Can you explain in simple terms why the internal rate of return is called “internal”?
The word “internal” indicates that the rate of return involved is determined entirely by the cash flows themselves, with no external influence. This distinguishes the IRR from other possible rates which could be calculated, and which involve assumptions about rates of interest not at all connected with the investment itself — ie external to it.
In order to fully appreciate the significance of this distinction, it is necessary to clarify the general meaning of “rate of return”, and to examine this in the context of different possible patterns of cash flows.
Meaning of “rate of return”
Generally, the “rate of return” on an investment is supposed to indicate how much annual income is generated per £100 of capital invested.
In come cases it is easy to see what the rate of return is, without having to do any significant calculations:
Example 1 Investment A requires an initial capital outlay of £1,000 and guarantees cash flows of £80 each, at the end of each year for five years, as well as the return of the £1,000 at the end of the five years.
It is clear in this case that the rate of return is 8%: £8 annual income is produced for every £100 invested, and the return of the capital outlay itself is dealt with as a separate issue.
Other patterns of cash flows may not reveal the rate of return so easily. Annual income may not be constant, may be accumulated into the form of a lump sum at the end, or may be combined with periodic repayments of capital. In such cases, various adjustments may be made to the actual cash flows received in order to facilitate a comparison with the simple pattern illustrated in Example 1. That pattern, of constant annual income payments free from returns of capital, may be regarded as a standard of comparison. The adjustments necessary to make this comparison in an actual example introduce the possibility of the use of external rates of interest. It is in the context of these more complex cases that “internal rate of return” has a distinctive role to play.
Example 2 Investment B requires an initial capital outlay of £700, and then guarantees payments of £200 each at the end of each of the next five years. These five payments of £200 each are all that is received by the investor.
In this case, of course, part of each cash flow of £200 is capital, and therefore the rate of return cannot be calculated as the ratio of these unadjusted cash flows, to the initial capital outlay.
The internal rate of return in this case is 13.2% (to two dp). We are not here concerned with the details of how this figure is arrived at, as this is adequately dealt with elsewhere, but we are very much concerned with what the figure means.
Referring to the general definition of “rate of return” given above, it should mean that the annual investment income (free of capital sums) is 13.2% of the initial capital outlay. That is, £92.41 pa for five years. The remaining balance of the £200 actually received each year, ie £200 — £92.41 = £107.59 is to be regarded as capital, and not as remunerative income. The five sums of £107.59 must be set aside each year to replace the original capital invested.
This interpretation of the IRR figure may be contrasted with the more usual one that it is the discount rate at which the present values of the cash flows exactly match the price paid. This is also perfectly correct, of course, but it is arguably not as practically useful to the investor as the interpretation given above. This interpretation also highlights a problem, which should be more widely recognised.
Recoupment of capital
Although the five annual sums of £107.59 in Example 2 are intended to replace the original capital outlay, it is easy to see that they will not do so unaided. 5 x £107.59 = £537.95, which falls well short of the £700 required.
Although this shortfall can be made up by reinvesting these annual capital sums as soon as received so as to attract compound interest, it is important to note that we are now going outside of what the investment itself provides, and relying on assistance from external sources of interest.
This represents a fundamental difference compared with Investment A (above). In that case the investment was quite self-contained in the sense that it provided both income and recovery of capital without external assistance. Investment B cannot do this. At least, not at 13.2% rate of return.
In order to accumulate to the required amount, viz £700, the annual capital sums must be reinvested at the internal (IRR) rate of 13.2%.
which confirms this, except for a slight rounding error)
This shows how the investor can obtain a remunerative income of 13.2% of capital invested, while maintaining his capital intact.
But it depends on the assumption that he will be able to re-invest the annual returns of capital, when received, at the IRR of 13.2%. There is no reason at all to suppose that this will be possible. It is true that Investment B itself offers the possibility of achieving this rate with sums invested now, but this does not mean that there will be similar opportunities available in one, two, three or four years’ time, when the need for re-investment will arise.
Perhaps we should be more cautious, and choose an interest rate for re-investment which, after due consideration, we feel can be relied on over the next few years. It should be emphasised that if we do this, we will be using an external figure — one which is quite independent of the cash flows of the investment itself. If we were doing this at the time of writing, it is unlikely that we would choose a rate any higher than 7%. As the following calculations show, the effect of this assumption is to reduce the rate of return shown by the investment to 11.18%, and to require the proviso that the figure is no longer a purely “internal” rate of return.
Assuming 7% re-investment rate:
Thus £121.72 must now be set aside each year to replace capital. This leaves a balance of £78.28 out of the £200 actually received each year, which can be regarded as remunerative income. Therefore,
Although this figure of 11.18% is not the IRR, it is at least arguable that it gives a truer indication of the return from the investment, when comparing it with others which may have a pattern of cash flows more like that shown by Investment A (as would, for example, short-dated gilt-edged securities).
Other patterns of cash flows may also invite the use of an “external” interest rate which is different from the “external” rate. One other case is illustrated by Example 3.
Example 3 Investment C requires a capital outlay of £1,000, and repays a single sum of £1,610.51 at the end of five years, and nothing more.
Here, it is not difficult to establish that the “rate of growth” (IRR) is 10% pa for the five years. The “income” is paid as a lump sum of £610.51 right at the end, simultaneously with return of the original capital.
Returning to the original general definition of rate of return given above, as the amount of annual income generated per £100 of capital invested, we would need to be satisfied that the lump sum of £610.51 is equivalent to an income of £100 pa for five years (10% of capital invested) if we are to accept that the deal represents a rate of return of 10% pa.
To decide whether they are equivalent, we must first ask ourselves the question: if we had an income of £100 pa for five years, how much of a lump sum could we accumulate with it, using whatever investment opportunities are generally available? This requires us (as with Example 2) to consider what interest rate, external to Investment C, we feel we could rely on obtaining with moneys to be placed over the next four years. Again, 7% seems realistic. At this rate, our £100 income would only accumulate to £1,402.55. The conclusion must be that in these circumstances, the lump sum which Investment C actually produces must in fact be equivalent to an annual income of more than £100 pa.
Calculation
Therefore the lump sum of £610.51 is really equivalent to an annual income of £106.16 pa or £10.62 per £100 of capital invested. Therefore rate of return = 10.62%.
These examples serve to illustrate that although there may be quite strong arguments in favour of using external interest rates in the calculation of rates of return, the IRR does not do so, relying on nothing other than the cash flows of the investment itself.