informatics

Mortgage Selection Using a Decision Tree Approach

ROBERT E . LUNA Sandia National Uboratory

Albuquerque, New Mexico 87185

RICHARD A. REID Anderson Schools of Management University of New Mexico Albuquerque, New Mexico 87131

People choose mortgage types to minimize their costs, basing their decisions largely on expected future interest rates. An analysis of past changes in interest rates shows surprising regularity. This result provided confidence that the potential benefits associated with a more structured approach could be realized. Using concepts from classical decision theory and a reasonable range of alternative future scenarios, a rational choice for financing a personal investment was identified.

y \ married couple plans to purchase a (ARM) which has constant payments X A . condominium at a mountain resort over the first three years of the mort- for use as a tax shelter and, coinciden- gage life, and then the interest rate is tally, as a vacation retreat. Their goal in adjusted triennially based on the Fed- purchasing the condominium is to maxi- eral Reserve's three-year treasury (T) mize their return on investment. Al- bill rates as adjusted for constant ma- though the net return involves many turiHes. The current index rate for this components, a major cost factor is the type of mortgage is WA percent, type and terms of the mortgage selected. (2) A five-year ARM which has constant Thus, mortgage cost minimization be- payments over the first five years of comes an important factor in this mortgage life and thereafter is ad- situation, justed every fifth year to reflect the

There are three principal choices: Federal Reserves five-year treasury (T) (1) A three-year adjustable rate mortgage bill rate adjusted to constant maturi-

Copyright C 1986, The Institule of Managemcnl Sciences DECISK)N ANALYSIS - APPLICATIONS 0092-2102/86/i603/0073$01.25 FINANCE - PERSONAL FINANCE This paper was refereed.

INTTERFACES 16: 3 May-June 1986 (pp. 73-81)

LUNA, REID

Type of Mortgage

Treasury-Bill Index Rate

Interest Rate Increment

Mortgage Interest Rate

Origination Fee

3-Year ARM 5-Year ARM Conventional

10V4% 123/4%

IVi 14V8

21/4

13/4

Table 1: Interest rale and origination fee differentials for three mortgage types as related to the treasury-bill index rates. Each of the ARMs has a suboplion that limits annual payment in- creases lo 7Vi percent. The payment stability achieved by this option may produce negative am- ortization which could increase (rather than decrease) the loan principal. This alternative was not considered because any negative amortization must be recouped when the condominium is sold.

ties. The current index rate for this type of mortgage is 10% percent.

(3) A conventional fixed rate mortgage which currently requires interest cal- culated at W/B percent.

Table 1 shows the different rates and orig- ination fees associated with each loan type. Even though the investor's planning horizon is between five and seven years, the tabular values are based on amortiza- tion over a 30-year loan life. The interest increment covers costs and represents some contribution to profit for the lender.

A major issue associated with the selec- tion of a mortgage alternative concerns present and future interest rates. Al- though borrowers use current interest rates to assess the attractiveness of a con- ventional mortgage, evaluating ARMs re- quires that they estimate future interest rates. If interest rates could be predicted with certainty, competitive market forces would rapidly remove any cost advantage of one financial vehicle over another. However, future interest rates cannot be predicted with any degree of certainty, and thus, the borrower's expectations of subsequent changes in interest rates be- come a major decision criterion. More- over, the market for alternative mortgage

instruments appears to shift in response to these expectations which also influence future interest rates.

In short, a decision is required between several alternatives where significant un- certainty is associated with future states of nature that will influence the final re- sult. This uncertainty results from the fact that the total cost of the mortgage can be significantly affected by changes in interest rates over the life of the invest- ment. Specifically, adjustments in the ARM'S interest rates at years three, five, and six after the mortgage loan is initi- ated need to be considered. Methodology and Results

We used a decision tree to help analyze this situation. Initially, this required a forecast of the range over which future interest rates could vary to characterize the relevant states of nature. After the de- cision tree was constructed, we used var- ious decision criteria to help assess the economic consequences of various mort- gage alternatives.

An examination of interest rates on three- and five-year T-bills for the past 30 years provided the basic data for analysis. We calculated statistical summary param- eters from these data (see Table 2). A

INTERFACES 16:3 74

MORTGAGE SELECTION

Calculated Values

Statistics Mean Standard Deviation Coefficient of Variation

Least Squares Parameters Correlation Coefficient Intercept Slope

3-Year T-Bill Rate

5.94% 3.05% 0.51

0.90 1.03% 0.32%/yr.

