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Financial mathematics deals with the problem of investing money or capital. If a company (or an individual investor) puts some capital into an investment, he/she will want a financial return for it.
E.g.: IF A puts Rs 1000 into account with a bank, he will expect a return in the form of interest, which will be added to the original investment in his account; If An enterprise invests Rs 10,000 in an item of equipment, the company will expect to make some sort of profit out of the project over the working life of equipment.
Investors may wish to know:
How much return will be obtained by investing money now for n years?
How much return will be obtained in n years' time by investing some money every year for n years?
Time is an important element in investment decisions. The longer an investment continues, the greater will be the return required by the investor. For example, if a bank lends Rs 20,000 to a company, it would expect a bigger interest payment if the loan last for two years than if it lasts for only one year.
This time factor in investment decision has a direct relation to inflation as well as the declining value of money over a period of time. The cost of interest would increase for a longer period of investment even if there is no inflation. The effect of inflation is simply to increase the size of the return required by the investor over any period of time.
The two major techniques of financial mathematics are compounding and discounting. These techniques are very closely related to each other, and are discussed hereunder.
Sn= P + nrP1. Simple Interest: Simple interest is the amount if interest, which is earned in an equal amount every year (or month), and which is a given proportion of the original investment or (principal). If a sum of money is invested for a period of time then the amount of simple interest, which accrues, is equal to the number of accounting periods * the interest rate * the amount invested, i.e.,
Where P=Original sum invested;
r= Interest rate; n=Number of periods; S=Sum after n periods, consisting of the original capital plus interest earned.
E.g. how much will an investor have after 5 years if he invests Rs 1000 at 10% simple interest per annum?
Sol: S5=Rs 1000 + (5 x 0.1 x Rs 1000) = Rs 1500
2. Compound Interest:
Interest is normally calculated by the means of compounding. If a sum of money, the principal, is invested at a fixed rate of interest and the interest is added to the principal and no withdrawals are made, then the amount invested grows at an increasing rate as time progresses, because interest earned in earlier periods itself also earns interest later on.
E.g. Rs 2,000 is invested to earn 10% interest. After 1 year, the original principal plus interest will amount to Rs 2,200.
Thus interest for 2nd year will be on Rs 2200. So interest will be earned on interest.
Formula for compounding:
Sn = P (1+r) n
Where P - original sum invested
r= interest rate, expressed as proportion (5%=0.05)
n=number of periods
Sn=sum after n periods.
Rs 2000 invested at 10% per annum for 3 years
S3 = 2000 (1+0.10)3
= 2000(1.10)3 (See the compound value table for 3years and 10 %)
=2000 x 1.331 = Rs 2,662.
The interest earned over 3 years is Rs 662.
IF the money were invested for 6 years, it would be worth
Rs 2,000(1.10)6 = Rs 2000 x 1.772 = Rs 3544.
The interest earned over 6 years is Rs 1544.
E.g. what would be the total value of Rs 5,000 invested now?
A. after 3 years, if the interest rate is 20% per annum.
B. after 4 years, if the interest rate is 15% per annum.
C. after 3 years if the interest rate is 6% per annum.
Sol: S3 = 5000 (1+0.20) 3 = 5000 x 1.728 = Rs 8,640.
S4 = 5000 (1+0.15) 4 = 5000 x 1.749 = Rs 8,745.
S3 = 5000 (1+0.06) 3 = 5000 x 1.191 = Rs 5,955.
Inflation: The same compounding formula can be used to predict future prices after allowing for price inflation. For e.g. if we wish to know how much a skilled employee will earn per annum in 5 years time, given that he earns Rs 8,000 p.a. now and wages inflation is expected to be 10% p.a.
Formula: Sn = P (1+r) n
= Rs 8,000 (1.10)5
= Rs 12,888 p.a.
Withdrawal of capital or interest
IF an investor takes money out of an investment, it will cease to earn interest. Thus, if an investor, A puts Rs 3,000 into a bank deposit account, which pays interest at 8% p.a. and makes no withdrawals except at the end of year 2, when he takes out Rs 1,000. What would be the balance in his account at the end of 4 years?
