Financial mathematics

1 Introduction

This chapter is the only substantial piece of arithmetic in BA1, and both of its syllabus outcomes are level 3 — you are asked to calculate, not to explain. It covers two things: what happens to a sum of money that is left to earn interest, and the reverse question, which is what a sum of money receivable in the future is worth today.

The second of those is discounting, and it is the foundation of investment appraisal. Almost every question in this area is won by setting the cash flows out in a table, in time order, with a factor against each one — so learn that layout now and use it every time, even when the numbers look easy enough to do in your head.

2 Interest

YouTube videoLoad YouTube video

A sum of money invested or borrowed is known as the principal. When money is invested it earns interest; when money is borrowed, interest is payable. (Note that interest is not permitted in Islamic finance.)

2.1 Simple interest

With simple interest, interest is received or paid each year but is not added to the principal, so it does not itself earn interest. Every year earns the same amount of interest as the first.

$600 invested at 10% simple interest grows like this:

2.2 Compound interest

With compound interest, the interest is added to the principal each year and the following year’s interest is calculated on the new total. Interest earns interest.

The same $600 at 10% compound:

$798.60 against $780 on simple interest — and the gap widens fast as the term lengthens. Building the table is fine for three years and hopeless for twenty, so use the formula:

So $600 for 3 years at 10% gives 600 × 1.1³ = $798.60, and $500 for 8 years at 4.5% gives 500 × 1.045⁸ = 500 × 1.4221 = $711.05.

3 Nominal and effective rates: APR and AER

So far interest has been calculated once a year. In practice it is often calculated monthly, quarterly or even daily, and that changes the answer.

Suppose you save $1,000 in an account paying 2% every quarter. The rate r = 0.02 is measured in quarters, so the period n must be measured in quarters too. One year is four quarters:

1,000 × (1 + 0.02)⁴ = $1,082.43

Interest of $82.43 has been added to a principal of $1,000, so the effective interest rate is 8.243% a year. The effective rate is also called the annual equivalent rate (AER) or the annual percentage rate (APR).

But a rate of 2% a quarter would usually be quoted as 8% a year. That 8% is the nominal rate, and it is not the rate actually charged. Nominal means here what it means in ordinary English: in name only. If somebody is described as the nominal head of an organisation, they are not really running it — and a nominal 8% is not really 8%.

4 Discounting and present value

YouTube videoLoad YouTube video

Here is a simple choice. Would you rather have $1,000 now, or $1,000 in one year’s time?

Most people would take it now, because it is safer, because it can be enjoyed now, and because — even if it is not needed now — it can be invested for a year.

That last reason is the one that can be measured. If the money could be placed on deposit at 6%, the two options really are:

So earlier cash flows are worth more than later ones. This is called the time value of money.

Rather than projecting everything forward to a terminal value, it is more useful to bring everything back to the present. For option 2, how much would have to be received today to be worth the same as $1,000 in a year? If p is received now and invested at 6%:

p × 1.06 = 1,000

p = 1,000 ÷ 1.06 = $943.40

$943.40 is the present value of $1,000 receivable in one year at a discount rate of 6%. Check it: 943.40 × 1.06 = $1,000.

So the original choice can be restated as: would you rather have $1,000 now, or $943.40 now? Put that way it answers itself.

5 Discount factors and discount tables

The examples above should make the general rule clear.

So in Example 6(a), the present value of $4,000 received in 4 years at a discount rate of 7% is:

4,000 × 1 ÷ (1.07)⁴ = 4,000 × 0.763 = $3,052

and 0.763 is the discount factor for 4 years at 7%.

Rather than working every factor out by hand, use the table below. It is the same table you are given in the exam. Read the rate across the top and the number of years down the side: at 7% and 4 years the factor is 0.763, exactly as calculated.

5.1 Annuities

An annuity is an equal cash flow occurring every year for a number of years — a rent, a lease payment, a constant stream of savings. Annuities are so common in investment appraisal that there is a shortcut for them.

If $1,500 has to be paid after one, two and three years, and the discount rate is 9%, the long way is:

Because the cash flow is the same every year, the three factors can be added first and the total applied once. That is written as:

The added-up factors are themselves tabulated, in what is called an annuity table or cumulative discount table. At 9% for three years the cumulative factor is 2.531, as calculated above.

Use it whenever you can. BA1 is assessed entirely by objective test, and a candidate who writes out five separate rows where one cumulative factor would do is not only spending time they have not got — every extra working is another chance to make the arithmetical slip that loses the whole mark.

