FOOL SCHOOL
The Time Value Of Money

January 27, 2003

Have you heard of the "time value of money"? In a nutshell, it means that the value of a particular amount of money depends on when you get it. The reason for this is that, until you get it the money is earning money for someone else. It is therefore worth more depending on when all this happens.

The Value of Future Cash Flows

Consider this example. For the moment, we'll assume that all money earns interest at 8% per annum and costs 8% per annum to borrow. If I have £100 now, what will it be worth in 10 years' time? The answer is:

£100 x 1.08^10 = £215.89   (^10 denotes to the power of 10)

Now, if someone offers to give you £215.89 in 10 years' time, how much will you pay them now to make them do it? The answer goes like this. The money you pay now is either money that won't be earning you 8% for the next 10 years, or it's money that you've borrowed and have to pay interest on at 8% for the next ten years. Either way, paying out money now costs you 8% per annum until you get it back. So, to buy a cash flow of £215.89 in 10 years' time, we'd pay up to £100 because, if we'd kept the £100 (or not borrowed it), we'd have turned it into £215.89 over 10 years (or saved ourselves that much).

Remember that if you lend someone money, then they have borrowed it. In fact if you put capital into any enterprise, then that enterprise has raised it from you. This is why, in theory anyway, we think of money getting the same return from being invested as it costs to borrow it. This is the time value of money.

Why 8%, though? Good question. It was nothing more than a stab in the dark really. People will argue until the cows come home about what figure to use. Essentially, it should represent the "opportunity cost of capital". Ideally, you'd come up with a different figure depending on what you might plan to be doing with the money. If you would have put it into a gilt, then you might use 5%. If you planned to use it on some exciting business enterprise, then you might use 20% (although any figure over 10% is pretty optimistic by most standards).

Whatever figure you use, using this approach, you can work out the value to you now (the 'present value') of any amount of money that you might receive or have to pay at any time in the future. For instance, if you were offered a deal which involved paying out £100 in three years' time and receiving £500 in 12 years' time, how much would you be prepared to pay to take it? To get the answer, you just add up the present value of the two cash flows (remember that the payment of £100 is a cash "outflow" and is therefore negative). So:

If PV is the present value of £500 in 12 years' time:

£500 = PV x 1.08^12

dividing both sides by 1.08^12, we get:

PV = £500 / 1.08^12 = £198.56

If PV is the present value of a £100 outflow in 3 years' time:

-£100 = PV x 1.08^3

dividing both sides by 1.08^3, we get:

PV = -£100 / 1.08^3 = —£79.38

So, the present value of that deal to you is £198.56 - £79.38, which comes to £119.18. You would therefore pay up to £119.18 to enter the deal now.

Internal Rates of Return

Now think of the £119.18 that you're paying out now as an additional cash outflow.That makes three cash flows. The following table sets them out. "Discount factor" means the number that you'd divide by to get the present value.

Year    Cash      Discount     Present
       Flow(£)     Factor      Value(£)

 0    -£119.18    1           -£119.18
 3    -£100.00    1.08^3       -£79.38
12     £500.00    1.08^12      £198.56
                               -------
Total                             0.00

The present value of all these cash flows is 0. Don't tell me you're surprised by this! We worked out the most we could pay (£119.18) to break even for these future cash flows, paid it and, unsurprisingly, we broke even. The crucial fact is that we break even for these cash flows only if we assume a "discount rate" of 8%. If we picked 7%, then we'd have made £21.20, because the sum is dominated by that future inflow of £500 which is discounted less and therefore worth more. If we picked 9%, then we'd have lost £18.63, because that future inflow of £500 is discounted more and therefore worth less.

This tells us all sorts of things about money. One thing it tells us, for instance, is that if the 'discount rate' falls, then the things with distant future cash flows are worth relatively more. So, when the yield on long-dated gilts (which basically sets the parameters for our discount rate) falls, then "growth" shares, with their distant cash flows, become relatively more attractive. This was very much a phenomenon of the 1990s. When the yield on long-dated gilts rises, then "value" shares become relatively more attractive because money in the here and now has become relatively more important. Gilt yields peaked at around 17% in 1974. In those dark days, the pound in your pocket was pretty important.

Perhaps the most useful thing about this time value thing is that we can flip it on its head. Assuming we know what all the cash flows are, it lets us work out what the discount rate needs to be for us to break even. This basically tells you what your rate of return is from any particular investment. Imagine that there is a product which is on sale for £199. It offers to give you £80 per year for the next three years. We can set the cash flows out in the following table:

Year    Cash      Discount     Present
       Flow(£)     Factor      Value(£)

 0   - £199.00     1         - £199.00
 1      £80.00    (1+r)        £80/(1+r)
 2      £80.00    (1+r)^2      £80/(1+r)^2
 3      £80.00    (1+r)^3      £80/(1+r)^3
                               -------
Total                             0.00

where "r" represents our rate of return.

This translates into the following sum:

            80      80       80
0 = -199 + ---- + ------ + ------
           1+r   (1+r)^2  (1+r)^3

You can work out that r equals about 9.98%. This means that if you spend £199 on the product, you will be getting an annual return of 9.98%. I worked it out by trial and error. I started by trying 8%, then 9%, then 9.7%, then 9.9%, then 9.93%. It was only at this point that I finally put in 10% and found that it was too much. There is a technique for this trial and error stuff and it seems I haven't got it. Anyway, these sums can become pretty infuriating. Especially if you have lots of different cash flows at lots of awkward times. Really you need some sort of spreadsheet -- in fact, Excel has an automatic tool for it.

Doing the sums isn't really the important bit, though. The important thing is understanding how they are done. What we have just done is the most fundamental thing in finance. It essentially determines the rate of return on any investment or the cost of any borrowing. Any financial transaction can (and probably should) be considered in this way. That makes it pretty damn useful. The financial whizzes call it the "Internal Rate of Return". If they're really showing off, they'll call it "IRR" for short. It's a good enough way to describe it but it does make it sound rather more scary than it is.