Time Preference Rate and Required Rate of Return
The time preference for money is generally expressed by an interest rate. This rate will be positive even in the absence of any risk. It is called the riskfree rate. For example, if an individual’s time preference is 8%, it implies that he is willing to forego Rs. 100 today to receive Rs. 108 after a period of one year. Thus he considers Rs. 100 and Rs. 108 are equivalent in value. But in reality this is not the only factor he considers. There is an amount of risk involved in such investment. He therefore requires another rate for compensating him with this which is called the risk premium.
Required rate of return=Risk free rate + Risk Premium
There are two methods by which time value of money can be calculated – compounding and discounting.
3.2.1
Compounding Technique
: Under this method of compounding, the future values of all cash inflows at the end of the time horizon at a particular rate of interest are found. Interest is compounded when the amount earned on an initial deposit becomes part of the principal at the end of the first compounding period. If Mr. A invests Rs. 1000 in a bank which offers him 5% interest compounded annually, he has Rs. 1050 in his account at the end of the first year. The total of the interest and principal Rs. 1050 constitutes the principal for the next year. He thus earns Rs. 1102.50 for the second year. This becomes the principal for the third year. This compounding procedure will continue for an indefinite number of years. The compounding of interest can be calculated by the following equation:A=P (1+i ) n
Where A = Amount at the end of the period P = Principal at the end of the period i =rate of interest
n = number of years
The amount of money in the account at the end of various years is calculated as under, using the equation:
Amount at the end of year 1=Rs. 1000 (1+0.05)==Rs. 1050 Amount at the end of year 2=Rs. 1050 (1+0.05)==Rs. 1102.50 Amount at the end of year 3=Rs. 1102.50 (1+0.05)==Rs. 1157.63
Year 1 2 3
Beginning amount Rs. 1000 Rs. 1050 Rs. 1102.50
Interest rate 5% 5% 5%
Amount of interest 50 52.50 55.13
Beginning principal
1000 Rs. 1050 Rs. 1102.50 Ending principal Rs. 1050 Rs. 1102.50 Rs. 1157.63
The amount at the end of year 2 can be ascertained by substituting Rs. 1000 (1+0.05) for R.
1050, that is, Rs. 1000(1+0.05) (1+0.05)= Rs. 1102.50.
Similarly, the amount at the end of year 3 can be ascertained by substituting Rs. Rs. 1000(1+0.05) (1+0.05) (1+0.05) =Rs. 1157.63.
Thus by substituting the actual figures for the investment or Rs. 1000 in the formula A=P (1+i ) n , we arrive at the result shown above in Table.
3.2.2
Discounting Technique
: Under the method of discounting, we find the time value of money now, that is, at time 0 on the time line. It is concerned with determining the present value of a future amount. This is in contrast to the compounding approach where we convert present amounts into future amounts; in discounting approach we convert the future value to present sums. For example, if Mr. A requires to have Rs. 1050 at the end of year 1, given the rate of interest as 5%, he would like to know how much he should invest today to earn this amount. If P is the unknown amount and using the equation we get P (1+0.5)=1050. Solving the equation, we get P=Rs. 1050/1.05=Rs. 1000.Thus Rs. 1000 would be the required principal investment to have Rs. 1050 at the end of year 1 at 5% interest rate. In other words, the present value of Rs. 1050 received one year from now, rate of interest 5%, is Rs. 1000. The present value of money is the reciprocal of the compounding value.
Mathematically, we have P=A {1/(1+i) n } in which P is the present value for the future sum to be received, A is the sum to be received in future, i is the interest rate and n is the number of years.
3.2.3
Future Value of a Single Flow
(lump sum): The process of calculating future value will become very cumbersome if they have to be calculated over long maturity periods of 10 or 20 years.A generalized procedure for calculating the future value of a single cash flow compounded annually is as follows:
FVn = PV(1+i) n
Where FVn = Future value of the initial flow in n years hence PV = Initial cash flow
I = Annual rate of interest N = Life of investment
The expression (1+i) n represents the future value of the initial investment of Re. 1 at the end of n number of years at the interest rate i, referred to as the Future Value Interest Factor (FVIF). To help ease in calculations, the various combinations of “I” and “n” can be referred to in the table. To calculate the future value of any investment, the corresponding value of (1+i) n from the table is multiplied with the initial investment.
Example: The fixed deposit scheme of a bank offers the following interest rates:
Period of deposit Rate per annum
<45 days 9%
46 days to 179 days 10%
180 days to 365 days 10.5%
365 days and above 11%
How much does an investment of R. 10000 invested today grow to in 3 years?
Solution: FVn=PV(1+i) n or PV*FVIF(11%, 3y)
=10000*1.368 (from the tables)
=Rs. 13680
Doubling period: A very common question arising in the minds of an investor is “how long will it take for the amount invested to double for a given rate of interest”. There are 2 ways of answering this question. One is called ‘rule of 72’. This rule states that the period within which the amount doubles is obtained by dividing 72 by the rate of interest. For instance, if the given rate of interest is 10%, the doubling period is 72/10, that is, 7.2 years.
A much accurate way of calculating doubling period is the ‘rule of 69’, which is expressed as 0.35+69/interest rate. Going by the same example given above, we get the number of years as 7.25 years {0.35 + 69/10 (0.35 +6.9)}.
