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Topic 3: Fixed Rate Mortgage Mechanics Fixed Rate Mortgage Mechanics • Recall that to the investor (lender), the fixed rate mortgage is a type of annuity. – The investor pays the borrower an up-front amount in return for a promised stream of future cash flows. – At time zero (i.e. origination) the present value of the annuity must equal the cash the investor pays the borrower, i.e. • Casho = PV0(Future Cash Flows) • If the cash were worth more than the PV of the future cash flows, the lender would not be willing to make the loan, since they would be paying more for the annuity than it was worth. • If the cash were worth less than the PV of the future cash flows, the borrower would not be willing to accept the loan because they would be taking on a liability that is higher than the asset they would receive (the cash), thus reducing their wealth. 2 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics • Therefore, at time 0, the only way the two parties will come to an agreement is if the exchange is equal: the lender must give the investor an amount in cash that is equal to the present value of the remaining future cash flows. • After time 0, of course, this relationship does not hold. 3 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics • The mortgage contract specifies how to calculate the various cash flows associated with the mortgage. This includes: – The “Principal” amount of the loan determines the monthly payments. This is normally set to the amount of cash the investor gives the borrower at time 0. (unless the loan includes points); – C - the contract rate of the mortgage; – n - the number of monthly payments; and – Pmt – the monthly payment on the mortgage. 4 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Balance • At time 0 we know that the value of the mortgage is equal to the cash received. For now, we will assume that the principal is set to that same amount. • Thus, the value of the mortgage must have the following relationship: 1 1 n Prin Pmt * 1 c 12 c 12 5 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Balance • Thus, we know that if the contract rate were 8%, with a 20 year (240 payments) term and monthly payments of $850, the principal amount must be 101,621.15 1 1 1 .08 240 12 101,621.15 850 * .08 12 – You can confirm this on your financial calculator by entering the following: N=240, i=8/12, Pmt=-850.00, and FV=0, and then solving for PV. 6 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Balance • Note that this formula actually works for any point during the life of the mortgage – that is, if you know the remaining term, the contract rate, and the monthly payment, this formula tells you the current outstanding principal balance. 1 1 1 c n 12 Prin Pmt * c 12 7 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Payments • While knowing how to determine the principal amount is important, it is perhaps more interesting (from a potential homeowner’s standpoint) to know how to calculate the payment that will be required given a known balance or loan amount. • This just requires simple algebraic manipulation of the balance formula. 8 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics – Payments Pmt Prin * c / 12 1 1 1 c n 12 – So, for a $100,000 loan at 10% for 30 years, the payment is $877.57. 877.57 100,000* .10 / 12 1 1 1 .10 360 12 – Again, you can confirm this on your calculator by setting: • N=360, i=10/12, PV=100,000, and FV=0. 9 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgages Mechanics - Payments Pmt Prin * c / 12 1 1 1 c n 12 • This formula also works at any point in time. That is, if you know the balance, remaining term, and contract rate, you can plug those numbers into the above formula and determine the monthly payment. 10 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Amortization • The mortgage contract terms determine the order in which payments are attributed to the account. The usual way this occurs is: • Overdue interest and penalties are paid first; • Current interest is paid second; • Overdue principal is paid third; • Current principal is paid fourth; and • Any remaining cash pre-pays principal are paid last. • Thus, normally (i.e. when scheduled payments are made on time), the investor (lender) takes the interest out of the payment first, and then takes the principal. • The interest amount is found by multiplying the balance at the beginning of the month by the monthly interest rate: • Interest due = Beginning Balance * c/12. 