A loan is being repaid with 20 annual installments of 1. Interest is at effective rate i for the first 10 years, and j for the last 10 years. Find an expression for each of the following:
a)The amount of principal repaid in the eighth installment.
I am pretty sure the answer is $v_i^3 (1-i\times a_{10|j} )$ but I am not sure how to get to this. Is this general knowledge or is there some proof or something I need to do.
b) The amount of interest paid in the last installment.
I am pretty sure this ones answer is $\frac {j} {1+j}$ but once again, not sure if this is general knowledge or there is someway to find this.
Thank you
The loan $L$ is $$ L=a_{\overline{10}|i}+v_i^{10}\, a_{\overline{10}|j} $$ The outstanding debt $D_7$ at time $7$ is $$ D_7=a_{\overline{3}|i}+v_i^{3}\, a_{\overline{10}|j} $$ and then the principal repaid in the eight installment $K_8$ is $$ K_8=1-iD_{7}=1-i\,a_{\overline{3}|i}-i\,v_i^{3}\, a_{\overline{10}|j}=v_i^3\left(1-i\,a_{\overline{10}|j}\right) $$ The interest $I_{20}$ paid in the last installment is $$ I_{20}=j\,a_{\overline{1}|j}=j\frac{1-v_j}{j}=1-\frac{1}{1+j}=\frac{j}{1+j} $$