FV of an annuity-immediate

Question

The question is "Find the FV of an annuity-immediate with payments of $50$ every $6$ months for $8$ years at a nominal rate of interest of $4\%$ compounded semiannually."

$p\times s_{\overline{n}|i}$. When computing $i$, I would do $(1+0.04) = (1+ \frac{i^{(2)}}{2})^2$ and solve for $i^{(2)}$, but that is the wrong method since the given interest rate in the solution is $2\%$.

What would the correct set-up be to find the interest rate, and what is the intuition for getting it the right way vs. my approach?

Answer 1

When the cash flow occurs at the same interval as the compounding of interest, then the calculation is simple. In this case, payments are made every 6 months--i.e., semiannually--and the nominal rate of interest also is specified as being compounded semiannually. So, the effective semiannual rate of interest is simply half of the nominal rate; i.e., $$i^{(2)}/2 = 0.04/2 = 0.02$$ is the effective rate, and the desired future (or accumulated) value is $$50 s_{\overline{16}\rceil 0.02}.$$ The number of payments is simply $8$ years times $2$ payments per year.

The calculation is not so simple if the compounding period is not the same as the rate of cash flow; e.g., if the payments were to occur every month, rather than every six months, how would the accumulated value calculation change? Here, we need the monthly effective rate of interest, not the semiannual rate. The effective semiannual rate is $0.02$ as we said before, so in order to find the effective monthly rate, we need a rate $j$ such that $$(1+j)^6 = 1 + 0.02,$$ because we need to know what rate, if it were compounded over a $6$-month term, would be equivalent to ("effectively") the semiannual rate.

But what if the compounding frequencies are not integer multiples? For example, what if the effective semiannual rate was still $0.02$, but the payments are made every $4$ months (i.e. $3$ times a year)? Then we should consider the smallest period common to both compounding rates; i.e., $$1+i = \left(1 + \frac{i^{(2)}}{2}\right)^2 = (1.02)^2,$$ and then solve $$(1 + j)^3 = (1.02)^2,$$ which gives the effective $4$-month rate.

Answer 2

$i^{(2)}=4\%$ so we have that the semiannual effective interest rate is $i_s=2\%$ and the number of payment is $n=2\times m=2\times 8=16$ where $m=8$ is the number of years, the payment is $p=50$. So the future value is $$ FV=p\times s_{\overline{n}|i_s}=p\times\frac{(1+i_s)^{16}-1}{i_s}=50\times\frac{(1.02)^{16}-1}{0.02}=931.96 $$

Observe that we have $(1+i_s)^2=\left(1+\frac{i^{(2)}}{2}\right)^2=1+i$ where $i$ is the effective annual interest rate and then $$ FV=p\times s_{\overline{n}|i_s}=p\times\frac{(1+i_s)^{n}-1}{i_s}=p\times\frac{(1+i)^{m}-1}{i}\times \frac{i}{i_s}=p\times s_{\overline{m}|i}\times \frac{i}{i_s} $$