11 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Amortization • The principal due can then be found by subtracting the interest due from the payment: • Principal Due = Pmt – Interest Due • From this information we can create an amortization chart or schedule, which is a table detailing the periodic payments of the loan. 12 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Amortization • For a 30 year, 9% mortgage, original balance of $200,000, the amortization chart is as follows: Principal $200,000.00 Payments 360 Contract rate 9.00% Beginning Interest Ending Month Balance Payment Due Principal Due Balance 1 $200,000.00 1609.245 $1,500.00 $109.25 $199,890.75 2 $199,890.75 1609.245 $1,499.18 $110.06 $199,780.69 3 $199,780.69 1609.245 $1,498.36 $110.89 $199,669.80 4 $199,669.80 1609.245 $1,497.52 $111.72 $199,558.08 5 $199,558.08 1609.245 $1,496.69 $112.56 $199,445.52 6 $199,445.52 1609.245 $1,495.84 $113.40 $199,332.11 7 $199,332.11 1609.245 $1,494.99 $114.25 $199,217.86 8 $199,217.86 1609.245 $1,494.13 $115.11 $199,102.75 9 $199,102.75 1609.245 $1,493.27 $115.97 $198,986.77 Note that the above is an Excel spreadsheet – you should be able to “click” on it and actually use it. 13 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Amortization • Notice the relationships between principal payment, interest payment and total payment. 1800 1600 1400 1200 Payment 1000 Interest Due 800 600 Principal Due 400 200 0 0 100 200 300 400 14 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Price • At origination, the contract rate of the mortgage equals the market interest rate for the type of loan and creditworthiness of the borrower. – It is the equality of the market and contract rates which forces the balance and value of the mortgage to be the same at time 0. • Over time, since the contract rate is fixed, the contract and mortgage rates will diverge. Thus, the value and balance of the mortgage will diverge over time. – This means we have to concern ourselves with determining the value (price) of the mortgage at times other than time t. 15 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Price • To do this we simply take the present value of the remaining payments using the current market rate, which is virtually the same formula that we used to determine the balance: 1 1 1 1 Value Pmt * 1 r n 12 1 c n 12 Balance Pmt * r c 12 12 • The difference is that we use the market rate, r, instead of the contract rate, C. 16 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Example • At this point it might be useful to look at an extended example. • Consider a borrower who originally took out a $200,000 loan for 30 years at 9%. Five years have passed and the market rate is now 7%. – What is the monthly payment on the loan? – What is the balance of the loan? – What is the value of the loan? – The monthly payment is $1,609.25: $1,609.25 200,000* .09 / 12 1 1 1 .09 360 12 – Confirm using calculator: N=360, i=9/12, PV = 200,000, FV = 0, solve for Pmt. 17 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Example • Example (continued) – After 5 years the balance is $191,760: 1 1 1 .09 300 12 191,760.27 1609.25* .09 12 – You can confirm with a calculator in two ways: • N = 60, i=9/12, PV = 200,000, Pmt = -1609.25, solve for FV – Or • N = 300, i=9/12, Pmt = -1609.25, FV=0, solve for PV. 18 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Example • Example (continued) – Also note that you can determine the payment from ONLY the current balance, contract rate, and remaining term. You do not need to know the original term: $1,609.25 191,760* .09 / 12 1 1 1 .09 12 300 – On a calculator: • N=300, i=9/12, PV = 191,760, FV=0, solve for PMT. 19 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Example • Example (continued) – The value of the mortgage, at the 7% contract rate is: $227,687.12, 1 1 1 .07 300 12 227,687.12 1609.25* .07 12 • Confirm with calculator: N=300, i=7/12, PMT = 1609.25, FV=0, solve for PV. – Contrast this with the balance which is still: 1 1 1 .09 300 12 191,760.27 1609.25* .09 20 Copyright 2009 - Dr. Richard Buttimer 12 Fixed Rate Mortgage Mechanics - Effective Yield • Frequently, we will know the price of a mortgage, and its contractual details, but we will not know the market discount rate. – Fortunately, we can use the present value of an annuity formula to solve for the discount rate. – We simply have to solve for the effective yield (y) in the equation below. This can be done through a search algorithm or by use of a financial calculator. 