(tt = 30) Fractional Change

0.2595 0.3359 1.29

- 0 . 0 3 0.2745

- 0.0011

5-Year T-Bill Rate

5.60% 3.11% 0.56

0.92 0.47% 0.30%/yr.

{n = 33) Fractional Change

0.3993 0.3330 0.83

-0.10 0.4601

-0.0042

Table 2: Calculated statistical and parameter values for interest rates and their fractional changes. The data were collected as follows: three-year T-bill interest rates were obtained from the Economic Report of the President — February 1983; five-year T-bill interest rates were ob- tained from the Statistical Abstract of the United States: 1984 for years 1972-1982; and earlier (1950-1971) five-year rate data were obtained from the Federal Reserve Bulletin. Fractional change values are calculated by [i, (n + r) - iin)y[i,(n)] where i. represents the T-bill interest rate for the period shown in parenthesis, r = 3- or 5-year intervals, and « = 1,. . .,30 or 1,. . .,33 annual period numbers, respectively.

simple linear regression equation (t, = 1.03 + 0.32«) for the three-year T-bill in- terest rates over time had a high correla- tion coefficient (0.90). The regression {u, = 0.47 -)- 0.30n) for the five-year T-bill interest rate data showed an even greater correlation coefficient (0.92). In these equations, f^represents the estimated in- terest rate at time period n for each of the T-bill series. An examination of the frac- tional changes in annual interest rates (percentage rate change between succes- sive years) over both three- and five-year time intervals indicated relatively little correlation (-0.03 and -0.10, respec- tively) with time. Moreover, the fitted regression lines possessed slopes which were very small (-0.0011 and -0.0042, respectively) in comparisor\ with the mag- nitudes of the average fractional differ- ences (0.2595 and 0.3993, respectively). These combined results provided a sound rationale for calculating expected changes

in the T-bill interest rates of successive three- and five-year time intervals. Al- though this analysis produced a logical plan, we noted that since both of the de- rived interest rate fractional changes have coefficients of variation close to unity

Procedural Steps

1. T-bill index rate 2. Fractional change in

interest rate Mean Standard deviation

3. Estimated new index rates Mean Standard deviation

4. Range of index rate values Low Mean High

3-Year ARM

10'/4%

0.2595 0.3359

2.66% 3.44%

- V4% + 2%% + 6'/B%

5-Year ARM

10%%

0.3993 0.3330

4.147c 3.54%

- y 4 % + 4Vt%

Table 3: A sequential procedure for estimating a range of index rate percentages. All index rate values are rounded to the nearest one- eighth percent. High and low index rate val- ues correspond to the mean value plus and minus one standard deviation, respectively.

May-June 1986 75

LUNA, REID

PLANNING HORIZON

© I =P(High hterest Rates) ©2= P( Average Interest Rates) ©3=P{Low Interest Rates)

$737 $881 $1025

Figure 1: This decision tree graphically illustrates the mortgage alternatives with costs re- flecting $1,000 investment increments. The decision maker must select one of three mortgage types: (1) a three-year adjustable rate mortgage (ARM), (2) a five-year ARM, or (3) a fixed-rate conventional mortgage. The probabilities of high, medium, and low future interest rates occur- ring and costs associated with these events are recorded respectively on appropriate tree branches for three different planning horizons (five, six, and seven years).

INTERFACES 16:3 76

MORTGAGE SELECTION

(1.29 and 0.83, respectively), significant variability in these values can be expected to occur.

Table 3 presents the procedure we used to select the range of T-bill index rate possibilities that could be associated with ARM future interest rates. We calculated the expected increase in the mean and the standard deviation of the ARMs by multiplying the original T-bill index rate by the expected fractional changes in the index rate. The range over which the new index rate may be expected to vary is plus or minus one standard deviation. We rounded these values to the nearest one- eighth percent to reflect traditional inter- est rate change increments.

A decision tree provides a schematic il- lustration of the interaction between alter- native decisions and probable states of nature which produce various outcomes. Its structure helps the decision maker un- derstand the options available and possi-

ble outcomes. Figure 1 shows a decision tree that identifies the relative costs asso- ciated with the different mortgage ar- rangements in terms of $1,000 investment increments. This basic unit of borrowing permits the results to be generalized for any amount of investment. The tree pre- sents the three mortgage alternatives and their associated interest costs under a probable range of interest rates.

We used four decision criteria to assess cost differences between the three mort- gage types over a five-, six-, and a seven- year planning horizon (see Table 4). The first three decision criteria represent dif- ferent managerial attitudes toward deci- sion making while the last criterion incorporates the probability of different interest rates prevailing.

