If I know that the present value today of £75 per year for three years is £171.13, how can I calculate what the three-year rate of interest is?
I know (well, think at least) that the formula for calculating the present value of an annuity is
Present Value = C(1/r - 1/r(1+r)^t)
where C is the payment, r is the rate of interest and t is the number of years.
I imagine it would be possible to solve this for r using the values I have indicated above (i.e. Cu and t=3), but am unsure how to rearrange the expression to make r the subject. I also have a feeling that I am probably making this more complicated than it needs to be!
You can't. You have to do it iteratively or graphically.
Doing it graphically involves plotting PV as a function of r for a fixed t, and then seeing for which ordinal r the abscissa takes the value you want. I get 15.0394% that way.
Doing it iteratively involves guessing a value for r, working out the PV and seeing how close you get to the required value. If it's too high, try a higher value for r. Keep refining your guesses until you get the right answer, or near enough.
You might guess r=0.1 and get £186.51. You might guess r=0.2 and get £157.99. Then try r=0.15 and get £171.24. Then try r=0.1505 to get £171.10. Then try r=0.1504 to get £171.13.
Would it make a difference if I knew some other information. For example, what if the one-year interest rate was 13% and the two-year interest rate was 15%?
No, not unless there is no three-year rate as such, and if the three year scenario uses the one-year rate in year 1, the two-year rate in year 2, and only leaves you one variable with no fancy powers to solve for for year 3.
The extra information might give a good guide for your first guess in the iterative method, though. But you needn't rely too heavily on a good initial guess. You can do what computing dweebs call a binary search by starting with two bounding rates, both ridiculously extreme, such as 0% and 100%. At each step, try the value halfway between the bounds (i.e. your initial guess would be 50%). If the answer is too high (or low) the way you adjust your guess is to replace either the lower or the upper bounding rate by the last value tried.
The basic idea is that if you know the right value has to be between A and B, try halfway (call it C). If it's wrong, you should know whether you need next to look between A and C or between C and B.
This relies on the assumption being correct, that the function is monotonic, i.e. it either rises or falls but not both in the whole interval between A and B.
The fundamental problem with solving your equation is the same as that of solving the standard equation for working out periodic mortgage payments.
In this form it's already solved for Payment, but you can easily solve for Capital too, and with logarithms even for Periods. But you can't solve for Rate.
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