I would really appreciate if someone can guide me through this
A 90,000 mortgage is repaid by payment at end of each month for the next 25 years. The rate of interest is 11.5% convertible semiannually
(a) divide the first payment into principal and interest.
(b) find the outstanding principal immediately after the 75th payment
(c) Divide the 76th payment into principal and interest
(d) Find the total amount of interest paid during the life of the mortgage
For (a) i used the present value formula but i don't know how convertible semiannually gonna affect things:
i used:
p = r(1-(1+i)^-n)/i solve for r...
p = 90,000, and i = 11.5%/12 and n = 25*12? I don't feel like this is correct because i didn't use the 'convert semiannually' anywhere...
Let be $L=90,000$, $n=25$ years, $i=i^{(2)}=11.5\%$ nominal interest rate compounded semiannually, $P$ the monthly payment.
First of all we need the effective monthly interest rate $i_m$ from the semiannual interest rate $i_s=\frac{i}{12}$; that is $$ 1+i_s=(1+i_m)^{6}\qquad \Longrightarrow\quad i_m=\left(1+i_s\right)^{1/6}-1\approx 0.936149\% $$ The number of monthly payment in $n=25$ is $m=12n=12\times 25=300$. Thus the monthly payment is $P$: $$ L=P\,a_{\overline{m}|i_m}=P\,\frac{1-(1+i_m)^{-m}}{i_m}\qquad \Longrightarrow\quad P=\frac{L}{a_{\overline{m}|i_m}}\approx 897.35 $$ The first paymemt is then the sum of interest $I_1$ and principal $K_1$ where $I_1=i_mL=842.53$ and $K_1=P-I_1=54.82$.
The outstanding principal immediately after the 75th payment is $$ B_{75}=P\,a_{\overline{m-75}|i_m}=84,077.08 $$
The 76th interest is $$ I_{76}=i_mB_{75}=787.09 $$ and the 76th principal $$ K_{76}=P-I_{76}=110.27 $$ The total interest payed during during the life of the mortgage is $$ I=mP-L=179,206.35 $$
You can check here (click View Amortization Table)