Interest

CIMA-BA2 · Chapter 19

Interest

Money has a time value: $1 today is worth more than $1 in the future, because today’s $1 can be invested to earn interest. Compounding adds interest to find a future value; discounting removes interest to find the present value of a future amount.

This chapter covers simple and compound interest, effective annual rates (APR), present values using discount factors, annuities (the same amount every year) and perpetuities (the same amount every year for ever) – the tools used for investment appraisal in the next chapter.

1 Introduction

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The purpose of this chapter and the next chapter is to consider a key area for management accountants – the appraisal of capital investments.

In this chapter we will look at interest on capital and continue in the next chapter with the use of these techniques in investment appraisal.

The time value of money

Would you rather receive $1,000 today or $1,000 in a year’s time? Today – because money received now can be invested to earn interest. At 10% a year, $1,000 today will have grown to $1,100 in a year, so $1,000 today is worth more than $1,000 in a year. Money received later is also exposed to risk (it might not arrive) and to inflation (it may buy less). This is the time value of money.

So cash flows that arise at different times cannot simply be added together. To compare them, they must be expressed at the same point in time – either by adding interest to find their value in the future, or by removing interest to find their value today (their present value).

2 Simple interest

A sum of money invested or borrowed is known as the principal.

When money is invested it earns interest; similarly when money is borrowed, interest is payable.

With simple interest, the interest is receivable or payable each year, but is not added to the principal.

Simple interest for the year = principal × rate of interest, so over several years the interest is principal × rate × number of years.

A man invests $200 on 1 January each year. On 31 December each year simple interest is credited at 15% but this interest is put in a separate account and does not itself earn interest.

Find the total amount standing to his credit on 31 December following his fourth payment of $200.

	Capital account ($)	Interest account ($)
Payment – 1 Jan year 1	200
Interest – 31 Dec year 1 (15% × 200)		30
Payment – 1 Jan year 2	200
Balance	400
Interest – 31 Dec year 2 (15% × 400)		60
Payment – 1 Jan year 3	200
Balance	600
Interest – 31 Dec year 3 (15% × 600)		90
Payment – 1 Jan year 4	200
Balance	800
Interest – 31 Dec year 4 (15% × 800)		120
Totals	800	300

Total amount standing to his credit = $800 + $300 = $1,100

3 Compound interest

With compound interest the interest is added each year to the principal and in the following year the interest is calculated on the total.

A man invests $500 now for 3 years with interest at 10% p.a.

How much will be in his account after 3 years?

	$
Payment now	500.00
Year 1 interest (10% × 500)	50.00
	550.00
Year 2 interest (10% × 550)	55.00
	605.00
Year 3 interest (10% × 605)	60.50
Amount after 3 years	665.50

Or: $500 × 1.1³ = $665.50. Adding 10% is the same as multiplying by 1.1, so three years of interest means multiplying by 1.1 three times.

The amount (A) at the end of the n’th year is given by:

A = P(1 + r)ⁿ

where P is the principal, r the rate of interest per year (as a decimal: 10% = 0.10) and n the number of years.

This is also known as the future value (or terminal value)

A man invests $800 at 6%p.a. for 5 years.

How much will be in his account at the end of 5 years?

A = P(1 + r)ⁿ = 800 × 1.06⁵ = $1,070.58

4 Effective Rate

For simplicity, the previous compound interest examples have assumed that interest is calculated only once a year.

However in practice interest may be calculated on a monthly or even daily basis. The same formula can still be used, but we need to distinguish between the nominal and annual percentage rates.

There are usually two rates quoted by financial institutions. The first is the nominal rate and the other, the rate actually earned, is known as the effective or the annual percentage rate (APR).

A credit card company charges a nominal rate of 2% per month.

If a customer has purchased $100 worth of goods on her credit card, calculate the amount she will owe after one year, and also the annual percentage rate (APR)

Interest of 2% is added every month, so the amount owed is multiplied by 1.02 twelve times.

Amount owed after 12 months = P(1 + r)ⁿ = 100 × 1.02¹² = $126.82

APR = actual interest over the year ÷ amount borrowed × 100% = 26.82 ÷ 100 × 100% = 26.82%

Simply multiplying 2% by 12 (24%) understates the true annual rate, because each month’s interest itself earns interest in later months.

In general, if interest is charged at a rate r per month, the effective annual rate (APR) = (1 + r)¹² − 1.

5 Discounting

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In the previous examples we calculated the future value of cash flows by adding on (or compounding) the interest.

We can do the same exercise in reverse to calculate the amount now that is equivalent to future flows, by removing interest.

This exercise is known as discounting and the equivalent amount is known as the present value.

What amount now is equivalent to $800 in 4 years’ time, with interest at 10% p.a.?