1 1 y n 1 12 Known Price Pmt * y 12 21 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Effective Yield • Continuing with our example, let us say that a bank could purchase the mortgage for $180,000. What would be the effective yield if a bank purchased it at that price? – It would be 9.79% 1 1 1 .0979 300 12 $180,000 1609.25* .0979 12 – Confirming this is simple: N=300, PV = 180,000, PMT = -1609.25, FV=0, solve for I. Might have to multiply the found i by 12. 22 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Effective Yield • Note that in the absence of prepayment penalties or points, the effective yield on a mortgage (to the borrower) is always the contract rate of the loan. 23 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Prepayment • This extended example raises an interesting point. • The borrower is scheduled to make payments that are worth, at the current market rate of 7%, $227,687.12. • The mortgage contract, however, grants them the right to pay off the loan at any time by repaying the balance, which is the $191,760.27. – Thus, by taking out a new loan for $191,760.27, at the current market rate (7%) and using the proceeds to pay off the original loan, the borrower would increase its wealth by $35,926.85. • In essence they would be replacing one liability of $227,687.12 with a new liability of $191,760.27, therefore, reducing their overall liabilities by $35,926.85. 24 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Prepayment • Discounting the remaining payments at the market rate and comparing that figure to the balance allows us to quantify the benefits of prepaying the loan. – Frequently it is costly to refinance a loan. There are often closing costs and other fees charged to refinance. Optimally, a borrower will not refinance if the gain from refinancing is less than the refinancing costs, i.e. (Value – Balance> Cost of Refinance). 25 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Prepayment • When we talk about the “value” of this loan to the lender, we have to realize that they factor in the borrower’s right to “call” the loan. – In the previous example the “value” of the loan is not really $227,687.12 because the lender knows the borrower is going to prepay it. They realize the value is probably no more than $191,760.27. 26 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Prepayment • If we denote the value of the promised payments as “A”, and the value of the call option as “C”, and any transaction costs of refinancing as “T”, then the true value of the mortgage will be: • V = A – (C-T). – Since in the previous example we had no transaction costs, i.e. T=0, then • $191,760.27 = 227,687.12 – 35,926.85 27 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Prepayment • It is useful to examine what happens to the value of the mortgage if rates changed instantaneously. – To do this let’s use the same data from our previous example but assume it will cost the borrower $2500 to refinance. – We assume the borrower will only prepay when it is financially beneficial to do so, i.e. when: – A – Balance – T > 0 28 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Prepayment – Graphically, the value of A, i.e. the PV of the remaining payments, (V if you ignore the value of C), looks like this to the lender: $450,000.00 $400,000.00 $350,000.00 $300,000.00 $250,000.00 $200,000.00 $150,000.00 $100,000.00 $50,000.00 $0.00 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 29 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Prepayment – Graphically, the value of C, i.e. the value of the borrower exercising their call option, is given by (again, to the lender!): $50,000.00 $0.00 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 ($50,000.00) ($100,000.00) ($150,000.00) ($200,000.00) ($250,000.00) 30 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Prepayment – Combining these two shows the value of the mortgage to the bank (V). Note the spike in value just below the contract rate. $300,000.00 $250,000.00 $200,000.00 $150,000.00 $100,000.00 $50,000.00 $0.00 0.06 0.07 0.08 0.09 0.1 0.11 0.12 ($50,000.00) ($100,000.00) 31 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Prepayment – It may be easier to see this by looking only at the graph of V: $200,000.00 $190,000.00 $180,000.00 $170,000.00 $160,000.00 $150,000.00 $140,000.00 $130,000.00 $120,000.00 $110,000.00 $100,000.00 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 32 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics – Prepay Penalties • One idea to remember is that banks understand, and explicitly build into mortgage rates, the risk of prepayments. • Some borrowers, primarily commercial borrowers, but, some residential borrowers as well (notably subprime borrowers) are (or were) willing to contractually agree not to prepay in order to secure a lower contract rate. By doing so, they helped to lock in the “annuity” for the lender, at the initial rate based on the fixed-rate mortgage. 