effective rate is 8.24%. This means that the rate 8.24% compounded annualhy isla.
the same interest as the nominal rate 8% compounded quarterly.
Example 18. Findthe effective rate equivalent to the nominal rate 6% converted
|Delhi Univ. B. Com. (H) 2014
()monthly, (ii) continuously.
Solution. (i) Here r= 0.06 and m= 12. Thus the equivalent efiective rate r, is given by
-|1+) -l=|1+:
0.06)12
0.08
12
, = 0.0616 or 6.16%
,= 0.0618
8.24%
quarterly ?
(ii) Using Formula (2), the effective rate equivalent to the nominal rate 6% converted
continuously is given by
or 6.18%.
r, =e-1=e-|= 1,0618-1 = 0.0618
-1 =(1.005) - l = 1.0616 -1 = 0.0616

Mathematics fFinance
12.9 PRESENT VALUE
When makingfuture plans we would often like to know how much money we must invest
nOWtoreceivea certain desired amount Sat some later date. In other words, we are asking
forthe originalinvested principal, which is caled the presentvalue or capitalvalue ofthe
amount. Thusif money IS worth i per period, the present value ofSdue in n periods isthat
principalwhich, invested now at the rate i per period, will amount to Sin nperiods.
FormulaforPresent Value
Toobtain a formula for the present value, we solve the compound amount formula
S = P(1+i)) "for Pby dividing both sides by (1 +i)". This gives P = S(1+i)"
Thusthe present value of Sdue nperiods hence atthe rate iper periodis given by
P = S (l +i)"
various n and i are given in Table II.
The quantity(1+ i) "inthe above formula is called the discount factor. It represents the
Note. It should be noted that P and S represent the value of the same obligation at different
dates. P is the present value of agiven obligation, while S is the future value of the same
obligation. P (now) is just as good as S(n periods hence).
To obtain a formula for the present value in the case of continuous compounding, we solve
the equation S = Pe for P. This givesP = Sen
P= Se
12.19
Thus the present valueofS due at the end oftyears at the annual rate ofr compounded
continuously is given by rt
We use the formula
) When the interest is compounded half yearly
...(1)
Example 27. Find the present value of? 500 due 10 years hence when interest of 10% is
|Delhi Univ. B.A. Eco. (H) 1989]
compounded (i) half yearly (i) continuously.
P= S(l+ )"
Solution. In thisproblem we want to find the principal P when we know that the amount S
after 10 years is going to be?500.
Where S = 500, i = 0.10/2 = 0.05, n =10(2) = 20
n for
P= 500(L05)-20
= 500 (0.3768)
= 188.40
... (2)
(using TableI)
present value of 1due n periods hence atthe rate i per period. Values of (1 +i)

Application Of mathematics in Finance- Time Value of Money

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