Capitulo III. Propuesta de la solución al problema
3.3 Elementos que conforman la propuesta
n-1
—r R
The present value of the annuity involves 'moving' each of the payments R to the present. Not an easy task, for the 300 monthly payments of a 25 year loan. Hence, the following mathematical formula can help:
A = R[1 -(1 +rTn/r]
This represents the present value A, of an annuity of Rs. R per payment period, for n periods, at the rate r per period.
Or, R = Ar/l -(1 + r)~n Which gives the periodic payment R of an annuity whose present value is A.
For an illustration, if the plan is to get paid Rs. 20,000 a year for 20 years and do it with an annuity, when the interest rate is 5 per cent, the amount you would need to invest in the annuity is
A = 20,000 [1 - (1 + 0.05)~20]/.05 = 4,00,000(1 - 0.37689) = Rs. 2,49,244
The present value of quarterly payments of Rs. 250 for 5 years at 12 per cent compounded quarterly, is r =0.12/4 = 0.03 n = 5 x 4 = 20
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The regular payments over 5 years are equivalent to having Rs. 3,719.37 now.
Rs. 25,000 is borrowed over 8 years. What will be the monthly repayments at 18 per cent compounded monthly?
r =0.18/12 = 0.015 n =8 x 12 = 96
R = 25,000 x 0.015/[l - (1 + 0.015)"96] = 493.08
Monthly repayments should be Rs. 493.08
A person wishes to borrow Rs. 5,000 now and Rs. 4,000 two years from now. Both loans require a repayment of equal monthly payments made at the end of the month for the next five years. What is the monthly payment? (Assume 10 per cent compounded monthly)
Bring everything back to the present value. Loans are presently worth = 5,000 + 4,000(1 + 0.1/12)24
= 5,000 + 3,277.64 = 8,277.64 The present value of the repayments is
r = 0.10/12 n= 12 x 5
A
=
R* [i -(i + o.io/n^/o.mi]
Then, 8,277.64 = R * [1 - (1 + 0.10/12) 60/0.1/12] R = 175.88 (i.e., Monthly repayment)
If Arlene thinks in terms of living exactly 15 years from today, how much money should she spend per year? It turns out that we can calculate this; using a loan amortisation formula. We can think of Arlene as lending the bank Rs. 3,00,000 for 15 years, and the bank paying her back in equal annual instalments at a rate of 3 per cent interest. When a loan is repaid in equal instalments, part of the payment covers interest and the rest covers principal. The formula for paying back a loan in equal instalments is known as the amortisation formula. The amortisation formula is
R = Ar/[l -(1 +r)"n]
Where R is the annual instalment, r is the annual interest rate, A is the initial loan balance, and n is the number of years to repay the loan. Plugging in Rs. 3,00,000 for A, 0.03 for r, and 15 for n, we have R = Rs. 25,130. This says that by lending (investing) her Rs. 3,00,000 at an interest rate of 3 per cent, Arlene can live for 15 years on Rs. 25,130 per year.
If there were no inflation, then Arlene would receive exactly Rs. 25,130 a year. If there is inflation of, say, 2 per cent per year, then the nominal interest rate will be 5 per cent and the real interest rate will be 3 per cent. Arlene will receive Rs. 25,130 the first year, Rs. 25,130 (1 + .02) the second year, and so on. That is, each year, her annuity payment will rise 2 per cent, in order to keep up with inflation. Adjusting for inflation is what makes this a real annuity.
In the real world, there are some complications. First, not all annuities are adjusted for inflation. Although inflation is important, all too often the elderly live on fixed incomes, which are annuities that do not adjust for inflation. Second, insurance companies need to earn a profit. If the insurance company earns 0.5 per cent, then Arlene will receive an annuity based on 3.0-0.5 or 2.5 per cent real interest. This will
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Finally, converting a fixed sum of money to an annuity leads to an exchange of risk between Arlene and the insurance company. If Arlene dies early, say in 5 years, she will not have collected her annuity and the insurance company earns a windfall gain. Conversely, if she defies the actuarial tables and lives for 25 years, the insurance company may take a loss, because the Rs. 3,00,000 will not earn enough interest to cover the additional payments.
When interest-bearing debts are amortised by means of a series of equal payments at equal intervals, it is important to know how much goes for interest from each payment and how much goes for the reduction in principal. For an illustration, this may be a necessary part of determining one's taxable income or tax deductions. We construct an amortisation schedule, which shows the progress of the amortisation of the debt.
Illustration
A debt of Rs. 22,000 with interest /4= 10% is to be amortised by payments of Rs. 5,000 at the end of each quarter for as long as necessary. Make out an amortisation schedule showing the distribution of the payments as to interest and the repayment of principal.
Solution
The interest due at the end of the first quarter is 2.5 per cent of Rs. 22,000 or Rs. 550.00. The first payment of Rs. 5,000 at this time will pay the interest and will reduce the outstanding principal balance by Rs. 4,450. Thus, the outstanding principal after the first payment is reduced to Rs. 17,550. The interest due at the end of the second quarter is 2.5 per cent of Rs. 17,550 or Rs. 438.75. The second payment of Rs. 5,000 pays the interest and reduces the indebtedness by Rs. 4,561.25. The outstanding principal now becomes Rs. 12,988.75. This procedure is repeated and the results are tabulated below in the amortisation schedule.
Table 1.4 Payment Number Periodic Payment Payment of Int @ 2.5% Principal Repaid Outstanding Balance 1 5,000 550.00 4,450 22,000 17,550 2 5,000 438.75 4,561.25 12,988.75 3 5,000 324.72 4,675.28 8,313.47 4 5,000 207.84 4,792.16 3,521.31 5 3,609.34 88.03 3,521.31 0 Total 23,609.34 1,609.34 22,000
It should be noted that the fifth payment is only Rs. 3,609.34, which is the sum of the outstanding principal at the end of the fourth quarter plus the interest due at 2.5 per cent. The totals at the bottom of the schedule are for checking purposes. The total amount of principal repaid must equal the original debt. In addition, the total of the periodic payments must equal the total interest and the total principal returned. Note that the entries in the principal repaid column (except the final payment) are in the ratio
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