33 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics – Prepay Penalties • A common way for the borrower to signal to the lender their willingness to forgo the prepayment option is by accepting a prepayment penalty. – A prepayment penalty is simply an additional fee that the borrower agrees to pay, in addition to the outstanding balance, should they prepay the loan. – Frequently, these prepayment penalties end after some specified period of time (5, 10 or 15 years for example). – Some common prepayment penalties include • A flat fee; • A percentage of the outstanding balance; and • The sum of the previous six months interest. 34 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics – Prepay Penalties • The real effect of the prepayment penalty is to raise the borrower’s effective interest rate should they prepay. • Consider the following example. – A borrower takes out a loan with a contract rate of 10%, a term of 30 years, and an initial balance of $100,000. The loan provides that there is a prepayment penalty of 2% of the outstanding balance if they choose to prepay the loan. 35 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics – Prepay Penalties • If the borrower prepays after 5 years, what is the effective interest rate on the loan? – To determine this we must first determine the cash flows related to the loan. – The original payment is simply $877.57/month. 877.57 100,000* .10 / 12 1 1 1 .10 360 12 36 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics – Prepay Penalties • After 5 years the balance will be 96,574.14. 1 1 1 .10 12 300 96,574.14 877.57 * .10 / 12 • Thus, to pay off the loan the borrower will have to pay a lump sum of $98,505.63. – 98,505.63 = 96,574.14 * 1.02 37 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics – Prepay Penalties • Therefore, to the borrower, the cash flows are: • Positive cash flow of $100,000 at time 0 • 60 negative cash flows of $877.57 • One final negative cash flow at month 60 of 98,505.63 • Their effective yield is the yield that makes this equation true, which is 10.30%. 1 1 1 y / 12 60 100,000 877.57 * 98,505.63 y / 12 (1 y / 12) 60 – Confirm: N=60, PV = 100,000, PMT=-877.57, FV=-98,505.63, solve for i. 38 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics – Prepay Penalties • One has to be careful in this analysis, however. – To the borrower making the decision to prepay at time 60, the previous 59 payments already made are sunk costs and must be ignored. – Where this enters the borrower’s decision making is at origination. • A borrower that expects to prepay at month 60, would opt not to take out a mortgage with a prepayment penalty. 39 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics – Prepay Penalties • Consider at origination if the borrower suspected that they would likely prepay within 5 years. – If they were offered two loans, one at a contract rate of 10% and a 2% prepayment penalty or one at 10.25% and no prepayment penalty, then they should select the 10.25% loan. – The reason lenders charge prepayment penalties is to induce borrowers to reveal their expectations about their future prepayment patterns. 40 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Points • One unusual feature of the mortgage market related to prepayment penalties is the practice of charging borrowers “points”. – Technically a point is a fee that the borrower pays the bank at origination. For each point charged, the borrower pays one percent (1%) of the initial loan balance to the bank. • The effect of this, of course, is to reduce the actual cash received by the borrower. – Thus if a borrower took out a $100,000 loan, but was charged 2 points, they would receive $100,000 from the bank and then write a check to the bank for $2000, making their net proceeds $98,000. Therefore, the lender is actually only paying out $98,000 on the $100,000 loan. 41 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Points • Of course since the borrower only received, net, $98,000, at origination, the value of the mortgage at origination can only be $98,000. – The payments, however, will be based on the nominal principal amount of $100,000. The only way for the PV of the future payments to be worth $98,000, therefore, is to reduce the contract rate. • This is, of course, exactly what happens – the more points you pay, the lower your contract