# Annuity problem in Finance

1. Jun 11, 2017

### issacnewton

1. The problem statement, all variables and given/known data
Mike Polanski is 30 years of age and his salary next year will be $40000. Mike forecasts that his salary will increase at a steady rate of 5% per annum until his retirement at age 60. If Mike saves 5% of his salary each year and invests these savings at an interest rate of 8%, how much will he have saved by age 60 ? Problem is from Principles of Corporate Finance (10 edition) by Brealey, Myers, Allen 2. Relevant equations Annuity equation, Compound interest equation 3. The attempt at a solution Here is my attempt at a solution. Mike is saving 5% of his salary. At the end of first year, he will get$40000. So he will save $2000 at the end of the first year. Now let $r=0.08$ and let $g = 0.05$. So in the first year, Mike saves$2000. Now in the second year, the salary becomes $40000(1+g)$. Mike saves $40000g = 2000$ out of this. So in the second year, Mike will save $2000 again. So saving each of the 30 years is$2000. So we have a saving cash flow of $2000 for 30 years. I will now calculate the present value of this annuity and then use the compound interest formula to get the future value. Now 30 year annuity factor is $$\mbox{AF} = \frac{1}{r}\left[1-\frac{1}{(1+r)^{30}}\right] = 11.25778$$ So the present value of the savings cash flow is $$\mbox{PV} = 2000 \times \mbox{AF} = 22515.57$$ We want to find out how much is the savings when Mike is 60 years old. So we can now just use the compound interest formula to forward this value. So future value of this is $$\mbox{FV} = 22515.57 (1+r)^{30} = 226566.4$$ So Mike will save$226,566.40 when he is 60. But the correct answer given is \$382,714.30. So I don't know where I have gone wrong. Please help.

Thanx

2. Jun 11, 2017

### andrewkirk

This is the problem. The amount saved is not the same every year. It grows at 5%pa.

Do you know how to calculate the final value of a geometrically increasing annuity?

3. Jun 11, 2017

### issacnewton

In the second year, his salary would be $40000(1+g)$. And 5% of this would be $40000g(1+g)$. Right ? $40000g(1+g) = 2100$. So yes, the savings are also increasing with rate $g$. I think I made a mistake. I will recalculate and report again.

4. Jun 11, 2017

### issacnewton

So first year, Mike will save $40000g$. In the second year, salary will increase to $40000(1+g)$ and Mile will save $40000g(1+g)$. Likewise, in the third year, Mile will save $40000g(1+g)^2$ and so on. So the saving cash flow is $40000g$, $40000g(1+g)$, $40000g(1+g)^2 \cdots$. This is growing 30 year annuity. The annuity factor in this case is given by $$\mbox{AF} = \frac{1}{(r-g)}\left[ 1- \frac{(1+g)^{30}}{(1+r)^{30}}\right] = 19.01656$$ So the present value of this growing annuity is $$\mbox{PV} = 40000g \times \mbox{AF} = 38033.12635$$ Now to calculate the saving done at the end of 30 year period , we forward this value in time at year 30 using the compound interest formula. $$\mbox{FV} = \mbox{PV} (1+r)^{30} = 382714.30$$ So Mike will save $382714.30$ at the age 60. And now my answer matches with the correct answer. So thanks Andrew