Before we get down to the maths, let me digress to explain why there is a still from BBC’s motoring magazine Top Gear at the top of this page. In the last few series of the show, I have noticed an interesting tendency for BBC to include attractive women in the camera shot whilst the presenters are sitting in the foreground. The image above shows a prime example of this.

Whether this is a conscious attempt at subliminal messaging targeted at the 18-35 male demographic that are the primary viewers of the show, or just a result of them putting shorter people towards the front, I don’t know. Still, I would like to see whether there is some correlation between the female effect and viewing figures for the programme.

Observation over. Let’s get down to business.

What do I need present discounted values for?

Let’s explain what the term actually means first. A present value the is value of something now. The value of a pen may well just be its sticker price, for example. But what if you have something that gives you £100 (like a bond) every year for 20 years? What is the value of that? It is not £100 x 20 = £2000 because of interest rates.

Hence, the present discounted value (PDV) tells us how much something is worth in total to us now, discounting interest (or anything else) gained. Of course, we don’t know what the interest rates are going to be in future, so we estimate them (we work out the expected future interest rates).

Okay, give me an example then.

Let’s say you have £100. Let us presume that the annual interest rate of your savings account is not the lowly 0.5% of today, but a nice 5%. How much will your £100 be worth in a year? The answer is 100 x 1.05 = £105.

But say now you will get a £100 gift exactly one year from today. How much would that money be worth to you today? In essence, we are now working backwards, and so we must divide by the interest rate rather than multiplying by it. 100/1.05 = £95.24. This means that if we had £95.24 today, it would be worth £100 a year from now (as long as we saved it and not spent it all on overpriced perfume).

£95.24 is therefore our PDV of £100 next year.

This is quite useful in estimating how much money we need to save to reach some figure in a number of years. Suppose we are saving to buy a car in 10 years, and we need £20,000 for it.

Let iy be the interest rate in year y – so that for example, i7 is the interest rate in year 7. Also, note that i is the decimal form of the interest rate e.g. 5% = 0.05. Then the PDV would be worked out as follows:

PDV = 20000/[(1+i1)(1+i2)(1+i3) (1+i4)(1+i5)(1+i6)(1+i7) (1+i8) (1+i9)(1+i10)]

We know what i1 is because it is the interest rate now. But we don’t know the future interest rates, so we need to estimate them. However, to simplify matters, let’s assume that the interest rate is going to be the same for the next 10 years. Then:

PDV = 20000/[(1+i)(1+i)(1+i)(1+i)(1+i)(1+i)(1+i)(1+i)(1+i)(1+i)]

= 20000/[(1+i)10]

So if the interest rate was at a constant 5%, I would need 20000/1.0510 = £12,278.27 in my savings account to buy that Skoda Superb of my dreams in 10 years.

So what if I have a stream of income per year?

Simples. Suppose your working life consists of n years. You get paid £Yt annually for year t. The real interest rate (which is the actual interest rate minus the inflation rate) is rt for the year t.

Then the formula for the PDV of your lifetime income is:

Y0 + Y1 /(1+r1) + Y2 /[(1+r1)(1+r2)] + … + Yn /[(1+r1)(1+r2)… (1+rn)]

That’s what your life looks like. Scary, huh?

To help you understand, let’s take a simplified example where we know all the values. Say you are a young sprightly fellow like myself of age 22. You estimate that you’ve got about 40 years of working life left, providing you don’t die beforehand or stop working. Your employer is mean and denies the existence of inflation, so you will get paid the same amount of money per year for 40 years – say a tidy £40,000.

The Bank of England, meanwhile, is having none of this monetary policy nonsense, and so announce that they will keep interest rates at 5% for the next 40 years. Luckily, you can predict the future, and you know that inflation will be 2.5%, so the real interest rate is just 2.5% for the next 40 years (don’t you wish life was this simple?).

First, let us simplify the formula above. We know that income is fixed, so we just need one fixed value for Y. Similarly, the real interest rate is also fixed, so we have one value for r. Then the formula above simplifies to:

Y + Y/(1+r) + Y/[(1+r)2] + … + Y/[(1+r)n]

Now, we can take out Y, as it is common to all the terms of the expression:

Y{ 1 + 1/(1+r) + 1/[(1+r)2] + … + 1/[(1+r)n] }

Here’s where we can use a swishy trick. The bit inside the curly brackets is called a geometric series. It simplifies to something very nice (although I won’t explain it here for fear of giving you maths overload), making the formula end up as this:

(I had to use equation editor to put it on more than one line, otherwise I probably would have confused myself as well as you.)

Now we have a nice ‘neat’ formula to work with, we can put in all our figures. Y=40000, r=0.025 and n=40.

So 1/(1+r) = 1/1.025 = 0.9756097561. Then 1 – 0.9756097561 = 0.0243902439

1/(1+r)n+1 = 1/1.02541 = 0.36334695. Then 1 – 0.36334695 = 0.63665305

Hence, the PDV of your (real) lifetime earnings is:

40000 x (0.63665305/0.0243902439) = £1,044,111

Congratulations! With your new found knowledge, you can act rationally by calculating things in advance to work out the optimal outcome for you, such as which mortgage works out the cheapest. And you can also rest assured that Cristiano Ronaldo will probably earn more in a couple of months than you will in your lifetime. Ahh, the joys of mathematics.