rate. • Note, however, that the effective interest rate is not constant, it is a function of when the loan is paid off. 42 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Points • To calculate the effective interest rate on a mortgage with points you must go through multiple steps: – First, use the contract parameters (i.e. contract principal, term, and contract rate) to determine the cash flows. You will use the contract balance to do this. – Second, find the effective yield which equates the present value of the future cash flows to the amount of cash (net) received at the closing. You will use the cash received at closing to do this. 43 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Points • Again, this may be best illustrated with an example. – A borrower takes out a loan with a 10% contract rate, a balance of $150,000, and 30 year term. The bank charges two points. • If the borrower never prepays, what is the effective interest rate on the loan? – Step 1: Determine the actual cash flows • The monthly payment is 1313.36: $1,316.36 150,000* .10 / 12 1 1 1 .10 360 12 – Step 2: Since the borrower never prepays, just solve for the effective interest rate using the cash received at closing. 44 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Points • Since the bank charges 2 points on a $150,000 loan, the borrower receives $147,000. • 147,000 = 150,000 * (1-.02) • The final step is to determine the yield which equates the cash received at time 0 with the present value of the monthly payments: 1 1 360 1 y 12 $147,000 1,316.36* y / 12 • This is 10.24%, which you can confirm on your calculator as: N=360, PV=147,000, PMT=-1,316.36, FV=0, solve for i. 45 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Points • What would be the yield if the borrower prepaid after 10 years? • Obviously the time 0 cash and the monthly payments are the same. The only additional item we need to know is the balance of the loan after 10 years, which is given by discounting the remaining payments at the contract rate. 1 1 $136,407.0 2 1,316.36 * 1 .10 12 240 .10 / 12 – You can confirm this in two ways on your calculator: N=120, I = 10/12, PV=150,000, PMT = -1,316.36, Solve for FV, or N=240, I = 10/12, PMT = -1,316.36, FV=0, and Solve for PV 46 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Points • Now, we again determine the yield which sets present value of the future cash flows equal to the cash received at time 0: 1 1 120 1 y 12 136,407.02 $147,000 1,316.36* y / 12 (1 y/12)120 • The answer is y = 10.33159%, which you can confirm as: N=120, PV = 147,000, PMT = -1,316.36, FV = -136,407.02 and then solving for i. 47 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Points • The effect of the points on the effective cost of financing changes with the amount of time the borrower holds the loan. – The chart below illustrates the effective yield given the date at which the borrower prepays the loan 35 30 Effective Yield 25 20 15 10 5 0 0 60 120 180 240 300 360 Prepayment Month 48 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Points • Finally, consider if at origination the borrower had a choice between two loans. – Loan A is the mortgage with points we just examined. – Loan B is for $147,000, at 10.3% and no points. • Which loan should the borrower take? – The answer to that depends upon the borrowers expectations regarding their tenure in the mortgage. 49 Copyright 2009 - Dr. Richard Buttimer Fixed Rate Mortgage Mechanics - Points – Clearly if the borrower expects to never prepay the mortgage, they should take loan A, because the effective rate on the loan will be 10.24%, well below the 10.30% of loan B. – If, however, the borrower expects to prepay after 10 years (or before), they should take loan B, since with a 10 year prepayment horizon loan A has an effective interest rate of 10.33%. 50 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Incremental Cost • The final issue we will examine is the incremental cost of financing. • Frequently, borrowers of equal creditworthiness will observe different interest rates for different sized loans. That is, an 80% loan to value (LTV) mortgage will have a lower contract rate than a 90% LTV loan. • The question is, what is the effective interest rate on that marginal increment of 10%? 