An investor wishes to purchase a $100,000 mortgage taken out at 5% compounded semi-annually for 30 years. Since the mortgage was issued, 42 monthly payments have been made. The loan is to be paid in full (a balloon payment) at the end of its fifth year. What is the yield if the purchase price of the mortgage is $90,000? Any help would be great. Thanks a lot.
First thing will have to do is to calculate the monthly payment, since it is not given, which is a very simple matter. But, before we do that, we have to convert 5% from semi-annual to monthly compound to match the payments, which comes to=4.95%.
The monthly payment at this rate, amortized over 30 years comes to =$533.69. Since the mortgage has to be paid off in full at the end of the 5th year, we have to find the balance of the mortgage at maturity, which comes to=$91,761.76.
But, since 42 payments have already been made on this mortgage, the investor has only 60 - 42=18 months left to worry about. Since he is willing to pay $90,000 for the mortgage, then all we have to do is to find an interest rate, through interpolation and iteration, that will equate the PV of the mortgage at maturity plus the PV of the remaining stream of 18 monthly payments, to his offer of $90,000. By doing that, we can see that a rate of 8.35% comp. monthly, will do just that. And this is the yield to the investor.
P.S. If you wish to know the various TVM formulas used to calculate any of the above, just let us know.
Thank you very much for this. One thing that I don't understand is this: How did you get the balance at the end of the 5th year, or $91,761.76?.
That is very simple:
1- First you find the FV of $100,000 for 5 years or 60 months, using this very common TVM formula:FV=PV[1 + R]^N=FV OF $1 TODAY..........(1)
2- Then you find the FV of the 60 monthly payments of $533.69 each, using this common TVM formula:FV=P{[1 + R]^N - 1/ R}=FV OF $1 PER PERIOD.............(2)
Where R=Interest rate per period, N=number of periods, P=periodic payment, PV=Present value, FV=Future value.
3- Then you simply subtract (2) from (1) above and you should get the balance of the mortgage after 5 years, or 60 months.
Good luck to you.
