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 (or pay it). The reason for this is that, until you get it (or pay it), the money is earning money for someone else (or you). It is therefore worth more (or less) depending on when all this happens.
The Value of Future Cash Flows
Consider this example. For the moment, let's 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
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 for this. Essentially, it should represent the "opportunity cost of capital". To get to 8% I just sort of took an average of what you might get from equities (around 10 or 11%) and what you might get from gilts (around 5 or 6%). It may be that you should use a different figure depending on what you might plan to be doing with the money. For instance, if you would otherwise have put it into a gilt, then you might use 5%. If you might otherwise have put it into some racy technology stock, you might use 20% (although using any figure over 10% is pretty optimistic by historical standards).
Whatever figure we use, with this approach we can work out the value to us now (the "present value") of any amount of money that we might receive or have to pay at any time in the future. For instance, if we were offered a deal which involved us having to pay out £100 in three years' time and receive £500 in 12 years' time, how much would we 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 us is £198.56 - £79.38, which comes to £119.18. We would therefore pay up to £119.18 to get this deal now.
Internal Rates of Return
Now, let's think of this £119.18 that we're paying as an additional cash outflow. We now have 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) £50/(1+r)
2 £80.00 (1+r)^2 £50/(1+r)^2
3 £80.00 (1+r)^3 £50/(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, I think 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. I'd say 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, I suppose, but it does make it sound rather more scary than it is.
Last year, in a couple of articles called "Grappling With Gilts", I introduced a thing called the redemption yield. This is effectively the implied rate of interest on any gilt. I have, of course, already referred to it a few times in this article. How do you think it is worked out? Yep, you've got it, it's the IRR (sounds good doesn't it). Say you pay £107.63 for a gilt with a "coupon" of 8%. The £107.63 is the negative cash flow now. The positive cash flows are the half-yearly payments of £4 and the final £100 that you get back at the end (the "redemption"). The interest rate which makes the present value of these cash flows add up to £0 is the IRR of the Gilt and is called the "yield to redemption" or "redemption yield". It's as simple as that, although I don't fancy doing the sums much.
Questions and comments on this article should be addressed to the Personal Finance message board.
Related Links
• Investment Strategies Message Board -- Discount Rates
• Personal Finance -- Coping With Compound Interest
• Personal Finance -- Grappling With Gilts, Part One
• Personal Finance -- Grappling With Gilts, Part Two
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