Financial Mathematics of Ordinary Simple Annuity

Example

Upon the birth of their first child, a couple decide to deposit $ X at the end of each month ( commencing in one months time) so that upon the child’s 18 th birth day, immediately after the final deposit, they may give her $ 100000. Calculate X, if the interest rate is (5 convertible monthly

As well, ,they alternatively deposit $ Y at a rate of 9 % convertible quarterly, Calculate Y. Explore why Y is greater than 3X and give possible reasons.

Solution

In this S = 100000, i= 0.09/12 =0.0075 n = 18*12 = 216

hence x = 100000/ s 0.0075/216 = $ 186.44.

In the second case, Y = 10000/ s 0.0225/72 = $ 567.73

Y is greater than 3X because in the first case the interest is 0.0075 and it is greater than the quarterly interest rate of 0.0225 as the equivalent rate of 0.0075 per month is approximately equal to 2.27%. In addition, Y is deposited end of the quarter such that it does not earn interest during that quarter. In contrast, the two payments of X deposited before the end of the quarter earn some interest in that quarter. Because of these reasons, Y is greater than 3X.

Finding the interest rate

A very practical applications of the above formulas is finding the interest rate. In many business transactions the effective interest rate is unknown or concealed by the complexities of the arrangements. In order to compare alternative investments or loans, it is necessary to determine the effective interest rate in respect of each proposition and make the decision based on these effective (or true) interest rates.

When R, n and either S or A are given the interest rate I may be determined approximately by linear interpolation. For most practical purposes linear interpolation gives sufficient accuracy. It is an approximate method.

Example

A used car sells for $600 cash or $100 deposit and $90 a month for 6 months. Find the effective interest rate if the purchaser buys the car on installment plan.

Solution

For the above transaction the following equation of value must hod true to have the cash option equivalent to the installment plan.

Cash price = deposit + discounted value of installment

There fore 600 = 100 + 90 i/6

That is,a i/6 = 500/90 = 5.5556

The relevant equations for linear interpolations are:

a 0.02/6 = 5.6014

a 1/6 = 5.5556

a 0.025/6 = 5.5081

That is, i– 0.02/ 0.025 – 0.02 = 5.556 – 5.6014/ 5.5081 – 5.6014

i-0.2/ 0.005 = -0.0458/ -0.0933 = 0.00245

i= 0.00245 per month

There fore effective rate per year = 0.02245*100*12 = 26.95% per year

Finding the term of an annuity

In some situations the accumulated value S, and present value A, the annuity payment R and interest rates are known. However one must determine the number of term of an ordinary simple annuity. In most circumstances for the formula S = R s i/n or A =R a i/n will not give a full number. In this instance a last payment for the next period is paid as to have an equivalent value different from R in most practical circumstances.

Example

A man dies and leaves his wife an estate of $50000. The money is invested at 12% per year. How many monthly payments of $750 would the widow receive and what would be the size of the concluding payment?

Solution

In this problem, A = 50000, R =750 per month i=12/12/100 = 0.01

There fore 750 * a0.01/n = 50000

a 0.01/n = 50000/750 = 66.666

1- (1.01)-n/0.01 = 66.666

1 – (1.01)-n = 0.6666

(1.01)-n = 0.3333

-n log 1.01 = log 0.3333

n = 110. 40963

In this n is not a full number. There fore one must be able have a payment for the 111 th payment. Say it is X.

There fore, 750* a 0.01/110* (1.01) + X = 50000* (1.01)111

150575.63 + X = 150863. 76

X = 308. 13

Changes in the interest rate and the calculation of accumulated value and present value of an annuity

In real life the interest rate can change over an annuities term. In this situation one must calculate for separate periods of the annuities and must have a focal date closer to the final term and compound the earlier periods to the final date or term. This systematic process is important sothat the focal dates are correctly chosen and the number of periods for separate annuities are correctly selected.

Example

An annuity is having interest rate change over its term. If that annuity has the rates for year 1,2,3 as 8% and for year 4 and 5 interest rate is 7% and 6, 7, 8, 9 and 10 having an interest rate of 6%. What is the present value of the 10 year annuity if the annuity is 50 monthly for 10 years?

Solution

First draw a time line for the above annuity. The time line is as follows for different periods of interest rate changes.

_50_50_50_50_50_50_50_50_50_50_50_50_________50_50

0 1    2   3     4    5    6   7     8   9   10 11     12                19   20

|———–0.04———|——0.035——|————0.03———-|

Intermediate value at the end of year 5 = 50 a 0.03/10 = 426.51

Intermediate value at year 3 = 50 a 0.035/4 + 426.51* (1+0.035)-4

= 183.65 + 371.68 = 555.33

Present value = 50 a 0.04/6 + 555.33* (1+0.04)-6

= 262.10+ 438.89 = 700.99