Sally lends $10000$ to Tim. Tim agrees to pay back the loan over $5$ years with monthly payments at the end of each month. Sally can reinvest the the monthly payments from Tim in a savings account paying interest at $6$% compounded monthly. The yield rate earned on Sally's investment over the five-year period turned out to be $7.45$%,compounded semi-annually. What nominal rate of interest, compounded monthly, did Sally charge Tim on the loan?
To find the interest Tim pays to Sally
$$P=\frac{1000}{a_{n|j}}$$
It is this loan amount that is reinvested, so
$$P\left(\frac{1.02^{60}-1}{0.02}\right)=10000(1+0.00375)^{10}$$
$$\therefore P=126.39$$
$$a_{60|i^{(12)}/12}=79.12$$
$$\frac{1-(\frac{1+i^{(12)}}{12})^{-60}}{\frac{i^{(12)}}{12}}=79.12$$
Then I start to have problems to solve this.
The value of the end of the 5 years of Sally’s investment is $$10000\left(1+\frac{0.0745}{2}\right)^{10}=14415.65$$
Let $P$ be the amount of the payments Sally receives from Tim. Then, $$ P\, s_{\overline{60}|6\%/12}=14415.65\qquad \Longrightarrow\quad P=206.6167$$ Let $i^{(12)}$ be the nominal rate of interest, compounded monthly, which Sally charged Tim on the loan. Then, $$10000 = 206.6167\,a_{\overline{60}|i^{(12)}/12}$$ So $\frac{i^{(12)}}{12}=0.7333$ and $i^{(12)}=8.801\%$.