- 40 Marks
FM – L2 – Q80 – Simple interest and compound interest
Question
(a) Kofi borrowed GH¢120,000 for eight months at 15% simple interest per annum. How much interest would he pay?
Compute the annual rate of interest that, if compounded continuously monthly, would result in the payment of the same amount of interest.
(b) Ama paid GH¢500,000 into a fund which yielded 8% per annum compounded annually.
How much amount will she have in the fund after 10 years?
(c) At the start of each 6 month period for 7 years Kwame paid GH¢25,000 into a fund earning annual interest at 6% compounded semi-annually.
What amount would be in the fund at the end of 7 years?
(d) Yaw deposited an amount into a bank which will be doubled in eight years. Find the rate of interest on the basis that the amount is compounded annually.
(e) How much should Adwoa deposit now to yield GH¢600,000 at the end of five years at 10% per annum simple interest?
(f) How much should Esi deposit now to yield GH¢600,000 at the end of five years at 10% per annum compound interest?
(g) How much should Kwesi deposit now to yield GH¢600,000 at the end of five years at an annual interest of 10% compounded half yearly?
(h) Nii wants to purchase an annuity that will provide GH¢6,000 per annum at the end of each year for 10 years.
How much will he need to invest in a fund with a return of 6% per annum?
How much interest would he earn over the period of 10 years?
(i) Two years ago, Apex Limited borrowed an amount at an annual interest rate of 10%. The amount of the loan today is GH¢100,000 and the final amount will be paid back in four years.
Apex Limited is to set up a sinking fund which yields a return of 8% per annum compounded quarterly (by parts).
How much will Apex Limited need to deposit at the end of each quarter, in the sinking fund, to settle the loan at the end of four years?
(j) Star Limited wants to invest equal annual amounts in a bank for five years starting from January 1, 20X1 in order to have the following amounts available:
- GH¢1.0 million for the purchase of land on January 01, 20X6.
- Enough cash to buy an annuity of GH¢240,000 per annum for 4 years commencing from January 1, 20X7 at an interest rate of 8%.
How much will Star Limited need to deposit at the end of each year in a fund that generates a return of 10% per annum compounded annually?
Answer
(a) Kofi
Interest paid
GH¢120,000 × 15% × 8/12 = GH¢12,000
Period interest rate
15% × 8/12 = 10% (or GH¢12,000 / GH¢120,000)
The interest rate that, if compounded monthly would lead to the same
Period rate = (1+r)^n – 1
Monthly rate = (1.1)^(1/8) – 1 = 0.011985 or 1.11985%
The equivalent annual rate can also be found using the equivalent period interest rate formula.
Annual rate = (1.011985)^12 – 1 = 0.1537 or 15.37%
(b) Ama
Sn = So × (1+r)^n
Sn = GH¢500,000 × (1.08)^10 = GH¢1,079,462
(c) Kwame
The payments are in advance so this is an annuity due.
Sn = [X(1+r)^n – 1] / r × (1+r)
Sn = [25,000(1.03)^14 – 1] / 0.03 × (1.03) = GH¢439,973
(d) Yaw
Using: Sn = So × (1+r)^n
If the initial deposit = x, then the amount after 8 years will be 2x
2x = x(1+r)^8
2x / x = (1+r)^8
2 = (1+r)^8
√[8]{2} = 1+r
2^(1/8) = 1+r
1.0905 = 1+r
r = 0.0905 or 9.05%
(e) Adwoa
Adwoa’s initial investment would earn 10% of the amount invested each year over the period of the investment.
Let the initial investment = x
x + (5 × 0.1x) = GH¢600,000
x = GH¢600,000 / 1.5 = GH¢400,000
Proof:
GH¢400,000 + (5 × 0.1 × GH¢400,000) = GH¢600,000
(f) Esi
Sn = So × (1+r)^n
GH¢600,000 = So × (1.1)^5
GH¢600,000 / (1.1)^5 = So
So = GH¢372,552
Note that this question is simply asking the present value of GH¢600,000 received in 5 years at an annual interest rate of 10%.
(g) Kwesi
If the annual interest rate is 10%, the 6 monthly interest rate compounded by parts is 10% / 2 = 5%.
Sn = So × (1+r)^n
GH¢600,000 = So × (1.05)^10
GH¢600,000 / (1.05)^10 = So
So = GH¢368,348
Note that this question is simply asking the present value of GH¢600,000 received in 5 years at a 6 monthly interest rate of 5%.
(h) Nii
What amount invested now is the equivalent of GH¢6,000 per annum at an interest rate of 6%?
This is the same as asking for the present value of 10 annual
Amount now = GH¢6,000 × [1/0.06 (1 – 1/(1.06)^10)]
Amount now = GH¢6,000 × 7.36 = GH¢44,161
Interest earned
Total receipts – initial investment
(10 years × GH¢6,000) – GH¢44,161 = GH¢15,839
(i) Apex Limited
Step 1: Calculate the final amount of the loan to be repaid
Sn = So × (1+r)^n
Sn = GH¢100,000 × (1.1)^4 = GH¢146,410
Step 2: Calculate the periodic sinking fund payment
Sn = [X(1+i)^n – 1] / i
Note that the quarterly interest rate is 8% / 4 = 2% and there are 16 quarters in 4 years
GH¢146,410 = [X((1.02)^16 – 1)] / 0.02
GH¢146,410 = [X(1.3728 – 1)] / 0.02
GH¢146,410 = [0.3728X] / 0.02
X = (146,410 × 0.02) / 0.3728 = GH¢7,855
(j) Star Limited
Step 1: Calculate the final amount needed at 1 January 20X6 (31 December 20X5)
For purchase of land: GH¢1,000,000
Present value of annuity
PV at 1 Jan 20X6 = GH¢240,000 × [1/0.08 (1 – 1/(1.08)^4)] = GH¢794,880
Total amount needed = GH¢1,000,000 + GH¢794,880 = GH¢1,794,880
Step 2: Calculate the periodic sinking fund payment
Sn = [X(1+i)^n – 1] / i
GH¢1,794,880 = [X((1.1)^5 – 1)] / 0.1
GH¢1,794,880 = [X(1.61051 – 1)] / 0.1
GH¢1,794,880 = [0.61051X] / 0.1
X = (GH¢1,794,880 × 0.1) / 0.61051 = GH¢293,996
The sinking fund calculation assumes that the first payment is at the end of the first period. In this example, the payments start at the beginning of the first period. Therefore, the payments must be discounted by one year.
Sinking fund payment = GH¢293,996 × 1/1.1 = GH¢267,269
Proof:
| Payment made on | Compounding factor | Final value |
|---|---|---|
| 01/01/20X1 | 1.1^5 | 430,447 |
| 01/01/20X2 | 1.1^4 | 391,312 |
| 01/01/20X3 | 1.1^3 | 355,742 |
| 01/01/20X4 | 1.1^2 | 323,402 |
| 01/01/20X5 | 1.1 | 294,001 |
| Final value on 31/12/20X5 | 1,794,904 |
Alternative approach
Step 2 above could have been carried out by discounting the required amount to its present value and then dividing that by the 5 year, 10% annuity factor (which is 3.791) in order to calculate an equivalent annual cost.
GH¢1,794,910 / (1.1)^5 = GH¢1,114,498
GH¢1,114,498 / 3.791 = GH¢293,985
This would then need to be discounted by one year to reflect the fact that the payments are in advance not arrears (as before).
Sinking fund payment = GH¢293,985 × 1/1.1 = GH¢267,259
Note that the small differences in this approach are due to rounding differences in the discount and compounding factors.
- Uploader: Samuel Duah