Fair Value of a monthly payment given two Bank Payment structures

The question below was given in class and I have doubts about the use of the word "fair value". THe provided solution is to have Bank B make an overall return on the mortgage of 12%, but I don't agree with this and I feel the question is incomplete and the "fair value" can be computed only with a risk-free rate. The question is provided below

QUESTION

￼You are looking to finance your first home purchase. The price of your dream home is \$250,000.You have \$25,000 cash but need to finance the rest. Bank A has offered to loan you the rest of the value at 12% per annum compounded monthly for 5 years. On the other hand, bank B has offered you a deal for 40% of the initial loan, amortized at 6% per annum compounded monthly over 2.5 years, after which, they would set your monthly payments to amount X for the remaining 2.5 years to pay off the rest of the loan. What would be a ‘fair’ amount that bank B should charge for X? As a consultant to bank B, would you recommend an amount great or less than X – why?

It would be awesome if someone could show their calculations, but mostly I want to know which interpretation of "fair value" they agree with. If mine (hopefully), how I would go about best arguing it in a logical manner.

• "Fair value" would appear to mean "just as fair as that other bank" – DJohnM Nov 21 '13 at 4:27
• An overall return of 12% (assuming that you mean 12% total for the entire loan) is actually not that much over 5 years... depending on how you back it out that's really only a few percent per year. – THEAO Nov 21 '13 at 8:38
• OP wrote 12%/annum. Not really a common US word, are rates this high anywhere? – JTP - Apologise to Monica Nov 21 '13 at 11:07
• Is there any penalty for refinancing the Bank B mortgage with another company in 30 months? – mhoran_psprep Nov 21 '13 at 13:22
• There's no Penalty, Bank B is offering you a two-step plan (they finance the entire 60 months in two periods of 30 months, each financed differently). – Borat.Sagdiyev Nov 21 '13 at 15:58

There is one basic principle to apply here: to compare money paid at different times, all the amounts must be compounded or discounted to the same point in time.

In this case, the moment of the initial \$225,000 loan is convenient.

At that moment, you get \$225,000

You then make 30 payments on the 40% mortgage. The amount of these payments has to be calculated; they're paying off a \$90,000 mortgage with 30 monthly payments at a monthly rate of 0.5%

Finally, you make 30 payments of an amount X, starting one month after the 40% mortgage ends.

So far we've just listed the amount and time of all the payments back and forth. A time-line type diagram is a huge help here.

Finally, use compound interest and annuity formulas to bring all the payments to the starting point, using an interest rate of 1% a month! Equate money in with money out and solve for X

• Yes, this is exactly as the solution states. Could you expand on why "using an interest rate of 1% a month gives you the "fair value". I've mulled this over for a while and I've even created an analagous question, which is simpler and easier to discuss.Bank A offer to finance \$1000 at 10% per month compounded monthly for 2 months. Bank B offers to finance 40% of \$1000 at 5% per month compounded monthly for 1 month and the 60% by 1 payment of an annuity X. – Borat.Sagdiyev Nov 21 '13 at 16:01
• Solving this question, using the method you provided, gives cash flow streams of A: (576.19, 576.19) and B: (405,764.5). I wold think that in order to accurately state that both streams are "fair", you would have to know a risk-free rate(discount rate). I don't fully understand why its justified to set the risk-free rate to be 10%. Sorry if I'm stretching this too far, but I am not really grasping the concept of "fair value" in this question. – Borat.Sagdiyev Nov 21 '13 at 16:04
• I took "fair" to mean that the two cash flows (from the one mortgage scheme and from the two consecutive mortgage scheme) should generate the same interest rate of return to the two lenders. Since the one-mortgage scheme is returning 12% per annum compounded monthly, I used the same interest rate to value the second cash flow. A third lender who said,"Here's the \$ 225K; pay me \$408756.76 five years from today" would have a very different cashflow, but would still be earning 12% per annum compounded monthly. Just as fair? – DJohnM Nov 21 '13 at 20:00