51 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Incremental Cost • An example may be useful: – In February of 2000, 80% LTV 30 year mortgages had a contract rate of approximately 8.25% while 95% LTV mortgages had a contract rate of approximately 8.75%. • If you were a borrower with a $200,000 house you could borrow $160,000 at 8.25% or $190,000 at 8.75%. – So to borrow the incremental $30,000 your overall interest rate goes up by .5%. • One way of looking at this is that you are borrowing the first $160,000 at 8.25%, and the remaining $30,000 at some effective rate. The question is, what is that effective rate? – To solve this, let’s consider the cash flows. 52 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Incremental Cost – The payments on the 80% and 95% loans are (respectively): Payment80% loan 160,000* .0825 / 12 1,202.02 1 1 1 .0825 12 360 Payment95% loan 190,000* .0875 / 12 1,494.73 1 1 1 .0875 12 360 – Thus, there is a $292.70/month differential between the two loans. 53 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Incremental Cost • One way to view this is that you are paying $292.70/month to borrow $30,000 for 30 years. This implies the following statement must be true: 1 1 360 1 y 12 30,000 292.70 * y / 12 Solving for y we find that the incremental cost of financing is: 11.308%. • Again, you can confirm this with your calculator by: – N=360, PV=30,000, PMT = -292.70, FV=0,and solving for i. 54 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Incremental Cost • What this means is that if you can borrow $30,000 for less than 11.308%, you should take the 80% LTV loan and then borrow the remaining funds from that other source. If you cannot borrow $30,000 for less than 11.308%, you should take the 95% loan. • A closely related issue is the idea of a second mortgage. 55 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Second Mortgages • It is not at all uncommon for a borrower to take out two mortgages on the same property. The first will typically be for 80% or 90% LTV, with the second being for 20%, 10% or 5%. – Occasionally, it will be the case that it is cheaper, in terms of the effective cost of financing, to take out an 80% LTV first loan, and then a 10% second loan, than it would be to take out a single 90% LTV loan. • In the past, borrowers could do this to avoid having to have a Private Mortgage Insurance contract, since lenders only looked at the first mortgage balance. Now, they are increasingly looking at the Cumulative LTV (CLTV) when deciding whether to require PMI. 56 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Second Mortgages • Calculating the effective cost of financing when there are two mortgages is not particularly difficult. – You first determine each mortgage’s monthly payments individually. – You then combine their initial balances and monthly payments and solve for the effective interest rate. – An example may make this easy to see. 57 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Second Mortgages – Example: Bob wishes to buy a house for $100,000. He will put 10% down, take out an 80% LTV first loan at 5.5%, and a Pmt1 80,000 * .055 / 12 454.23 10% second loan at 7%. Each 1 1 mortgage is for 30 years. What 1 .055 / 12360 will be his total cost of financing if he keeps each mortgage for Pmt2 10,000 * .07 / 12 66.53 the full 30 year term? 1 – First, let’s calculate each 1 mortgage payment: 1 .07 / 12360 58 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Second Mortgages – Now we can combine the two mortgages, and find the interest rate that sets the initial balances equal to the present value of the total monthly payments. 1 1 1 y /12 360 90, 000 (520.76) * ( y /12) Solving for y, yields: y 5.67% – You can confirm this on your calculator by: N=360, PV = 90,000, PMT = -520.76, FV=0, and solving for i. 59 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Second Mortgages • Notice that the effective interest rate is not simply the average, or even the arithmetically weighted average of the two contract rates. – You must use the procedure on the previous two slides to determine the effective interest rate. If you try to take an arithmetic average of the two contract rates you will get the wrong answer. – This is because the mortgage payment equations are non- linear due to the exponents in the formulas. 60 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Second Mortgages • Now, what happens if the two mortgages are for unequal terms? – You simply have to deal with two sets of cash flows. This means you will have to use your cash flow keys on your calculator instead of your Time Value of Money keys, but once you become familiar with that procedure its not too difficult. – Let’s return to our previous example, but now assume that the second mortgage was only for 10 years. 