- In 25 years, Mack wants to have \$25 000. He plans to invest less than $8000 now. Which of these investment options would allow him to invest the least and still meet his goal? Justify your choice.
A. 4.8%, compounded semi-annually $$P=A/(1+r/n)^{nt}$$ $$r= .048, t= 25, n= 2, A= 25000$$ $$P=25000/(1+(.048/2))^{2\times25}$$ $$P=7637.34$$
B. 4.3%, compounded monthly
$$P=A/(1+r/n)^{nt}$$
$$r= .043, n= 12, A=25000$$
$$P= 25000/(1+(.043/12))^{12\times25}$$
$$P= 7491.73$$
C. 4.65%, compounded quarterly $$P=A/(1+r/n)^{nt}$$ $$r=.0465, n=4, A=25000$$ $$P=25000/(1+(.0465/4))^{4\times25}$$ $$P=7870.18$$
D. 4.25%, compounded weekly $$P=A/(1+r/n)^{nt}$$ $$r=.0425, n=52, A=25000$$ $$P=25000/(1+(.0425/52))^{52\times25}$$ $$P= 8643.52$$
The best option is B as is allows Mark to invest the lowest amount of starting money and reach his goal in 25 years.
Another way is to compare the effective annual interest rate $i_n=\left(1+\frac{r^{(n)}}{n}\right)^{n}-1$ and select the highest.
So if $r^{(2)}=4.8\%$, $r^{(12)}=4.3\%$, $r^{(4)}=4.65\%$, $r^{(52)}=4.25\%$ we have $$i_{52}=4.34\%<i_{12}=4.39\%<i_{4}=4.73\%<i_2=4.86\%$$
and then the correct answer is $A.\, 4.8\%$, compounded semi-annually.
Observe that $P=8548.85$ in B and not $7491.73$.