Question

Time value of money

1. A parent is setting up a trust fund for his daughter which is expected to be worth GH¢200,000 in ten years when she is ready to marry. Given that the relevant discount rate is 12% per annum, what should be the onetime investment that this parent should make today to achieve his perceived target. Assume monthly compounding.
2. The parent has the alternative of making ten annual contributions into the fund that will grow to GHS 200,000 ten years from now. Under this arrangement, she makes the first payment immediately and then nine subsequent payments. How much should the annual payments be under this arrangement? A timeline may help you conceptualize this scenario correctly.                                                        
3. An investment firm has assured you that their returns are attractive, and that any funds invested with them would double in 5 years. How much is this attractive rate of return?

Answer 1

Solution to i.

The value of the trust fund after 10 years is GHS 200,000

So let future value, FV = 200,000,

Rate of interest p.a. = 12% but since compounding is monthly, monthly rate, r = 12%/12 = 1% or 0.01

Number of years = 10 but since compounding is monthly the number of compounding periods, n = 12 x 10 = 120 months

We have to find the investment to be made now i.e. present value or PV

We have the equation for future value of single payment

FV = PV (1+r)n ...........(a)

200,000 = PV x (1+0.01)120

200,000 = PV x (1.01)120

200,000 = PV x 3.300387

PV = 200,000 / 3.300387

PV = 60,598.95

Hence, the amount to be invested now by the parent is GHS 60,598.95

Solution to ii.

This time the investments are made annually with the first and subsequent investments made at the start of each year.

Had this been investments made at end of each year, the future value would be

FV = A x [(1+r)n - 1] / r

Modifying this formula for the first and last investments since there is a first payment at point of time 0 (instead of at 1 in formula above) and there is no payment at point of time n, let's assume the future value is denoted by

and the modified formula which is shown in below diagram of the timeline and explained below is:

= {A0 x (1+r)n } + {Ai x [(1+r)n - 1] / r} - A10

The first set of curly brace brackets show the future value of the first payment at point of time 0 and the second set, that of the annuity of payments including the last payment at point of time n; finally, A is subtracted as there is no final payment at point of time n. Since all As are equal we have ignored the Ai notation below and just shown it as A.

= {A x (1+0.12)10 } + {A x [(1+0.12)10 - 1] / 0.12} - A

= {A x (1.12)10 } + {A x [(1.12)10 - 1] / 0.12} - A

= {A x 3.105848 } + {A x [3.105848 - 1] / 0.12} - A

= {A x 3.105848 } + {A x [2.105848] / 0.12} - A

= 3.105848A + 17.548733A - A

= 3.105848A + 17.548733A - A

= 19.654581A

Since = 200,000, hence we have:

200,000 = 19.654581A

A = 200,000 / 19.654581

A = 10,175.74

Hence, the amount to be invested at the beginning of each year by the parent is GHS 10,175.74

Solution to iii.

An investment firm has assured you that any funds invested with them would double in 5 years.

GHS 1 (value of PV) invested would become GHS 2 (value of FV) at the end of 5 years (value of n)

Using the formula in (a) above and substituting these values, we have:

2 = 1 x (1+r)5

(1+r)5 = 2

1+r =

r = - 1

r = 1.14869836 - 1

r = 0.14869836

r = 14.87%

Hence, the attractive rate of return that doubles an investment in 5 years is 14.87%.