Case Study 1: Adjustable-Rate Home Mortgage Loan Concepts illustrated: Time value of money, equivalence calculation, and loan analysis. Required readings: Chapters 4 and 5. 1 Background Buying a home today is much more confusing than in the past. The mortgage market has changed completely from the days when \home mortgage" meant a 30- year, xed-rate loan. Then, the mortgage market was something like the ice cream market - all you could get was \vanilla," but today wehave \3l avors. As with ice cream, the more choices a consumer has, the better. For example, the current mortgage market oers nancing plans such as the traditional xed rate mortgage (FRM) the adjustable rate mortgage (ARM), the graduated payment mortgage (GPM), the growing equity mortgage (GEM), the wraparound mortgage (WM), and the buy-down mortgage (BDM). These choices may seem confusing initially, but they provide a real opportunity for the homebuyer to tailor a loan to his or her particular nancial circumstances. c2001 by ChanS.Park, Department of Industrial & Systems Engineering, Auburn University - This case is developed for classroom discussion purpose only. 1

A xed-rate mortgage is one in which the interest rate and monthly payments remain constant for the loan term, which has traditionally been 30 years. The advantage of this method is that homebuyers are protected against rising interest rates and the monthly payment does not change. The disadvantage is that lenders usually charge a higher interest rate for these loans than for adjustable rate loans and homebuyers are locked in at those rates for the duration of the loan. In recent years, adjustable rate mortgages (ARMs), for which the interest rate or monthly payment changes at specic intervals to an agreed-upon market rate, have gained popularity. The benet of this type of mortgage is that the initial mortgage interest rate is usually lower than the prevailing rate for a long-term, xed-rate mortgage. Therefore, the monthly payment is lower, and less income is needed to qualify for a loan. If interest rates decline, the mortgage payment is adjusted downward when the adjustment interval is reached. The disadvantage of the ARM is that it is dicult to predict whether interest rates will rise or fall, and thus it is hard to plan for future payments. Since future mortgage interest rates are bound to uctuate according to prevailing economic conditions, probable variations in the mortgage rate have been of concern to most homebuyers when considering this type of mortgage. 2 Description of the ARM Features There are several features that characterize the ARM. 2.1 Basic Features There are as many dierent types of ARMs as there are styles of houses. All ARMs, however, have four basic features: (1) initial interest rate, (2) adjustment interval, (3) index, and (4) margin. 2.1.1 Initial Interest Rate The initial interest rate is the beginning interest rate on any ARM. It will stay the same until it is adjusted either up or down for the adjustment period specied in the mortgage. 2

2.1.2 Adjustment Interval The adjustment interval is a period of time between changes in either the interest rate or the monthly payment. The interest rate on most ARMs could change after one, three, or ve years. 2.1.3 Index The index is a published measure of interest rates on certain types of borrowing or investments. There are several types of indexes, and they are used by lenders to determine the mortgage loans's new interest rate at the time of adjustment. The two most common indexes are: 1. One-, three-, and ve-year U.S. Treasury security (T-bill) yields (interest rates), 2. The monthly average cost-of-funds incurred by savings and loan institutions. Most lenders use one-year T-bill yields as their indexes. The T-bill rate tends to uctuate much more dramatically than the cost-of-funds rate, which reects the average price a savings institution must pay for its money. Because indexes reect the general movement of interest rates in the national economy, lenders rely upon them to make certain their rates are competitive with other lenders and yet not so low that they are likely to lose money. 2.2 Margin This is an additional amount the lender adds to the index to establish the interest rate on the loan. Typically, the interest rate charged on an ARM is calculated by adding a margin of between 2 and 3 percentage points to the index rate. For example, suppose your loan is indexed to a one-year T-bill and your lender's margin is two points. If the current one-year T-bill is 8.14%, the new rate on your loan is 10.14%. 2.3 Optional Features There are several features that can be added to the ARM, if the lender oers them. Common optional features include (1) interest rate caps, (2) monthly payment caps, and (3) negative amortization. 3

