Topic: Simple interest and compound interest

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  • 10 Marks

FM – May 2020 – L2 – Q3a – Simple interest and compound interest

Calculate the quarterly loan installment and prepare an amortization schedule for a loan with compound interest and equal quarterly payments.

Odapagyan Foods Ltd is borrowing GH¢500,000 to finance a project involving an expansion of its existing factory. It has obtained an offer from Sika Bank. The terms of the loan facility are as follows:

  • Annual interest rate: 22%
  • Duration: 2 years
  • Interest method: compound interest with quarterly compounding
  • Payment plan: equal installments at the end of each quarter

Required:

i) Compute the quarterly installment.
(3 marks)

ii) Prepare a loan amortization schedule to show the periodic interest charges, installment payments, principal payments, and balance of the loan at the end of each quarter.
(7 marks)

i) Calculation of Quarterly Installment:
The present value of the payments, PVAn = Loan principal = GH¢500,000
Annual interest, i = 22%
Frequency, m = 4
Term (in years), t = 2
Number of periods, n = Term x Frequency = 2 x 4 = 8
The formula for the present value of an annuity (PVA) is:
Plugging in the values:

Thus, the quarterly installment is GH¢78,932.01.
(3 marks)

ii) Amortization Schedule:

Period Interest (GH¢) Installment (GH¢) Principal Repayment (GH¢) Outstanding Balance (GH¢)
0 – – – 500,000.00
1 27,500.00 78,932.01 51,432.01 448,567.99
2 24,671.24 78,932.01 54,260.77 394,307.23
3 21,686.90 78,932.01 57,245.11 337,062.12
4 18,538.42 78,932.01 60,393.59 276,668.53
5 15,216.77 78,932.01 63,715.24 212,953.29
6 11,712.43 78,932.01 67,219.57 145,733.72
7 8,015.35 78,932.01 70,916.65 74,817.07
8 4,114.94 78,932.01 74,817.07 –

(7 marks)

  • 20 Marks

FM – July 2023 – L2 – Q3 – Discounted cash flow | Simple interest and compound interest

Compute the maturity value of a loan, determine the size of quarterly sinking fund deposits, and compare annuity due with ordinary annuity.

a) Osiadan Contractors Ltd plans to construct housing units in the Kpone-Katamanso District of Ghana for sale to young professionals. It has obtained a US$4 million loan facility from the International Finance Corporation (IFC). The total principal granted will be released in two equal tranches: the first tranche to be released now and the second tranche at the beginning of the fourth year. Interest will be charged on the loan at 12% per annum with semi-annual compounding. The total principal granted plus the total interest is to be paid at the end of the duration of the loan in six years’ time.

To raise money toward the settlement of the maturity value of the loan, Osiadan Contractors Ltd plans to establish a sinking fund by which it will deposit equal amounts of U.S. dollars at the beginning of every quarter starting from the third year until the end of the loan duration. The dollar deposits will be invested at 8% per annum with quarterly compounding.

Required:
i) Compute the maturity value of the loan at the end of six years. (3 marks)
ii) Compute the size of the quarterly deposits to be invested in the sinking fund to raise the amount needed to settle the maturity value of the loan. (4 marks)
iii) Compare and contrast annuity due and ordinary annuity. (3 marks)

b) Klessy Beverages Inc (Klessy), an American malt drink producer, imports barley from Australia and pays for the import in Australian dollars. It has bought a consignment of barley with an invoice value of AUD2.5 million, and payment is due next month. The exchange rate between the United States dollar (USD) and the Australian dollar (AUD) is currently trading at USD0.7265/AUD. The Managers of Klessy fear that the Australian dollar may strengthen against the US dollar. The Treasury Manager has recommended using currency options to address the potential currency risk.

