I'm reading through Marcel B. Finan's A Basic Course in the Theory of Interest and Derivatives Markets: A Preparation for the Actuarial Exam FM/2 on my own and am unsure how to proceed with a question (page 103, question 12.4). It is:
Brian and Jennifer each take out a loan of $X$. Jennifer will repay her loan by making one payment of 800 at the end of year 10. Brian will repay his loan by making one payment of 1,120 at the end of year 10. The nominal semi-annual rate being charged to Jennifer is exactly one-half the nominal semi-annual rate being charged to Brian. Calculate $X$.
I'm assuming "nominal semi-annual rate" in the question actually means a nominal annual rate convertible semiannually. (If that's wrong, please correct me!)
Then letting Brian's nominal annual rate be denoted $B$ (so Jennifer's is $B/2$), we have $$X=800\left(1+\frac B4\right)^{-20}=1120\left(1+\frac B2\right)^{-20}$$ but I am at a loss as to how to solve this for $B$ (without expanding each (binomial)$^{20}$). Can anyone help please?
Best Answer
Hint: $$800\left(1+\frac B4\right)^{-20}=1120\left(1+\frac B2\right)^{-20}$$ might suggest $$800^{-1/20}\left(1+\frac B4\right)=1120^{-1/20}\left(1+\frac B2\right)$$ which might be easier to solve for $B$.