2.3.1 Interest Rate Caps There are two types of interest rate caps. The rst is a limit on how much the interest rate on an ARM can increase or decrease at the time of each adjustment, usually annually. The second type is a limit on how much the interest rate can increase or decrease over the life of the loan. 2.3.2 Monthly Payment Caps The monthly payment cap is a limit on how much the monthly payment can increase or decrease at the time of each adjustment, for a specic number of years. 2.3.3 Negative Amortization Amortization means that the monthly payments are large enough to pay the interest and reduce the principal on the loan. Negative amortization occurs when the monthly payment is not enough to pay all the interest due on the loan. The unpaid interest is added to the unpaid balance of the loan. Stated dierently, instead of reducing the amount you owe, negative amortization means you owe more than you initially borrowed. Negative amortization may occur when a monthly payment cap is in eect. Lenders usually will not allow negative amortization to exceed 125 % of the original loan balance. If that happens, the lender will adjust the mortgage payments upward to a level sucient to amortize or pay o the new balance in full over the remaining term of the loan. 3 Monthly-Payment Calculations Under A One- Year ARM The purpose of this case study is to demonstrate how economic equivalence principles can help prospective homebuyers understand the consequences of choosing the ARM as opposed to the conventional xed-rate mortgage plan. We rst briey describe the features common to ARMs. Several examples are introduced to determine the monthly payments under various scenarios. In this section, we will illustrate how to compute the monthly payments under 4

one-year ARM loans. Three examples are presented with varying features of the ARM. 3.1 Example 1: An ARM Containing Only the Four Basic Features This type of ARM has no \cap" or limit on the amount your payment could increase. Consider the following specic mortgage containing only the four basic ARM features. Loan amount $100,000 Loan term 30 years Initial interest rate 12% Adjustment interval Annually Index One-year Treasury Bill Margin 2% Monthly payment (not including taxes, insurance and mortgage insurance premiums, if any) for the rst year is A = 100 000(A=P 1% 360) = $1 028:61: Assume that, at the end of the rst year, the index is 11%. You add the index rate (11%) and the margin (2%) to get the new interest rate (13%). At the end of rst (right after making 12th payment), the remaining loan balance is 1 028:61(P=A 1% 348) = $99 637:12 This indicates that, during the rst year, you have paid only $362.88 on your loan principal. Then, your new rate will be gured on a balance of $99,637.12, and on a term of 29 years, because your loan is one year old. Your new monthly payment is A = 99 637:12(A=P 13%=12 348) = $1 105:41: Thus, your monthly payment has increased $76.80. Just a 1% increase in the index over a one-year period can have a rather dramatic impact on your monthly payments. 5

3.2 Example 2: An ARM with Interest Rate Cap As mentioned previously, an interest rate cap limits how much the interest rate on your mortgage can increase or decrease at the time of each adjustment, or over the life of the loan. The most common experience is that lenders who oer these optional features have a 2% limit on each annual adjustment and a 5% limit over the loan life. The interest cap is a consumer safeguard because it keeps payments within predictable limits and protects you against dramatic increases in your monthly payments. We will use the previous basic ARM example to show what happens to the ARM with an interest rate cap. This way, we can show you the worst that can happen if you accepted an ARM with an initial rate of 12 % that included these important consumer safeguards. Loan amount $100,000 Loan term 30 years Initial interest rate 12% Adjustment interval Annually Index One-year Treasury Bill Margin 2% Interest rate cap 2% annually 5% over the life of the loan Monthly payment (not including taxes, insurance, and mortgage insurance premium, if any) for the rst year is A = 100 000(A=P 1% 360) = $1 028:61: Assume that, at the end of the rst year, the index rate is 12.5%. Adding the 2% margin to the index rate gives you the new rate of 14.5%. As shown previously, during the rst year, you have paid only $362.88 on your mortgage principal, the new rate will be applied to the remaining balance $99,637.12, and on a term of 29 years. At 14.5%, your new monthly payment would have been A = 99 637:12(A=P 14:5=12% 348) = $1 222 66: 6