Original investment Rs. 3,000.00
Interest in year 1 (8%) 240.00
Investment at the end of year 1 3,240.00
Interest in year 2 (8% on Rs 3,240) 259.20
Investment at the end of year 2 3,499.20
Less: Withdrawal 1,000.00
Investment at the start of year 3 2,499.20
Interest in year 3 (8% on Rs 2,499.20) 199.94
Investment at the end of year 3 2,699.14
Interest in year 4 (8% on Rs 2,699.14) 215.93
Balance at the end of 4 years 2,915.07
A better approach to the solution would be as follows:
Rs 3,000 invested for 2 years at 8% would increase
In value to Rs 3,000 (1.08)2 Rs. 3,498
Less: withdrawal 1,000
Rs 2,498 invested further 2 years at 8% would
Increase in value to
Rs 2,498 (1.08)2 Rs 2,913
Changes in the rate of interest
If the rate of interest changes during the period of an investment, the compounding formula would be amended slightly as follows -
Sn = P (1+r1) x (1+r2) (n-x)
Where, r1 = initial rate of interest
x = number of years in which the interest rate r1 applies
r2 = Next rate of interest
n-x = balancing number of years in which the interest rate r2 applies
Illus: If Rs 8,000 is invested now, to earn 10% interest for 3 years and then 8% p.a. in the subsequent years, what would be the size of total investment at the end of 5 years?
Sol: Sn = P (1+r1) x (1+r2) (n-x)
= 8000(1.10)3 (1.08)2
= 8000(1.331) (1.166)
= Rs 12415.568
Illus: An investor puts Rs 10,000 into an investment for 10 years. The rate of interest earned is 15% p.a. for the first 4 years, 12% p.a. for the next 4 years and 9% p.a. for the last 2 years. How much will the investment be worth at the end of 10 years?
Sol: Sn = P (1+r1) x (1+r2) y (1+r2) (n-x-y)
= 10000 (1.15)4 (1.12)4 (1.09)2
= 10000(1.749) (1.574) (1.188)
Illus: An item of equipment costs Rs 61,000 now. The rate of inflation over the next 4 years is expected to be 16%, 20%, 15% and 10%. How much will the equipment cost after 4 years?
Sol: Rs 61,000 (1.16) (1.20) (1.15) (1.10) = Rs 107413.68
Semi-annual and other compounding periods:
Semi annual compounding means that there are two compounding periods within a year. Very often the interest rates are compounded more than once in a year. Savings institutions, particularly give compound interests semi-annually, quarterly and even monthly.
Assume Mr Y places his savings of Rs 1,000 in a two-year time deposit scheme of a bank with 6% interest compounded semi - annually. He will be paid 3% interest compounded over 4 periods - each of 6 months' duration.
6% twice a year for 2 years, i.e. 4 times totally
So, 6% divided by (m) 4 times = 1.5% x (n) 2 years = 3% for each 6 months.
Formula: - Sn = P 1 + r mn
= 1000 (1 + 0.06/2)2 x 2
= 1000 (1 + 0.03)4
In the compound value table see 3% in 4 year column
= 1000 x 1.125
= Rs 1125
(Difference between the two methods is due to rounding off done in compounding table)
Interest is paid in 4 equal installments after every 3 months.
E.g.: - How much does a deposit of Rs 5,000 grow to at the end of 6 years, if the nominal rate of interest is 12% and the frequency of compounding is 4 times a year? The amount after 6 years will be:
Sn = P 1 + r mn
= Rs 5,000 ( 1 + 0.12/4)4 x 6
= Rs 5,000 ( 1.03)24
In the compounding table we will see in 3% in 24th year
= Rs 5000 x 2.033= Rs 10,165
Effective versus nominal rate
We have seen above that Rs 1,000 grows to Rs 1,060.90 at the end of a year of nominal rate of interest is 6% and compounding is done semi-annually. This means that Rs 1,000 grows at the 6.09%.
A company may offer to investors 10% p.a. interest, pay 5% every 6 months at a compound rate of interest. The effective annual rate of interest would be:
r = 1+ k m _ 1
Where r = effective rate of interest; k = nominal rate of interest; m = frequency of compounding per year.
r = (1+ 0.1/2)2 - 1
= (1.05) 2 - 1
= 0.1025 or 10.25%
Similarly, if a bank offers to its depositors a nominal 12% p.a., with the interest payable quarterly, the effective rate of interest would be 3% compound every 3 months i.e.
Formula = (1+0.12/4)4 - 1
= (1.03) 4 - 1
= 0.126 or 12.6%
E.g. calculate effective annual rate of interest:
a. 15% compound quarterly
b. 18% compound monthly
r = (1+0.15/4) 4 - 1
= (1.0375) 4 - 1
= 0.1587 or 15.87%
r = (1+0.18/12) 12 - 1
= (1.015) 12 - 1
= 0.1956 or 19.56%
Future value of an Annuity
A general definition of an annuity would be a constant sum of money each year for a given number of years.
It is a stream of equal annual cash flows.
Ordinary annuity: Payments or receipts occur at the end of each period.
