Application Of mathematics in Finance- Time Value of Money

Application of Mathematics in Finance

1. 12.1 INTRODUCTION
The entire operation of our industrial civilization is based on the concept that money
belonging to one individual may 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.

2. 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 a year for which the principal is borrowed or
loaned.
The amount is the sum of the principal and the interest.
The rate of interest is the amount charged for the use of the principal for a given length of
time, usually on a yearly or per annum basis.
Rates of interest are usualy expressed as percentages : 5% per annum; 10% per annum and so
on. However, when using rates of interest 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 of the 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
thus computed on a principal which grows periodically, we are dealing with compound
interest.
12.3 SIMPLE INTEREST
The simple interest on a given principal at a given rate for a given time is the product of three
factors: the principal, the rate of interest (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 of money)
r= Rate of interest (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 a given time at
a given rate, and to find the amount.

3. Mathematics of Finance
Eample 1.
Find the simple interest on S00 for
and
Now,
Example
6 months ?
Example 2. Find the time required for 2500 to yield 300 in simple interest at 8%.
Solution.
From the formula I= Prt, we obtain
I= (500) (0.04) (3) = 60
t=
S = P+I= 500 + 60 =560
I = Prt
I= 300, P = 2500 and 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
Substituting I= 120, P= 3000 and t =
P=
Pt
P=
1
2
120
(3000)
we have
Example 4. What principal will amount to 645 in 1
= 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 a principal of ? 600 for 2 years at 5%.
12.3
A person invests 1000 in a savings bank paying 6% simple interest. What is the balance or
amount of the savings account after 6 months ?
nd the amount at 6% simple interest of 1200 due in 9 months.
r3 years at 4%, and find the amount.
Solution, Here eP= 500, r= 0.04 and t=3. Since

4. 12.4 COMPOUND INTEREST
In transactions involving compound interest, the interest earned by an invested amount oe
money (or principal) is reinvested 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
other regular periods of time. The frequency of conversion is a number indicating how 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 of conversion is 4 and the conversion period is 3 months. 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
of conversion periods in a year. Thus, if the quoted rate is 8?% compounded quarterly, the rate
per conversion period is 2% or 0.02. Further, when the frequency of conversion 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 FOR COMPOUND AMOUNT
In this section we shall develop a formula for computing the amount, when interest is
compounded. Let
P= the original principal
n = the number of conversion periods
i = the interest rate (in decimal form) per conversion period
S = the amount at the end ofn periods

5. Then
the
amount at the end of first
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 of third
conversion period
the amount at the end of the nth
conversion period = P(1+ i)"
once the compound amountS of a principal P at the end ofn conversion periods at the interest rate of i per 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 2 converSion periods per year and
i= 0.08/2 = 0.04
S= P1+
If the values of i and 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 i and 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 m times 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 S at the end of t years becomes
m/
mt
That is, the compound amount S at the end of years at an annual interest ofr
Compounded m timesa year is given by
...2)

6. 12.6
using the following formula:
Note 2. Once compound
amount is
obtained,
compound
interest may then be
obtained by
() Annually
Compound Interest =S-P
Example S. Find the compound
amount of
2000 for 4 years at 6%
converted:
(ii) Quarterly
(i) Semiannually
(iv) Monthly
Solution. (i) When the interest is
converted
annualy, 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 converted semiannually, 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
Mathematics jor
(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 i is 0.08/2 = 0.04. Substituting these values in Formula (1), we obtain

7. 12.7 COMPOUND AMOUNT AT CHANGING RATES
(hus far 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

8. Mathematics of Finance
g8% when a deposit 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
and two or 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 following examples.
Example 15. A man made a deposit of 2500 in a savings account. The deposit was left to
accumulate at 6% compounded quarterly for 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 at the end of 13 years is:
S= 2500 (L.015)20 (1.04)16
= 2500 (1.34685) (1.87298)
=6306.55 (app.)

9. NOMINAL 'AND EFFECTIVE 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
earned on 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 ? 100 is 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
a year,
the effective rate is the same as the stated annual rate.
dRelationship Between the Effective Rate and the Nominal Rate.Let r, denote the effective
rate corresponding to the nominal rater, converted m times 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)

10. 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 of 39%
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. Find the 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

11. Mathematics fFinance
12.9 PRESENT VALUE
When making future plans we would often like to know how much money we must invest
nOW to receive a certain desired amount S at some later date. In other words, we are asking
for the original invested principal, which is caled the present value or capital value of the
amount. Thus if money IS worth i per period, the present value ofS due in n periods is that
principal which, invested now at the rate i per period, will amount to S in n periods.
Formula for Present Value
To
obtain a formula for the present value, we solve the compound amount formula
S = P(1 +i) ) " for P by dividing both sides by (1 + i)". This gives P = S (1+i)"
Thus the present value of S due n periods hence at the rate i per period is given by
P = S (l + i)"
various n and i are given in Table II.
The quantity (1 + i) " in the 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 a given 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 value of S due at the end of t years 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 this problem 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 Table I)
present
value of 1 due n periods hence at the rate i per period. Values of (1 +i)