Below are the CME Group’s quotations for weekly options on the Australian dollar:

Contract Specification Call Put
Contract size AUD100,000 AUD100,000
Premium (US$) 0.0131 0.0031
Strike price (US$) 0.7275 0.7275
Expiration Week 4 Week 4

Required:
i) Justify which option Klessy should buy considering its underlying currency exposure. (3 marks)
ii) Compute the intrinsic value of your selected option and interpret the results. (4 marks)
iii) Distinguish between option premium and option strike (or exercise) price. (3 marks)

a)
i) Computation of the maturity value of the loan at the end of six years:


ii) Computation of the size of the quarterly deposits into the sinking fund:
The objective of the sinking fund is to raise money to settle the maturity value of the loan in six years’ time. The future value of the fund (F) should be equal to the maturity value of the loan:
F6 = MV6 = US$6,861,431
As the deposits are to be made at the beginning of each period, this case is an annuity due. The future value formula of an annuity due is used:

Where:

  • Future value = US$6,861,431
  • Annual interest rate,
  • Investment period, years (year 3 to year 6)
  • Frequency,

iii) Comparing and contrasting annuity due and ordinary annuity:
Annuity due and ordinary annuity are both forms of a series of equal cash flows. They are similar in the amount of cash flows in the series and the time interval between the cash flows. In both cases, the cash flows are equal in size and occur within the same time interval (e.g., monthly, quarterly).

The difference between the two lies in the timing of the cash flows:

  • Annuity Due: Cash flows occur at the beginning of each period.
  • Ordinary Annuity: Cash flows occur at the end of each period.

(Marks allocation: 3 marks)

b)

i) Justify the type of option Klessy should buy:
A call option gives the holder the right to buy the underlying asset at an agreed price before or on the expiration date. In contrast, a put option gives the holder the right to sell the underlying asset at an agreed price before or on the expiration date. Klessy should buy call options on the Australian dollar. Klessy needs to buy the Australian dollar next month to settle the invoice value of the consignment of barley bought. Thus, it needs to buy call options to get the right to buy the Australian dollar at the quoted strike price.

(Marks allocation: 3 marks)

ii) Computation and interpretation of the intrinsic value of the call option:
The intrinsic value of a call option is the maximum of the spot price minus the strike price and zero:

The intrinsic value of the call option is zero, meaning the call option is currently out-of-the-money and will not be profitable to exercise now.

(Marks allocation: 4 marks)

iii) The distinction between option premium and option strike price:

  • Option Premium: The price paid for the right to trade the underlying asset at a given strike price before or on a specified expiration date. For example, Klessy would have to pay US$0.0131 to obtain the right to buy one Australian dollar.
  • Option Strike Price: The agreed price for the underlying asset. If Klessy buys the call option and elects to exercise the option, it will pay US$0.7275 for every Australian dollar in the contract. The premium is payable when the option contract is bought, while the strike price is payable later when the option is exercised, and the asset is traded.

(Marks allocation: 3 marks)

  • 4 Marks

FM – March 2023 – L2 – Q3b – Simple interest and compound interest

Determine the length of time required to double an investment of GH¢100,000 at an annual interest rate of 15%.

Trimpo Ltd can invest funds in Ghana’s financial market at 15% per annum. Presently, the Treasury Manager of the company is considering investing GH¢100,000.

Required:
Determine the length of time it will take to double the investment (round your answer to the nearest year). (4 marks)

Using the compound interest formula:

Where:

  • (Future Value, since the investment doubles)
  • (Initial Investment)
  • (Annual interest rate)
  • is the number of years to double the investment.

From the future value interest factor table for a single amount, it can be seen that
it takes about five years for a 15% annual interest rate to compound to a FVIF of 2

Rounded to the nearest year: It will take approximately 5 years to double the investment.

Alternative Approach – Rule of 72:

Using the Rule of 72, which is a simplified formula to estimate the number of years required to double an investment at a fixed annual rate of interest:

Rounded to the nearest year:
It will take approximately 5 years to double the investment.

(Marks allocation: Computation using compound interest formula = 2 marks; Final answer = 2 marks)

 

  • 10 Marks

FM – April 2022 – L2 – Q3a – Simple interest and compound interest

Compute the terminal value of investments in Treasury bills and fixed deposits over a five-year period and analyze the investment strategy.