Increased frequency of compounding
It has been assumed that the compounding is done annually. If a scheme is offered where compounding is done more frequently, let us see its effect on interest earned. For example, if we have deposited Rs. 10000 in a bank which offers 10% interest per annum compounded semi
annually, the interest earned will be as follows:
Amount invested Rs. 10000
Interest earned for first 6 months
10000*10%*1/2 (for 6 months) Rs. 500 Amount at the end of 6 months Rs. 10500 Interest earned for second 6 months
10500*10%*1/2 Rs. 525
Amount at the end of the year Rs. 11025
If in the above case compounding is done only once a year the interest earned will be 10000*10%
which is equal to Rs. 1000 and we will have Rs. 11000 at the end of first year. We find that we get more interest if compounding is done on a more frequent basis. The generalized formula for shorter compounding periods is:
FVn=PV (1+i/m) m*n
Where, FVn= Future value after n years PV= Cash flow today
i= Nominal interest rate per annum
m= No. of times compounding is done during a year n= No. of years for which compounding is done.
Example: Under the Andhra Bank’s Cash Multiplier Scheme, deposits can be made for periods ranging from 3 months to 5 years. Every quarter, interest is added to the principal. The applicable rate of interest is 9% for deposits less than 23 months and 10% for periods more than 24 months.
What will the amount of Rs. 1000 today be after 2 years?
Solution:
FVn= PV (1+i/m) m*n 1000 (1+0.10/4) 4*2 1000 (1+0.10/4) 8 Rs. 1218
Effective vs. nominal rate of interest: We have just learnt that interest accumulation by frequent compounding is much more than the annual compounding. This means that the rate of interest given to us, that is, 10% is the nominal rate of interest per annum. If the compounding is done more frequently, say semiannually, the principal amount grows at 10.25% per annum. 0.25% is known as the “Effective Rate of Interest”. The general relationship between the effective and nominal rates of interest is as follows:
r = {(1+i/m) m }1 Where,
r= Effective rate of interest i= Nominal rate of interest
m= Frequency of compounding per year.
Example: Calculate the effective rate of interest if the nominal rate of interest is 12% and interest is compounded quarterly.
Solution:
r = {(1+i/m) m }1 r = {(1+0.12/4) 4 }1 r=0.126 or 12.6% p.a.
3.2.4
Future Value Of Series Of Cash Flows
We have considered only single payment made once and its accumulation effect. An investor may be interested in investing money in installments and wish to know the value of his savings after n years.
For example, Mr. Madan invests Rs. 500, Rs. 1000, Rs. 1500, Rs.2000 and Rs. 2500 at the end of each year for 5 years. Calculate the value at the end of 5 years compounded annually if the rate of interest is 5% p.a.
Solution:
3 1.158 1158
3 Rs.
1500
2 1.103 1654
4 Rs.2000 1 1.050 2100
5 Rs.
2500
0 1.000 2500
Amount at the end of 5 th Year Rs.
8020
3.2.5
Future Value Of An Annuity
Annuity refers to the periodic flows of equal amounts. These flows can be either termed as receipts or payments. For example, if you have subscribed to the Recurring Deposit Scheme of a bank requiring you to pay Rs. 5000 annually for 10 years, this stream of payouts can be called
“Annuities”. Annuities require calculations based on regular periodic contribution of a fixed sum of money.
The future value of a regular annuity for a period of n years at i rate of interest can be summed up as under:
FVAn = A{(1+i) n 1} / i
Where FVAn=Accumulation at the end of n years i= Rate of interest
n= Time horizon or no. of years
A= Amount deposit/invested at the end of every year for n years.
The expression {(1+i) n 1}/ i is called the Future Value Interest Factor for Annuity (FVIFA). This represents the accumulation of Re. 1 invested at the end of every year for n number of years at i rate of interest. The tables at the end of this book give us the calculations for different combinations of i and n. We just have to multiply the relevant value with A and get the accumulation in the formula given above.
Example: M. Ram Kumar deposits Rs. 3000 at the end of every year for 5 years into his account for 5 years, interest being 5% compounded annually. Determine the amount of money he will have at the end of the 5 th year.
End of
4 1.216 2432
2 Rs.
2000
3 1.158 2316
3 Rs.
2000
2 1.103 2206
4 Rs.2000 1 1.050 2100
5 Rs.
2000
0 1.000 2000
Amount at the end of 5 th Year Rs. 11054 OR Using formula and the tables we can find that:
Example: Calculate the value of an annuity flow of Rs. 5000 done on a yearly basis for 5 years, yielding an interest of 8% p.a.
Solution:
=5000 FVIFA(8%, 5y)
=5000* 5.867
=Rs. 29335
3.2.5.1
Sinking Fund
Sinking fund is a fund which is created out of fixed payments each period to accumulate to a future sum after a specified period. The sinking fund factor is useful in determining the annual amount to be put in a fund to repay bonds or debentures or to purchase a fixed asset or a property at the end of a specified period.
A=FVA*i / {(1+i) n 1}
i / {(1+i) n 1} is called the Sinking Fund Factor.
Self Assessment Questions 1
1. The important factors contributing to time value of money are __________, ________________
and _______.
2. During periods of inflation, a rupee has a ___________than a rupee in future.
3. As future is characterized by uncertainty, individuals prefer _________consumption to __________consumption.
4. There are two methods by which time value of money can be calculated by _________ and _________ techniques.