Future Values of Annuities

Question

Michelle has decided to invest $3000 at the end of each year for the next five years in a saving account that pays 8% annually, compounded semi-annually. How much is the annuity worth after 5 years? (Hint: be careful, interest conversion period, and investment period is not the same).

I got the correct answer by calculating the amount in the account after each year:

End of Year 1: $3000

End of Year 2: $3244.8+3000= 6244.80

End of Year 3: $9754.38

End of Year 4: $13550.33

End of Year 5: $17656.04

I was wondering if there was a formula that I could use. I know the formula for calculating the future value of an annuity, but the hint in the question confused me. So, i ended up calculating the amount after each year. The answer to this question should be $$17656.04.

Answer 1

I derived the formula and here it is

FV = $A*\dfrac{[(1+\frac{r}{2})^{(2*N)}) - 1]}{[(1+\frac{r}{2})^2 - 1]}$

Where A = Payment amount at the end of the year,

r = annual interest rate %

N = Number of years

It can further more generalized for any compounding period by

FV = $A*\dfrac{[(1+\frac{r}{n})^{(n*N)}) - 1]}{[(1+\frac{r}{n})^n - 1]}$

Where n = 2 for semi annual , n = 4 for quarterly, n= 12 for monthly.

Answer 2

Using actuarial notation, the quantity you want to calculate is $$K s_{\overline{n|}i}^{(m)},$$ where $K = 3000$, $n = 5$, $i = 0.08$, and $m = 2$. This represents the accumulated value of an annuity-immediate that pays $K$ at the end of each year for $n$ years, with a nominal annual interest rate of $i$ compounded $m$ times per year.

Let's look at the individual accumulated cash flows. The accumulated value of the last payment of $K$ is simply $K$, since it is made at the end of the term and has had no time to accrue interest. The second-to-last payment of $K$ has had one year to accrue interest at a nominal rate of $i$ compounded $m$ times, so its accumulated value is $K(1+\frac{i}{m})^m$. Similarly, the third-to-last payment has had two years to accrue interest and is compounded $2m$ times, so its accumulated value is $K(1+\frac{i}{m})^{2m}$, and so for the first payment, there are $(n-1)m$ compounding periods. The total accumulated value is therefore $$K s_{\overline{n|}i}^{(m)} = K \sum_{t=0}^{n-1}\left( 1 + \frac{i}{m} \right)^{tm} = \frac{(1 + \frac{i}{m})^{mn} - 1}{(1 + \frac{i}{m})^m - 1}.$$ In terms of the effective annual interest rate $r = (1 + \frac{i}{m})^m - 1$, this is simply $$\frac{(1+r)^n - 1}{r}.$$ So we can solve the question more easily by first computing the effective rate $r$, then treating the cash flow as annual payments at the effective rate, rather than as compounded every six months.