The first decision criterion, minimax, reflects a conservative or pessimistic ori- entation toward the future. For each mortgage type, the circumstances that

Decision Criteria

Minimax (pessimistic)

Minimin (optimistic)

Minimize the maximum regret

Expected Value (Bayes)

Type of Mortgage

3-yr ARM 5-yr ARM Conventional

3-yr ARM 5-yr ARM Conventional

3-yr ARM 5-yr ARM Conventional

3-yr ARM 5-yr ARM Conventional

5 Years

$792 695* 737

$664* 695 737

$ 97 31* 73

$727 695* 737

Planning Horizon 6 Years

$980 901 8 8 r

$789* 823 881

$ 99 34* 92

$883 867* 881

7 Years

$1228 1107 1025*

$ 907* 951

1025

$ 203 82*

118

$1065 1040 1025*

Table 4: Costs per $1,000 of mortgage value associated with each of the four standard decision criteria. The minimum cost alternative is designated by an asterisk {*) under each of the three planning horizons.

May-June 1986 77

LUNA, REID

produce the highest borrowing costs are assumed to prevail. The decision maker then selects the mortgage type which minimizes total costs. In this case, the five-year ARM is preferred for a planning horizon of five years. However, if the con- dominium is held for six or seven years, the conventional mortgage has the lowest total costs.

The second decision criterion, minimin, reflects an optimistic attitude regarding future events. Using this criterion, favora- ble situations leading to the lowest inter- est rates are assumed to occur. For this case, the decision maker will select the three-year ARM regardless of planning

horizon in order to minimize costs. Minimizing the maximum regret is the

third criterion considered. It refers to re- ducing lost opportunities that result from selecting a mortgage alternative that proves to be less than optimal. In other words, it seeks to minimize the incur- rence of additional mortgage costs associ- ated with making, what future hindsight shows to be, a poor or nonoptimal deci- sion. The five-year ARM prevails under each of the planning horizons when the decision maker seeks the alternative that guards against large opportunity losses.

The minimization of expected mortgage costs is the final criterion considered in

Type of Mortgage

First 3-year ARM

Second 3-Year ARM

5-Year ARM

Index rate

values

Low Mean High

Low Mean High

Low Mean High

Low Mean High

Lew Mean Hieh

Current index

rate (%)

ioy4 10'/4

9^2

9»/2

9y2

12% 12% 12%

163/8

16% 16%

10% 10% 10%

Interest rate {%)

differential

- 3/4%

+ 2% + 61/8

- 3 / 4

+ 2% + 6V8

+ 6'/8

- 3 / 4

+ 2% + 6'/8

- 3 / 4

+ 4% + 73/4

Future index

rate (%)

9'/2% 12% 16%

83/4

15%

15'/2 19

15% 19 22'/2

9% 14% 183/8

Future loan

rate (%)

12% 15% 18%

14% I8y8

14% 18 21 y2

181/8

25

123/8

173/8

20%

Probability of future loan rate

0.27 0.46 0.27

0.28 0.48 0.24

0.33 0.31 0.36

0.25 0.30 0.45

0.22 0.51 0.27

Table 5: Expected future index rates, their impact on future loan rates, and their probabilities of occurrence. The future loan rate values were calculated by first determining the normalized cu- mulative probabilities of the three index rate values IP (IRV = Low, Mean, High)} and then using the following formulas:

paRV= Low) - \1P(1RV= Low) + VURV= piIRV=Mean) = \2PiIRV=High) + P{mV= p{IRV=High = 1.00 - p(IRV= Low)~p(IRV= Mean).

INTERFACES 16:3 78

MORTGAGE SELEGTION

this problem. This approach requires the decision maker to assign probabilities that various interest rates will prevail in the future (see Table 5). By assuming that the index rate differentials are normally dis- tributed, the methodology for calculating these probabilities intentionally increases the chance of the mean values occurring. This conservative orientation results in the five-year ARM being the alternative of choice under both the five- and six-year planning horizons. The conventional mortgage has the lowest expected cost if the condominium is held for seven years.