An online university is setting up an endowment fund for the financing of scholarship grants. A total of GH¢200 million has been raised through fundraising events. This amount will be invested continuously for 5 years before disbursements will be made from the fund. The Trustees of the Endowment Fund have received a tentative investment strategy from the appointed Investment Manager.

Below is an extract from the tentative investment strategy:

“The seed money will be invested in fixed-income securities and negotiated short-term investments to secure the protection of the principal while earning stable returns over the 5-year gestation period. To achieve this objective, the seed money will be invested as follows: 60% in Government of Ghana 91-day Treasury Bills, 40% in 6-month fixed deposit accounts with top-class universal banks in Ghana. Over the 5-year gestation period, the maturity value of each round of investment will be rolled over as they mature.”

Being the only Trustee with expertise in finance, your fellow Trustees have asked you to do some simulations to inform them about the growth of the fund in the gestation period based on the tentative investment strategy.

Required:
i) Suppose the annual nominal interest rate on the Government of Ghana 91-day Treasury bills will be 15.2514% in year 1, 15.4814% in year 2, 15.7565% in year 3, 15.9478% in year 4, and 16.2146% in year 5. Compute the terminal value of that component of the investment at the end of the fifth year. (5 marks)
ii) Suppose the average nominal interest rate on the fixed deposits will be 16.5% over the next five years. Compute the terminal value of that component of the investment at the end of the fifth year. (3 marks)
iii) Considering the proposed strategy that the maturity value of each round of investment is rolled over as they mature, explain whether the interest that would accrue on the investment over the entire investment period would effectively be a simple interest. (2 marks)

i) The terminal value of the investment in the 91-day GoG Treasury bills:
Approximating the 91-day investment holding period to a quarter of a year, it can be concluded that the maturity value of the allocation to the 91-day GoG Treasury bills will be reinvested at the end of every quarter in each of the five years. Effectively, the allocated amount will be compounded quarterly over the next five years. The terminal value at the end of the fifth year may be calculated as under:

(Marks allocation: Amount allocated = 1; Computation of future value = 3; Final answer = 1)

ii) The terminal value of the investment in the 6-month fixed deposit:
The maturity value of the allocation to the fixed deposits will be reinvested at the end of every half in each of the five years. Effectively, the allocated amount will be compounded semi-annually over the next five years. The terminal value at the end of the fifth year may be calculated as under:

(Marks allocation: Computation of future value = 2 marks; Final answer = 1 mark)

iii) No. If the maturity value is reinvested, then it is both the principal and interest that are reinvested. Effectively, interest is earned on both principal and interest. In the case of simple interest, interest is earned only on the principal.

(Marks allocation: Conclusion = 1 mark; Explanation = 1 mark)

  • 4 Marks

FM – Nov 2019 – L2 – Q3a – Simple interest and compound interest

Calculate the balance of an endowment fund at the end of three years with varying annual interest rates.

Joy Mummy Ltd is establishing an endowment fund to finance a scholarship scheme to provide funding for the education of children of its employees. The company plans to make an initial deposit of GH¢500,000 into the fund now. The initial deposit will be invested for three years before any disbursements will be made from the fund. The effective annual rate of return on the fund is expected to be 14% in the first year, 15% in the second year, and 16.5% in the third year.

Required:
Compute the balance of the fund at the end of three years. (4 marks)

The future value of the endowment fund at the end of three years can be calculated using the formula for compound interest:
FV3 = P0(1+i1) (1+i2) (1+i3)
Where:

  • P0 = GH¢500,000 (Initial deposit)
  • i1 = 14% (Year 1 interest rate)
  • i2 = 15% (Year 2 interest rate)
  • i3 = 16.5% (Year 3 interest rate)

Calculation:

FV3 = GH¢500,000 × (1.14 × 1.15 × 1.165) 

FV3 = GH¢500,000 × 1.527315 = GH¢763,657.50 

Thus, the balance of the fund at the end of three years is GH¢763,657.50.

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