$x now will become $x × 1.10⁴ in 4 years. Therefore x × 1.10⁴ = 800, so

x = 800 ÷ 1.10⁴ = 800 ÷ 1.4641 = $546.41

Using the present value table (10%, 4 years): 800 × 0.683 = $546 – the table factors are rounded to three decimal places, so the answer can differ slightly.

The formula for this is:

P = A ÷ (1 + r)ⁿ

where A is the amount receivable in n years’ time and P is its present value. Tables are provided in the examination which give the discount factors 1 ÷ (1 + r)ⁿ at different rates of interest for different numbers of years, so the present value is simply the future amount × the discount factor.

The higher the rate of interest, or the further away the cash flow, the smaller its present value.

Present value table (extract)

Discount factor 1 ÷ (1 + r)ⁿ for an amount receivable or payable at the end of year n. The full table is given in the exam.

Year (n)	8%	10%	12%	13%
1	0.926	0.909	0.893	0.885
2	0.857	0.826	0.797	0.783
3	0.794	0.751	0.712	0.693
4	0.735	0.683	0.636	0.613
5	0.681	0.621	0.567	0.543
6	0.630	0.564	0.507	0.480
7	0.583	0.513	0.452	0.425
8	0.540	0.467	0.404	0.376
9	0.500	0.424	0.361	0.333
10	0.463	0.386	0.322	0.295
11	0.429	0.350	0.287	0.261
12	0.397	0.319	0.257	0.231

What is the present value of $1,200 receivable in 12 years’ time, with interest at 13%?

P.V. = 1,200 ÷ 1.13¹² = $277

or, using tables (13%, 12 years): P.V. = 1,200 × 0.231 = $277

$277 invested now at 13% would grow to $1,200 in 12 years.

6 Annuities

An annuity is a regular payment of the same amount each year.

Each payment could be discounted separately and the results added up, but because annuities are so common there are tables of annuity factors: the total of the discount factors for years 1 to n. The present value of an annuity = annual amount × annuity factor.

The present value of an annuity is given by the formula:

P = A × [1 − 1 ÷ (1 + r)ⁿ] ÷ r

but again, tables are provided for this in the examination.

Annuity table (extract)

Present value of $1 a year receivable or payable at the end of each of years 1 to n. The full table is given in the exam.

Years (n)	8%	10%	12%
1	0.926	0.909	0.893
2	1.783	1.736	1.690
3	2.577	2.487	2.402
4	3.312	3.170	3.037
5	3.993	3.791	3.605
6	4.623	4.355	4.111
7	5.206	4.868	4.564
8	5.747	5.335	4.968
9	6.247	5.759	5.328
10	6.710	6.145	5.650
11	7.139	6.495	5.938
12	7.536	6.814	6.194

Interest rate is 12% p.a.

What is the present value of $500 receivable in 1 year’s time and thereafter every year for a total of 8 receipts?

$500 a year for years 1 to 8 is an annuity. Annuity factor (12%, 8 years) = 4.968 – the total of the eight discount factors 0.893 + 0.797 + 0.712 + 0.636 + 0.567 + 0.507 + 0.452 + 0.404.

Present value = 500 × 4.968 = $2,484

The annuity table always gives the total for years 1 to n. If an annuity starts later than year 1, take the factor for the last year and subtract the factor for the years before the first payment.

A man expects to receive $1,000 in each of 9 years, with the first receipt being in 4 years’ time.

What is the present value of the receipts if interest is 8% p.a.?

The receipts are in years 4 to 12 (nine receipts: 4, 5, 6 … 12).

Annuity factor at 8%
Years 1 – 12	7.536
Less: years 1 – 3	(2.577)
Years 4 – 12	4.959

Present value = 1,000 × 4.959 = $4,959

7 Perpetuities

A perpetuity is an annuity that is expected to continue for an indefinitely long period of time.

The present value of a perpetuity is given by the formula:

P = A ÷ r

where A is the amount receivable each year, starting in one year’s time, and r is the rate of interest. There are no tables for perpetuities.

Interest rate is 12% p.a.

What is the present value of $5,000 receivable in 1 year’s time and thereafter in perpetuity?

Present value = A ÷ r = 5,000 ÷ 0.12 = $41,667

Check: $41,667 invested at 12% earns 41,667 × 0.12 = $5,000 a year, for ever – so $41,667 now is worth the same as $5,000 a year in perpetuity.

Cash flow	Present value
A single amount in n years	Amount × discount factor (present value table)
The same amount each year for years 1 to n	Amount × annuity factor (annuity table)
The same amount each year starting later (e.g. years 4 to 12)	Amount × (annuity factor for the last year − annuity factor for the years before the first payment)
The same amount each year for ever, starting in one year	Amount ÷ r

8 Test your knowledge

Two quick checks before you move on: work through the flashcards to fix this chapter’s terms and definitions, then sit the objective questions for exam-style practice. Both mark themselves and explain the answers as you go.

Answer the questions one at a time. Your progress is saved so you can leave and come back.

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