But, because you have a 2% annual interest rate cap, your interest rate can only go from 12% to 14%. Therefore, your actual monthly payment is now A = 99 637:12(A=P 14=12% 348) = $1 183:33 With an interest cap, you saved $39.33 per monthly payment. Assume that, at the end of the second year, the index rate is 15%. Adding the 2% margin to this index yields the new rate 17%. Due to the 2% annual interest rate cap, your new interest rate at the end of the second year is 16%, rather than the 17%. During the second year, you have paid $267.47 on your loan principal. Your new monthly payment will be gured on a balance of $99,369.65 and a term of 28 years because your loan is now two years old. A = 99 369:65(A=P 16=12% 336) = $1 340:58 Your monthly payment fromthe second to the third year has gone up $157.25. This is a total increase in each monthly payment of $311.97 from the rst year's monthly payments. Assume that the index rate at the end of the third year is 16.5%. With the 2% margin, the new rate is 18.5%. Because your interest rate has already gone up 4%, and you have a 5% loan cap, your new interest rate can go up only one more percentage point, from 16% to 17%. Therefore, in the fourth year, you have \topped out." The worst has happened. During the rst three years, you have paid $832.44 on your loan principal, your new monthly payment is A = 99 167:56(A=P 17=12% 324) = $1 419:76: Until the index declines in the future, your new monthly payment stays over the remaining loan life. In other words, they cannot go any higher (except, of course, if taxes and insurance go up). 7

3.3 Example 3: An ARM with A Monthly Payment Cap The monthly payment cap is a limit on the amount by which the monthly mortgage payment can increase at each adjustment period. A common payment cap is a monthly payment increase equal to an amount 7.5% greater than the previous year's monthly payment, regardless of what happens to the interest rate. Loans with monthly payment caps can cause negative amortization to occur. The monthly payment cap provides you with certainty about the maximum amount your monthly payment could go up each time your loan adjusts. But, by controlling the amount ofyour loan payment, you may not be paying enough to cover the interest that would have been charged if you had not selected the monthly payment cap feature. Remember that your ARM interest rate remains the indexed rate plus the margin, regardless of what you pay. The lender is permitted to add the unpaid interest to the outstanding balance of your original loan. We will show how this monthly payment cap works in the following example. Loan amount $100,000 Loan term 30 years Initial interest rate 12% Adjustment interval Annually Index One-year Treasury Bill Margin 2% Interest rate cap No. Monthly payment cap Yes, 7.5% annually Negative amortization Yes, unless 125% of loan balance is reached. As shown in Example 2, the monthly payments during the rst year is $1,028.62. Assume that, at the end of the rst year, the index is 12.5%. Your new interest rate would be 14.5%. With the principal payment of $362.88 and no interest rate cap, the new monthly payment would have been A = 99 637:12(A=P 14:5=12% 348) = $1 183:33: 8

But, because you have a monthly payment cap that permits your payment to increase only by an amount equal to 7.5% of the previous year's monthly payment, your monthly payment during the second year is actually A = 1 028:62(1:075) = $1 105:77: The dierence between what your monthly payment would have been if you did not have a monthly cap ($1,183.33) and what it actually is because you do have the cap ($1,105.77) is $77.56. The unpaid interest that accrues each month is added to the outstanding balance of your loan, because your payment under the cap is not enough to cover the interest accruing on your loan. This results in negative amortization, causing the loan balance to increase rather than decrease. In our example, we compute the second year beginning balance 100 000(F=P 14=12% 12) ; 1 105:77(F=A 14=12% 12) = 100 779:52 that indicates a negative amortization amount of $779.52. This can go on each year during the period of capped payments, assuming interest rates keep going up, and the yearly 7.5% increase in monthly payments is not enough to pay the accruing interest due. If and when negative amortization causes the loan balance to reach 125% of the original loan balance ($125,000), your lender will require you to adjust your payment toalevel sucient to pay o the loan over the remaining loan term. 4 Eective Interest Rates Under One-Year ARM Loans We will consider a prospective homebuyer in metropolitan Atlanta, Georgia, considering a house worth $125,000. After a 20% down payment ($25,000), the remaining balance ($100,000) is to be nanced through a local nancial institution. To provide a logical basis for computing the eective interest rate on the mortgage, the following assumptions are made: 1. The choices are a 30-year xed-rate plan and a plan using a one year adjustable rate with the index tied to a one-year T-bill. 9