5.2 Adjusting the factor for a non-standard pattern

If $200 is to be received in years 4 to 10 with a 5% discount rate, take the factor for the whole run and subtract the factor for the years you do not want:

Present value = $200 × 4.999 = $999.80

The commonest error in the whole chapter is to subtract the factor for years 1 – 4. That removes year 4 as well and leaves a factor for years 5 – 10. To keep year 4, remove years 1 – 3.

Two other patterns come up constantly:

6 The cumulative factor formula and perpetuities

The formula printed at the top of the annuity table is:

cumulative discount factor = (1 − (1 + r)⁻ⁿ) ÷ r

which some people prefer to write as:

cumulative discount factor = 1 ÷ r × (1 − 1 ÷ (1 + r)ⁿ)

Reproducing a figure that is already in the table is a pointless exercise in itself, but the formula is needed when the rate or the number of periods is not on the table.

An annuity is an annual cash flow. As it gets longer, it comes closer and closer to a perpetuity — an equal cash flow for ever. No real set of flows lasts for ever, but after twenty years or so, at ordinary discount rates, the factors are so small that later flows barely matter: the 20-year 10% factor is only 0.149. So a long stream can conveniently be treated as a perpetuity.

Look at what happens to the cumulative factor formula as n becomes very large. The term (1 + r)⁻ⁿ becomes vanishingly small, so (1 − (1 + r)⁻ⁿ) tends to 1 and the whole formula collapses to:

As with annuities, a perpetuity factor from the formula assumes the flows start at time 1 and occur every year. Any other pattern has to be adjusted the same way as in §5.2 — and Examples 15 and 16 below both turn on exactly that.

7 Net present value

Cash flows arising at different times cannot be compared directly, so for comparison they are all discounted to their present values.

In investment appraisal the outflows (usually the initial investment) and the inflows (usually future income, and the sale of the asset at the end of its life) are all discounted. Outflows are negative, inflows are positive, and the total is the net present value (NPV).

8 Choosing which cash flows go in the table

An NPV is only as good as the figures put into it, and the figures have to be relevant cash flows: future, incremental movements of cash caused by the decision. Five rules follow, and questions in this area are usually testing one of them rather than the discounting.

9 Internal rate of return (IRR)

The IRR will therefore never tell you anything about accepting or rejecting a project that the NPV has not already told you. It is worth knowing because it is examined, and because a single percentage is easy to talk about.

Estimating it means working out the NPV of the project at two discount rates. There is nothing special about the two you pick — 5% and 15%, 10% and 15%, 10% and 20% are all common — although it is convenient if one gives a positive NPV and the other a negative one, because then you know the IRR lies between them.

Suppose:

At a low discount rate the future inflows stay large after discounting, so the NPV tends to be positive. At a high one they are cut down hard, so the NPV tends to be negative. Somewhere in between the NPV passes through zero, and that is the IRR.

Plotted, the two results look like this. The true relationship is a curve, but the estimate assumes a straight line between the two points, which is why the IRR obtained this way is an estimate and not an exact figure.

The line crosses zero a little over 9%. To estimate where, use similar triangles. Call the point on the axis at 5% A, the point vertically above it at NPV $1,000 B, the point on the axis at 15% C and the point vertically below it at NPV −$1,300 D. F is where the straight line BD crosses the axis. The triangles ABF and CDF are similar — each has a right angle on the axis, and the angles at F are equal — so their sides are in the same proportion:

AF ÷ CF = AB ÷ CD = 1,000 ÷ 1,300

AF + CF = AC = 15 − 5 = 10, so AF = 10 × 1,000 ÷ (1,000 + 1,300)

AF = 10,000 ÷ 2,300 = 4.3

So F is at 5 + 4.3 = 9.3%, and that is the estimate of the IRR. It is an estimate, not the exact rate, because the true NPV curve is not a straight line.

If you would rather not sketch the triangles, the same result comes from a formula — which is not given in the exam, so it has to be learned:

In the example above A = 5, B = 15, N_A = 1,000 and N_B = −1,300, so:

IRR = 5 + [1,000 ÷ (1,000 − (−1,300))] × (15 − 5) = 5 + (1,000 ÷ 2,300) × 10 = 9.3%

Note that had the NPVs been calculated at, say, 4% and 16% instead, the estimate would come out slightly different. That is a property of the straight-line assumption, not a mistake.

10 Tests

Work each one before reading the answer. Every one of them is a table, even the ones that look as though they are not.

11 Test your knowledge

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