Problem Set For Real Estate Finance
With answers highlighted in blue.

This set contains questions for both FRM and ARMs.

1. Suppose that you purchase a house for $150,000 with two mortgages. The first mortgage is for $135,000, and is a 5% mortgage with two points for 30 years. The second mortgage is for $15,000, and is a 8% mortgage with no points, also for 30 years.
   1. What is your total monthly payment?
      Obviously all you have to do is calculate the two mortgage payments. Note that you do not take into account the points in this part of the problem.
      
      
       
      
      
      
      
      
      
      
      So the total payment is $834.77/month.
      
       
   2. Assuming that you hold both loans until maturity, what would be the effective interest rate on your total financing?
      
      This is where you will begin to take into account the points. At origination of the first loan you receive (135,000 * (1-.02))= $132,300, and at origination of the second loan you receive $15,000. Thus you receive a total of $147,300 at origination. You then make monthly payments of $834.77 for 30 years. The effective interest rate is the one that equates the present value of the future payments with the amount you receive at time 0.
      
      
      
      
      
      
      
      
      Solving for r gives you an interest rate of 5.483%. (To set this up on a financial calculator, set n=360, PV=147300, PMT=-834.77, and FV=0, and solve for i. Make sure to convert from monthly to annual rates when you get your answer!)
      
      
       
   3. Assuming that you paid off both loans after 10 years, what would be the effective interest rate on your total financing?
      
      This is obviously similar to part b, except now we have to know how much you payoff at the end of 10 years. First, we calculate the payoff amount for each loan.
      
      
      
      
      
      
      
      
      
      Thus, the total balance outstanding is (109,811.79+13,158.71) = 122,970.50.
      
      Now, we find the interest rate that equates all of the cash flows together (again, notice that we do take the points into account in this case since we are working with cashflows.)
      
      
      
      
      
      
      
      
      Solving for r yields 5.562%. (Note that on a financial calculator this would be solved by: n=120, PV=147300, PMT=-834.77, FV=-147300, and solving for i.)
      
       
   4. Assuming that you paid off only the second loan after 10 years, but held the first loan until maturity, what would be the effective interest rate on the total financing?
      This is a slight modification of problem c. The only difference is that the first mortgage is not paid off. So what we are looking for is the interest rate that equates the present value of these cash flows:
      
      
      
      solving for r yields: 5.375%.
      (Note on a financial calculator, use the cash flow keys. That is: CF(0)=-147,300, CF(1) = 834.77, N(1)=119, CF(2)= 13,158.71+834.77, N(2)=1, CF(3)=724.71, N(3)=240, and solve for IRR. Remember to multiply by 12 to get the annual interest rate.
      
      
       
2. Consider the following offer. You are buying a house, and currently 20-year mortgage rates in the market are at 8%. The seller is willing to sell you the house for $100,000 if you provide your own financing (assume you would get a 20 year, $100,000 mortgage). The seller is willing to sell you the house for $105,000 if you assume their $100,000, 7.5% mortgage. What should you do, and why? If the price of the home were $200,000, and they would sell it to you for $205,000 if you assumed their mortgage, would it change your answer, and why?
   
   The key to this problem is to realize that the seller is charging you $5,000 to take over the below-market rate mortgage. What you have to do is figure out whether or not it is worth that much to get the below market rate.
   
   There are multiple ways you could determine the answer to this question. The way I am going to show here is the one that I think makes the most sense, but there are other, equally valid, ways of doing it.
   
   The way I will approach the problem is to discount the payments from the 7.5% mortgage at the current market rates. If you add together the present value of those payments (at 8%) with the $5,000 additional you have to pay to get this “deal”, then you can decide which is better for you. The logic behind this is that if you buy the house with your own mortgage you are really paying $100,000 for the house. If you take the below market assumable loan, you are paying $5,000 plus the present value of the payments you are obligated to make under the assumable mortgage.
   
   To do this, I first calculate the payment for the assumable mortgage:
   
   
   
   Then, I calculate the present value of those payments at the current market rate of 8%:
   
   
   
   
   
   
   
   So if you were to take the assumable loan, you would incur a liability of 96,312.12 and you would have to pay the additional $5,000 (either by paying cash or getting a second mortgage), thus costing you a total of $101,312.12. If you found your own financing, and thus only paid 100,000 for the house, you would be better off by $1,312.12. Do not take the assumable loan in this case!
   
   In the second part of this question, we work with $200,000 instead of $100,000, but the basic procedure is the same. First, we calculate the payment under the assumable loan, then discount it at the market rate to get the value of that assumable loan, add the $5,000 to it to determine the real cost of purchasing the house with the assumable loan, and compare that to the cost of buying the house with your own financing, which will be simply $200,000.
   
   So, first, calculate the payment.
   
   
   
   
   
   
   
   
   
   
   
   
   Then calculate the present value of those payments at 8%:
   
   
   
   
   
   
   
   
   
   
   Then add in the additional $5,000 in cash (or second mortgage) that you would have to pay:
   
   
   
   
   Cost of buying = 192,624.24 + 5,000 = $197,624.24.
   
   
   
   
   
   Given that this is less than the $200,000 you would pay if you purchased the house with your own financing, you should take the assumable loan with its below market financing!
   
   
    
3. You are considering buying a house with a $125,000 mortgage, and you bank is offering you a choice. You can take a 6%, 30-year mortgage with 2 points, or a 6.35% 15 year mortgage with 1 point. If your expected holding period is 15 years, which mortgage should you take and why? What is the effective interest rate of each mortgage, assuming a 15 year holding period?
   
   This is a relatively straightforward exercise in calculating effective interest rates, although we do have to consider that there are different terms for each loan. To calculate the effective interest rate on a loan with points, we always follow this procedure:
   1. Calculate the monthly payment, using the balance of the loan.
   2. Calculate the payoff amount, using the balance of the loan.
   3. Find the interest rate that equates the cash received at closing (i.e. balance less points) with the payments made during the life of the loan plus any payoff amount.
   
   Let’s do this for each mortgage.
   Mortgage 1:
   
   
   
   
   
   
   
   
   
   
   
   
   
   Solving for r yields: 6.228%. Notice that in step 3 we use 122,500 because that is the cash received at closing (125,000 * (1-.02)).
   
   Now at this point since the effective interest rate is below the contract rate on the second mortgage, you can tell that the first mortgage is the one to chose, especially since the second mortgage has points associated with it. Just to illustrate the process, however, this is what you would do for the second mortgage.
   
   
   
   
   
   
   
   
   
   
   Solving for r yields: 6.509%. Clearly you should chose the first mortgage.
   
   
    
4. You are going to take out an ARM. This loan will be for 10 years. Given the parameters below, what will be the interest rate in each year of the life of the loan? Note that the index rate for this mortgage is the LIBOR rate:
   L = 6; Y = 1; a=2; m=2
   
   
   
   Clearly the first thing we want to do is recall our basic formulas for calculating these rates:
   C₁= r₁+m-α
   and
   C_i=max(min(r_i+m,C_i-1+Y,C₁+L),C_i-1-Y)
   
   Where C₁ is the contract rate in year 1, C_i is the contract rate in year i, where i>1, r₁ is the index rate in year 1 and r_i is the index rate in year i, where i>1.