2. The prospective homebuyer can qualify for either loan. 3. The interest rate for the 30-year xed-rate plan considered is a 12% annual rate, and the discount point for this xed-rate loan is one point. One point is an amount equal to 1% of the principal of the loan. Points are charged by the lender to increase the yield on the mortgage loan to make it comparable with other types of investments. 4. The one-year ARM to be compared has a 9% starting interest rate with a 2% annual cap and a 5% loan cap. Two discount points are applied to this ARM loan. 5. The origination fee for either loan is 1% of the loan amount. This is a fee the lender charges for several administrative tasks: processing the loan application, determining your ability to meet the loan payments, assessing the value of the house, and preparing the loan documents. 4.1 Eective Interest Rate Under the Fixed-Rate Loan Since all future cash ows occur monthly, we compute the eective interest rate per month. To do so, we rst compute the discount point payment and the origination fee. With the $100,000 mortgage principal, we have Discount point payment = 100,000(0.01) = $1,000 Origination fee = 100,000(0.01) = $1,000. The net amountyou are receiving from the lender is $98,000, but the monthly payments will be gured on $100,000. Therefore, the eective interest rate per month is obtained by solving the following expression for i. 98 000(A=P i 360) = 100 000(A=P 1% 360) The annual percentage rate (APR) is then i = 1:0227% per month r = (1:0227)(12) = 12:2721% and the eective annual interest rate is i a = (1 + 0:010227) 12 ; 1 = 12:99%: 10

Table 1: ARM Interest Rates and Monthly Payments Under Best and Worst Economic Conditions Best Case Worst Case Year Interest Monthly Interest Monthly Rate (%) Payment ($) Rate (%) Payment ($) 1 9 804.62 9 804.62 2 7 667.54 11 950.09 3 5 543.74 13 1,099.09 4 4 487.90 14 1,175.91..... 30 4 487.90 14 1,175.91 4.2 Eective Interest Rate Under the One-year ARM Although we cannot predict with certainty what would happen to future interest rates, we can compute the eective interest rates under both the worst and the best situations. 4.2.1 Worst Case Suppose the interest rate keeps increasing at the maximum annual cap rate until it reaches the loan cap. The interest scenario is 9% for the rst year, 11% for the second year, 13% for the third year, 14% for the fourth year and remaining years. (They cannot go any higher than the 14% cap.) The monthly payments are shown in Table 1. The discount points and the origination fee for the ARM loan is Discount point payment = 100,000(0.02) = $2,000 Origination fee = 100,000(0.01) = $1,000. The net amount to receive from the lender is then $97,000, but as with the xed-rate loan, the monthly payments are gured on the $100,000 principal. The eective interest rate per month for this worst case is obtained by solving 11

the following expression for i. 97 000 = 804:62(P=A i 12) + 950:09(P=A i 12)(P=F i 12) +1 099:95(P=A i 12)(P=F i 24) +1 175(P=A i 324)(P=F i 36) i = 1:11% per month r = (1:11)(12) = 13:32% i a = 14:16%: As expected, the eective annual interest rate is 1.17% higher than that of the xed-rate plan. If the worst happens, the ARM plan is a costly nancing option. 4.2.2 Best Case Suppose the interest rate keeps decreasing at the maximum annual cap rate until it reaches the loan cap. Then, the future interest scenario is 9% for the rst year, 7% for the second year, 5% for the third year, 4% (the loan cap) for the fourth year, and remaining years of the loan. The monthly payments under this scenario are also shown in Table 1. The eective interest rate is obtained by solving the following expression for i. 97 000 = 804:62(P=A i 12) + 667:54(P=A i 12)(P=F i 12) +543:74(P=A i 12)(P=F i 24) +487:90(P=A i 324)(P=F i 36) i = 0:42% per month r = 5:04% i a = 5:16%: The result of our best and worst case analysis shows that the annual eective interest rate of the ARM plan would uctuate between 5.16% and 14.16%. 5 Concluding Remarks As a borrower, you will need to decide for yourself the likely future course of interest rates, and, therefore, the benets to you of choosing an ARM over 12

a xed-rate mortgage. It is important for you to consider both the \worst case" and the \best case" possibilities with the understanding that neither extreme may actually occur. Case Study 1: Issues for Consideration 1. What historical information, if you were considering home purchase with an ARM, might enable you to build a more accurate best-case/worstcase scenario on which to base your decision? 2. Whenever there is sucient decline for the market interest rate (such as a prime lending rate), home mortgage interest rates (xed rate loan as well as ARM) also decline. When this happens, many ARM homeowners do consider renancing their mortgages by switching from ARM to xed-rate loans. Under what situations do you think it is worth switching? 13