In this analysis, we have assumed that all of the mortgages would be held to ma- turity. However, a large decrease in inter- est rates should prompt the informed borrower to repay the original mortgage and refinance. This decision becomes via- ble when the cumulative monetary bene- fits of a new lower monthly payment over the remaining planning horizon equal the cost of refinancing. Table 6 shows the re- quired decrease between original and cur- rent interest rates to make refinancing an

attractive choice for situations where the refinancing fees vary between two and four percent. The table has been con- structed so that the present value of a se- ries of payments representing the monthly savings is equal to the total refi- nancing costs. These results are valid for initial interest rates that are similar to those considered in this illustration. Since average refinancing costs are approxi- mately 2.5 percent [Business Week 1985], refinancing becomes an attractive alterna- tive whenever the decrease in interest rates ranges between 2% and % percent. These values from the third row of Table 6 illustrate that the threshold percentage depends upon the time remaining to a change in ARM rates or the end of the in- vestment period. Discussion of Results

It is possible that a potential conflict between consumption and investment at- tributes in the condominium purchase could influence the mortgage decision. If the consumption component is large, then the couple may select a mortgage

Refinancing Costs (%)

12 24

Time remaining (months) to ARM rate change or in planning horizon

36 48 60 72 84

2.00% 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00

2'/4% 2y8 2%

3% 3% 3% 41/4

41/2

V/4%

VA V/H

2 2'/8 2% 2'/2

1

VA V/8 V/2 V/2 V/H

VA

3/4%

1 V/8 V/8 VA V/8 v/8

3/4

3/4

V2%

3/4

3/4

1 1 v/8

1/2%

3/4

1

Table 6: Minimal percentage decrease in interest rales needed to support the refinancing of Ihe original mortgage. It is noteworthy thai points required for refinancing a 30-year conventional fixed-rate mortgage varied between 1 and 3 white those required for a 30-year ARM ranged between 1.5 and 4 [Business Outlook 19851.

May-June 1986 79

LUNA, REID

that has less favorable terms than might be acceptable under a pure investment decision. Hov̂ êver, in tbis analysis, the married couple's consumptive use is lim- ited to less than two weeks per year, and consequently, the mortgage decision crite- ria will be based solely on investment attributes.

The results of our structured approach must be analyzed from the perspective of the decision makers. In this situation, the couple wished to avoid high mortgage costs if interest rates increased signifi- cantly and were willing to forego cost sav- ing if future rates became more favorable. In the latter case, the option of refinanc- ing would be availabie whereas no such opportunity exists should high rates pre- vail in the future. Their conservative in- vestment goals most closely match the first criterion of minimizing the maxin:ium mortgage costs or the third criterion of minimizing the maximum regret (poten- tial opportunity costs).

For these goals, the findings presented in Table 4 show that the five-year ARM is favored if the condominium is held for five years. If the investment is made for six or seven years, either the five-year ARM or the conventional mortgage is pre- ferred. The option of selling the property or simply refinancing (should prevailing interest rates warrant it) appeared very attractive to the decision makers and so they chose the five-year ARM.

Although the expected value criterion also supports the choice of the five-year ARM for a five- or six-year planning hori- zon, it is important to remember that this criterion is most appropriate when em- ployed in circumstances where a large

number of similar decisions will be made over a relatively long duration. This long- run perspective is required in order to dampen short-term variations which yield deviations from expected outcomes.

The use of a structured decision tree analysis for this situation assumes that the future will be relatively well behaved, that is, follow the general trend of the past. Whereas this assumption is sup- ported statistically, conventional wisdom assigns significant uncertainty to future interest rates. Although this uncertainty is reflected in a range of future interest rate possibilities through the decision tree, it is noteworthy that future interest

This approach provides, at a minimum, a rational frame- work for what is frequently decided in a very intuitive or subjective manner.

rates for ARMs depend on a single month's index rate rather than an annual average which would dampen extreme variability. Thus, although our analysis considered index rates ranging from 8% to 22V2 percent, we may not have cap- tured the total variability appropriately. While this same variability affects all deci- sion criteria, the analysis of extremes without probabilistic weightings may be better suited for this decision situation.

The decision makers are comfortable with the methodology and analysis pre- sented here as well as their final choice of mortgage type. While future realities re- flecting more extreme variations in index

INTERFACES 16:3 80

MORTGAGE SELEGTION

rates could challenge the validity of this structured approach to decision making under uncertainty, this approach pro- vides, at a minimum, a rational frame- work for what is frequently decided in a very intuitive or subjective manner.

References Board of Governors of the Federal Reserve

System, Federal Reserve Bulletin, 1950-1971, Volumes 36-57, Qu\y) Table P, Washington DC.

Business Week 1985, "How to lighten a heavy mortgage," April 29, p p . 124-125.

Business Outlook 1985, "Money rates," May 6, p. 16.

US Bureau of the Census, Statistical Abstract of the United Slates: 1984, Washington DC.

US Office of the President, Economic Report of the President Transmitted to the Congress — Feb- ruary 1983, p p . 240-241, Washington DC.

May-June 1986 81