61 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Second Mortgages – Of course this does not change the first payment, but it does change the Pmt1 80,000 * .055 / 12 454.23 second payment: 1 1 1 .055 / 12360 Pmt2 10,000 * .07 / 12 116.11 1 1 1 .07 / 12120 62 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Second Mortgages – Once again, we find the interest rate that sets the present value of the payments equal to the combined balances. Notice that we now have to deal with two streams of cash flows: 1 1 240 454.23 * 1 r / 12 1 (r / 12) 1 1 r / 12120 90,000 (570.34) * 120 (r / 12) r 1 12 Solving for r, yields : r 5.57% 63 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Second Mortgages – Note that the second annuity (the $454.23/month one) starts in 121 months, so using the PVA formula tells us its value at month 120, so we have to discount that value back to time 0. 1 1 240 454.23 * 1 r / 12 1 (r / 12) 1 1 r / 12120 90,000 (570.34) * 120 (r / 12) r 1 12 Solving for r, yields : r 5.57% 64 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Second Mortgages • Its actually pretty easy to do this on your calculator. Simply use your cash flow keys: CF0 = -90,000 CF1= 570.34 N1=120 CF2= 454.23 N2=240 – And the solve for IRR. Note that you IRR will be in monthly terms, don’t forget to multiply by 12. On an exam, you will lose points if you forget to multiply by 12! 65 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Second Mortgages – What if Bob prepaid both mortgages after 5 years? Let’s go back to the assumption that Bob had a 30 year second mortgage. Remember that our two mortgage payments, then, are: Pmt1=454.23, Pmt2 = 66.53. – We need the balance of each mortgage after 60 months: 1 1 300 Balance 454.23 * 1 .055 / 12 73,968.48 1 .055 / 12 1 1 300 Balance2 66.53 * 1 .07 / 12 9,413.16 .07 / 12 66 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Second Mortgages – So now we can combine all of the cash flows and determine the effective interest rate: 1 1 1 r / 12 60 83,381 .64 90 ,000 520 .76 * r / 12 1 r / 12 60 Solving for r yields : r 5.67% – Notice that you can use your time value of money keys for this: n=60; PV=90,000; PMT=-520.76; FV=-83,381.64 and solve for r. 67 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Second Mortgages • Finally, what if Bob had only paid off the second mortgage after 5 years, but had held the first mortgage for the full 30 years? – Again, all we really have to do is lay out the cash flows on a month by month basis: 1 1 1 r / 12300 454.23 * 1 r / 12 1 1 r / 1259 520.76 9413.16 90,000 520.76 * r / 12 1 r / 1260 1 r / 12 60 solving for r yields : 5.569% 68 Copyright 2009 - Dr. Richard Buttimer Rate Mortgage Mechanics – Second Mortgages – Again, its easiest to do this using your calculator’s cash flow keys. – The only real trick is to realize that you get 59 payments of 520.76, then one payment of (520.76+9,413.16) in month 60 when the second mortgage is paid off, followed by 300 payments of 454.23.To enter this do the following: CF0=-90,000 CF1=520.76 N1=59 CF2=9,933.92 N2=1 CF3=454.23 N3=300 And then solve for IRR, and multiply your answer by 12. • Note that we call this the effective cost of financing, even though we express it as a rate! 69 Copyright 2009 - Dr. Richard Buttimer Second Mortgages • It is important to note what is really going on with a second mortgage. • The lender of the first mortgage makes a loan that is valued based on the LTV and creditworthiness of the borrower. • The second lender makes a second loan, that is subordinate to the first, which normally has a higher interest rate higher to compensate for the fact that it is increasing the overall LTV and is subordinate to the first mortgage. 70 Copyright 2009 - Dr. Richard Buttimer Second Mortgages • The problem often comes in the fact that the borrower may not be able to afford both mortgages. • The lender of the first mortgage is in “first position” with respect to collateral priority. • The lender of the second mortgage is just rolling the dice, since because they are in a second position, they aren’t very well protected. 71 Copyright 2009 - Dr. Richard Buttimer Second Mortgages • However, with two different lenders, operating with different criteria to loan to the borrower, there often isn’t any single person making sure that the loans (collectively) should be structured this way, or even made at all. • Thus, by seperating the lenders, and maximizing the LTV, borrowers are often able to borrow more than they should. • This helped lead to the real estate crisis we are are currently experiencing. 72 Copyright 2009 - Dr. Richard Buttimer

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