# Derive expression for growing annuity

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From first principles, show that the formula for calculating the future value, at an interest rate of i, of an annuity payable for n years annually in advance, where the first payment is R1 and subsequent payments increase by j p.a., is ̈ an @l ×(1+i)^n, where l= (1+i)/(1+j)1

answered Mar 12 by (360 points)
edited Mar 12 by Connor

This question can be broken up into two parts. The first part is present valuing the payments to time 0, while accounting for the growth in payments at $$j$$% per year. While the second part is future valuing the value of the annuity at time 0 to time $$n$$ at $$i$$% to get the total future value of all the payments.

The present value of the increasing annuity in advance has the following cash flow stream:

$$PV = 1 + v^1 * (1+j) + v^2 * (1+j)^2 + v^3 * (1+j)^3 + ....+ v^{n-1} * (1+j)^{n-1}$$

Remember that $$v = \frac{1} {1+i}$$

Therefore we can re-write the equation as follows:

$$PV = 1 + \frac{(1+j)}{(1+i)}+ \frac{(1+j)^2} {(1+i)^2}+ .... + \frac{(1+j)^{n-1}}{(1+i)^{n-1}}$$

Therefore we can set  $$v_l$$ = $$\frac{1+j}{1+i}$$.

Since we know $$v_l = \frac{1}{1+l} = \frac{1+j}{1+i}$$. We can therefore calculate the value of $$l$$. This is $$l = \frac{1+i}{1+j} - 1.$$

The present value formula is therefore:

$$PV = 1+v_l^1 + v_l^2 +....+ v_l^{n-1}$$

Which is the formula for an annuity in advance at $$l$$%. ($$\ddot{a}_n$$)

This gives us the present value of the payments at time 0. We therefore just need to future value this amount to time $$n$$ using interest rate $$i$$. This gives us:

$$FV = \ddot{a}_n * (1+i)^n$$ where the annuity in advance has an interest of $$l$$%.

Hope this helps! Also, remember to draw timelines! They really help with understanding how payment streams look.

Alternative method:

The method above is easier to understand. However, this alternative method is probably what they are looking for in an exam when they say prove by "first principles".

$$FV = (1+i)^n + (1+j)*(1+i)^{n-1} + (1+j)^2 *(1+i)^{n-2} + ... + (1+j)^{n-1} * (1+i)^{n-(n-1)}$$

This is the formula for the future value of the payments at time $$n$$. The first payment of an annuity in advance occurs at time 0 and that it is why the first term above is $$(1+i)^n$$. The growth of R1 by $$j$$% occurs at the end of the first year, this is then future valued to time n, and that explains the second term of the equation. The rest of the equation follows this process.

Therefore the equation above can be simplified to:

$$FV = (1+i)^n + \frac{(1+j)}{(1+i)} * (1+i)^n + \frac{(1+j)^2}{(1+i)^2} * (1+i)^n + ... + \frac{(1+j)^{n-1}}{(1+i)^{n-1}} * (1+i)^n$$

$$FV = (1+i)^n \left[1 + \frac{(1+j)}{(1+i)} + \frac{(1+j)^2}{(1+i)^2} +...+ \frac{(1+j)^{n-1}}{(1+i)^{n-1}}\right]$$

As explained before, the equation in the square brackets is an annuity in advance at $$l$$%. This therefore provides us with the final answer:

$$FV = \ddot{a}_n * (1+i)^n$$