Suppose you deposit Rs 1,000 every year in a bank for 5 years, interest rate is 10% compound, then value of this series of deposits at the end of 5 years will be: -
Rs 1,000 (1.10)4 + 1,000 (1.10)3 + 1,000 (1.10)2 + 1,000(1.10)1 + 1000 (1.10)0
= Rs 1,000(1.4641) + 1,000(1.331) + 1,000(1.21) + 1,000(1.10) + 1,000
= Rs 6,105
Formula = FVAn = A(1+k) n-1 + A(1+k) n-2 A(1+k) n-3+…..+ A
= A (1+k) n-1
Where FVAn = future value of an annuity which has a duration of n periods
A = constant periodic flow
k = interest rate per period
n = duration of the annuity
the term (1+k) n-1 is referred to as the future value interest factor for an
annuity (FVIFAk,n). The value for this factor for combinations k and n is given in Table A.2 (future value interest factor for annuity)
Illus: 4 equal annual payments of Rs 2,000 are made into a deposit account that pays 8% interest per year. What is the future value of this annuity at the end of 4 years?
FVA4 = Rs 2,000 (FVIFA8%, 4) = Rs 2,000 (4.507) = Rs 9,014.
= Rs 2,000 (1.08)3 + 2,000 (1.08) 2 + 2,000 (1.08) 1 + 2,000
= Rs 2,000 x 1.260 + 2,000 x 1.166 + 2,000 x 1.080 + 2,000
= Rs 2520 + 2332 + 2160 + 2000
= Rs 9,012.
Annuity DUE: When the annual payments are at the beginning of each year so a change in formula.
FVAn = A(1+k) (1+k)n-1
so just extra is multiplying (1+k) one more time as payments are made at the beginning of each year.
Present (discounted) value of a single amount
A rupee today is worth more than a rupee to be received one, two or three years from now. Calculating the present value of future cash flows allows us to place all cash flows on a current footing so that comparisons can be made in terms of today's rupees.
For e.g. Which should you prefer - Rs 1,000 today or Rs 2,000 10 years from today? Assume that both sums are completely certain and your rate of interest is 8% p.a. (i.e. you could borrow or lend at 8%). Present worth of Rs 1,000 is easy - it is worth Rs 1,000. However, what is Rs 2,000 received at the end of 10 years worth to you today? Or we can ask the question what amount (today) would grow to be Rs 2,000 at the end of 10 years at 8% compound interest. This amount is called present value of Rs 2,000 payable in 10 years, discounted at 8%.
Finding the present value is simply the reverse of compounding.
Sn = P (1+r) n
P = Sn 1 n
The factor (1/ (1+r) n) is called the discounting factor or present value interest factor (PVIFk,n)
So we can rearrange the above equation as
P = Sn(PVIFk,n)
= 2,000 (PVIF8%, 10)
= 2,000 (0.463)
= Rs 926
So finally if we compare this present value amount of Rs 926 to the promise of Rs 1,000 to be received today, we should prefer to take Rs 1,000. In present value terms we would be better off by Rs 74 (Rs (1,000-926).
In the above problem every rupee is handicapped to such an extent that each was worth only about 46 paisa. The greater the disadvantage assigned to a future cash flow, the smaller the corresponding present value interest factor.
Present value of uneven series
A company has an option, whether to spend Rs 22,500 on an item of equipment, in order to obtain cash profit of:
Year 1 Rs 7,500
Year 2 Rs 10,000
Year 3 Rs 6,250
Year 4 Rs 1,250
If the company requires a return of 10% p.a., is the project worth while? Use Net present value method.
The net present value is negative; that means cost of project is more than its return, so it will be cheaper to invest elsewhere at 10% than to invest in the project.
Present (discounted) value of an annuity
To calculate the present value of a constant annual flow or annuity, multiply the annual cash flows by the sum of the discount factors for the relevant years.
For e.g. there is constant annual cash flow of Rs 8,000 p.a. for 5 years, year 1 to 5 at 11% interest.
8,000 x 0.9001
Plus 8,000 x 0.8116
Plus 8,000 x 0.7312
Plus 8,000 x 0.6587
Plus 8,000 x 0.5935
= 8,000 x 3.6951 = Rs 29,559
In the Present value interest factor for an annuity table see the present value factor of Re 1 p.a. for 5 years at 11%, which is 3.6951.
Multiply the annual cash flow with the PVIFA.
So, Rs 8,000 x 3.6951 = Rs 29,559.
If we go by formula method
(1+r) n - 1
PVA = a r(1+r) n
(1+0.11) 5 - 1
= 8000 0.11(1.11) 5
= 8000 x 1.68506 - 1
= 8000 x 3.6958
= Rs 29,566.