Nominal annual interest

Question

A bank offers the following certificates of deposit:

$$ \begin{array}{c|lcr} \text{Term in years} & \text{Nominal annual interest rate(convertible semi-annually)} \\ \hline 1 & 0.05 \\ 2 & 0.06 \\ 3 & 0.07 \\ 4 & 0.08 \\ \end{array} $$

The bank does not permit early withdrawal. The certificates mature at the end of the term. During the next six years the bank will continue to offer these certificates of deposit. An investor plans to invest 1000 in CDs. Calculate the maximum amount that can be withdrawn at the end of six years.

Case 1:Buy 6 successive 1-year CDs

$=1000(1+\frac{0.05}{2})^{2X6}=1344.8$

Case 2:

Buy 3 successive 2-year CDs

$=1000(1+\frac{0.06}{2})^{2X6}=1425.76$

Similar approach to a case of buying 2 successive 3-year CDs.

Case 4: Buying 1 successive 4 yr CDs + 1 successive 2-yr CDs

$=1000((1.04)^{2X4}+(1.03)^{2X2})=2494.0$

A similar approach was carried out for a case of buying 4 successive 4-yr CDs+2 successive 1-yr CDs to get 3469.8

Still, that does not simplify my task, I still cannot reach the answer.

Answer 1

Longer term interest rates are higher, so the investor should hold certificates for as long of a maturity as possible. The best choices are: three years followed by three years, resulting in accumulation of $$ 1000\cdot\left(1+\frac{0.07}{2}\right)^{2\times 3}\cdot\left(1+\frac{0.07}{2}\right)^{2\times 3}=1000\times 1.035^{12}=1511.0687\tag 1 $$ or four years followed by two years (equivalent to two years followed by four years), resulting in $$ 1000\cdot\left(1+\frac{0.08}{2}\right)^{2\times 4}\cdot\left(1+\frac{0.06}{2}\right)^{2\times 2}=1000\times 1.04^8 \times1.03^{4}=1540.3365\tag 2 $$ So the best choice is $(2)$.

Answer 2

Your try is not correct. The rates quoted are annual rates, not over the whole term. If I buy the three year CD, I am paid $\frac {0.07}2$ interest six times (twice a year for three years). The sensible choices are to buy six successive one year CDs, to buy three successive two year CDs, to buy two successive three year CDs or to buy (assuming the four year pays nominal 0.08/year) a four and a two. Two three year CDs multiply our money by $1.035^{12}$ because we get $0.035$ interest twice a year for six years. Compute them all and pick the best.