The Internal Rate of Return - IRR

by Michael Kelly

In the real world, interest rates are seldom fixed for long. There are also fees, charges and penalties to consider. As if compound interest wasn't complicated enough, we now have to consider a whole new set of variables.

Fortunately all is not lost as there is a very important and quite straightforward mathematical function called the Internal Rate of Return, or the IRR, which is basically like the true rate of interest but it can cope with varying interest rates and varying payments during the schedule as well as the occasional fee.

Take the £1,000 loan illustrated in Figure 12, which displays different interest rates each month. The repayments are re-calculated each month to amortise the loan from that point assuming the new rate is unchanged.

Now, let's guess what the interest rate would have to be if it was fixed, but where the monthly repayment schedule still stayed the same as in Figure 12, amortising the debt to zero after twelve months. It comes out to 1.2257% per month or 15.7414% true rate per annum. Figure 13, illustrates why.

The table shows the identical monthly repayment schedule as shown in Figure 12. But, the interest rate column is now a fixed rate. The only column that has changed is the theoretical monthly debt, which differs because the monthly interest rate is different, although it still amortises to zero. The borrower makes the same payments and still repays the loan, but the interest rate is fixed instead of variable.

This fixed interest rate is called the Internal Rate of Return, and is expressed as a true interest rate, usually annually. It is a very powerful tool when used to compare two different schedules, one of which might be more favourable than the other.

The IRR is a single number that represents the whole payment schedule, regardless of variations caused by numerous actual interest rate changes. The IRR looks only at the payment schedule, and ignores the variable interest rates that produced it. So this one rate can represent many.

You will need to use a spreadsheet to calculate the IRR or use a specially written program. There are two ways of calculating it. If you have laid out the schedule as in the third column of Figure 13, you can use the spreadsheet function IRR (range, guess) to produce the answer. A more general approach is to use the Goal seek function which will calculate the rate required to amortise any given schedule, even if the schedule is defined by formulae rather than having to lay out the whole schedule - difficult for a 25 year mortgage with 300 different monthly payments. The techniques are easier to see with the sample spreadsheets in front of you.

It is important to use a sign convention when working with IRRs. Cash coming out could be negative with cash going in as positive - there must be at least one positive and negative figure in the range. Note the initial loan is shown as a negative figure whereas the repayments are positive.

Adding Fees and Charges

In summary, the IRR measures the fixed interest rate required to amortise any given loan schedule, regardless of how that schedule was derived. It is now really quite easy to add in any additional numbers in the schedule, such as fees and charges, at the appropriate point in time.

Take the example in Figure 12 and 13. Suppose the lender charged a £50 up-front fee. The starting cash payment was originally minus (-) £1,000 to reflect the initial £1,000 loan being received by you (remember the sign convention). We now simply change this to -£950 to take account of the £50 fee. We do not change the scheduled repayments in any way. In other words, of the £1,000 you applied for, you had to give the lender £50 so you only benefited from £950. But the actual repayment schedule is still calculated as if you borrowed £1,000.

The new IRR in this case is 27.6239% pa, up from 15.7414% pa, all because of a £50 (5% of the loan) up-front fee. If the fee were charged at the end of the loan instead of the beginning, the new IRR would be 25.08%, lower than when charged up front, illustrating the importance of timing. It can make quite a difference over a longer term.

The total amount paid out over the term, including all costs, will be the same whenever the fee is charged. But the IRR uncovers the truth and indicates that a later charge implies a cheaper loan. Although regulations insist on this total payable figure being quoted, it is not helpful when comparing one loan with another; only the IRR (and its cousin the APR - see later) accurately reflects the important effect of cash flow timing.

The IRR is a potent tool

So to summarise, the IRR is the most potent tool we have available to analyse and compare different sets of loan schedules. It takes into account varying interest rates and repayments as well as fees and charges along the way and importantly, when they are made. The IRR comes from the same family as true interest rate. But the IRR is a much more sophisticated and thorough measure, as it can account for any cash flow schedule, and is therefore particularly good for loan comparisons.

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