Im not sure if this situation is possible but can you calculate the interest rate or payments in the following situation:
Mr D is considering investing in an apartment block in town. The block consists of 20 units of equal size and is for sale for $16,600,000. The block also has 10 storage units that can be let out to tenants. Mr D wishes to invest for a period of 6 years. An 80% loan to value mortgage is available over a period of 20 years and requires monthly payments. Selling costs are expected to be 7% of the selling price at the time of sale. Assume the capitalisation rate remains unchanged for the entire period.
They give additional info of the mortgage balance being $14,870,000 at the end of year 6. (Surely though this is not correct?)
Yes, just build an amortization schedule in Excel with the starting balance, payment, interest, principal (which equals payment minus interest) and ending balance for each month (240 of them in this case).
There is a unique interest rate that will give you a mortgage balance at any period. So, if you know the mortgage balance at any period, you can find the interest rate. I don’t think there is a closed form solution for this, however. So, not only is solving for the mortgage payment for any interest rate a computationally iterative process, but you must perform this iteratively on interest rates as well.
There is a way to solve for the interest rate using the PVs at time 0 and time 6 simultaneously, but you’re gonna end up with a not-so-pretty polynomial expression which has to be iteratively solved, just like Ohai pointed out. Once you have the interest rate, then you can solve for the monthly payment.
Please tell me this is not a question on a CFAI mock or from one of the study providers. It’s a little beyond the material I’ve seen in the quant sections of the BoK.