Mortgage Selection Using a Decision-Tree Approach: An Extension
BETTY C . H E I A N
JAMES R . GALE
School of Business and Engineering Administration Michigan Technological University Houghton, Michigan 49931
School of Business and Engineering Administration Michigan Technological University
Mortgages are available at various interest rates and vary from traditional fixed-rate contracts to adjustable-rate con- tracts with a wide range of specific features. A method of comparison using decision-tree analysis recognizes the bor- rower’s concern with both the expected value and the varia- bility of possible outcomes. The expected value and the variance for each of three specific mortgages are calculated using plausible assumptions regarding time preferences. The rational choice among mortgages for the risk-averse borrower depends on the terms or features of the mortgage and the individual’s expectations and beliefs.
Borrowers, confronted with several al-ternative mortgage contracts, seek a systematic and consistent method for choosing a mortgage. In a recent article in Interfaces, Robert E. Luna and Richard A. Reid suggest a decision-tree approach to this problem and apply the approach to a specific case: a choice among a con- ventional fixed rate mortgage (FRM), an adjustable rate mortgage for which the
mortgage rate adjusts at three-year inter- vals (ARM-3), and an adjustable rate mortgage for which the mortgage rate a( justs at five-year intervals (ARM-5) [Lun and Reid 1986]. Their claim is ” . . . this approach provides, at a minimum, a ra- tional framework for what is frequently decided in a very intuitive or subjective manner.” As the following discussion w show, it is neither possible nor desirabl<
Copyright © 1988, The Institute of Management Sciences
0091-2102/88/1804/0072$01.25 This paper was refereed.
DECISION ANALYSIS — RISK FINANCE
INTERFACES 18: 4 July-August 1988 (pp. 72-83)
MORTGAGE SELECTION
to eliminate from the decision process in- dividual preferences, which are by nature subjective.
Typically, a decision-tree can be used if the consequences of each alternative de- pend on uncertain, discrete future events that can be described probabilistically. For the mortgage choice problem, the uncer- tain future events are the market interest rates to which the adjustable rate mort- gage (ARM) rates adjust. Thus, the objec- tive consequences of each mortgage choice, which depend on these market in- terest rates and the terms of the mort- gage contract, can also be described probabilistically. The borrower is assumed to choose among the available mortgage contracts or to not borrow using choice criteria applied to these probabilistic con- sequences. The operational validity of the approach depends critically on the proba- bilistic description of market interest rates, the simulation of the objective con- sequences for the borrower, and the borrower’s choice criteria.
The decision-tree approach, because it isolates the probabilistic element in the choice among mortgages, permits careful consideration of the borrower’s criteria for choice. Borrower concerns about the tim- ing of payments are defined as time pref- erences and are reflected in discount rates used to compute present values. Borrower concerns about the uncertainty of pay- ments are defined as risk preferences. Be- cause Luna and Reid ignore both the borrower’s time preferences and prefer- ences with respect to risk in their defini- tion of mortgage cost, they fail to address two of the central dilemmas facing the borrower. These are (1) how to choose
between two mortgages for which the sum of the payments is identical but one of which has lower payments in early years and higher payments in later years than the other; and (2) how to choose between two mortages, one of which has a higher expected obligation but less risk or uncer- tainty than the other. Luna and Reid im- plicitly make very specific assumptions regarding time preferences, the central is- sue in the first problem, and through the choice of decision rules treat only extreme cases of preferences with respect to risk.
This suggests the need for, first, a modified definition of the mortgage obli- gation, second, a method for valuing the obligation that incorporates time prefer- ences, and third, a method of defining risk to permit comparison among alternative mortgages. The Mortgage Obligation
A mortgage contract obligates the bor- rower to make a series of payments over time that will fully amortize the loan, if it is held to maturity. These payments will cover any origination fees and interest charges, which are generally regarded as the costs of borrowing, as well as the principal repayment, which frequently is not considered a cost of borrowing.
Luna and Reid, for the purposes of constructing their decision tree, define mortgage cost per $1,000 borrowed as the sum of the origination fees and the pay- ments made up to the termination of the mortgage. Using this definition, mortgage cost includes origination fees, interest charges, and payments on the principal made prior to the termination date but neglects the balance that must be paid when the loan is terminated. Mortgage
July-August 1988 73
HEIAN, GALE
obligation is a more appropriate term for payments that cover fees, interest costs, and principal repayment. For consistency with respect to various termination dates, the balance at the termination date should be included as part of the mort- gage obligation.
Thus the mortgage obligation can be defined as a series of payments, X(0; i, m) . . . X(T; i, m) given / and m where i = a specific state of the world, defined
here as a specific pattern of market interest rates;
m = a specific mortgage contract in this example taking values FRM, ARM-5, and ARM-3;
T = the termination year; X(0; I, m) = origination fees (paid at
f = 0); X(l; i,m)…. X ( r – 1 ; i, m) = annual
payments for years 1 through T – l , which may vary for adjustable rate mortgages; and
X(T; I, m) = the payment for year T plus the outstanding balance if any.
The mortgage obligation will depend on the terms of the contract and the state of the world as indicated by the market interest rates at each adjustment point. A specific contract may include a maximum rate, a limit on the periodic rate increase, or a payment cap with negative amortiza- tion (a provision for increasing the loan balance rather than increasing the pay- ment by the full amount implied by the mortgage rate increase). If negative amor- tization is significant, recognition of the balance outstanding at the termination of the mortgage can be important to the bor- rower. Otherwise, the balance outstand- ing declines as the period the mortgage is
held increases and becomes less impor- tant as the borrower’s time preferences rise. Time Preferences
Choosing among alternative mortgages involves comparing mortgage obligations that are streams of payments made through time. In order to be compared, each stream of payments must be valued at the same point in time. This can be done using the present value or possibly the terminal value. The discount rate used to compute the present value of the mortgage obligation for purposes of com- parison is a measure of how a borrower values money available at the beginning of the period relative to money available toward the end of the period, in other words, a measure of the borrower’s time preferences.
The present value of the mortgage obli- gation, Z{i,m,T,r), for the ith state of the world, for mortgage contract m, for termi- nation date T, using discount rate r is then
Z ( , , m , 7 ; r ) = S
where Z{i,m,T,r) = the present value of the mortgage obligation, and r is a dis- count rate which measures a borrower’s time preferences.
The value the borrower places on the mortgage obligation will also depend on the borrower’s time preferences. Thus, the present value of the mortgage obliga- tion may take on a different value for each possible pattern of interest rates, for each mortgage contract, for each discount rate, and for each termination date under consideration.
INTERFACES 18:4 74
MORTGAGE SELECTION
In general, under conditions of cer- tainty, a willingness to borrow implies a time preference rate at least as great as the loan rate. Under these assumptions, a rational individual is assumed to enter into a mortgage contract only if the value to the individual of what is received (the proceeds of the loan) is greater than or equal to the value to the individual of what is promised (the obligation under- taken). (For a brief discussion of time preferences in the context of investment decisions, see Harris [1981, p. 25].) Risk Preferences
Adjustable rate mortgages (ARM) are believed to transfer part of the risk inher- ent in the variation of market interest rates from the lender to the borrower. At
It is neither possible nor desirable to eliminate from the decision process individual preferences, which are by nature subjective.
the same time, borrowers are reluctant to accept ARM loans because of the uncer- tainty about the actual payments they will have to make. They seem to be concerned about the most likely level of payments and about the range of possible payments.
In the context of individual investment decisions, people are considered risk averse if they will accept additional risk only in return for additional compensa- tion. (See Radcliffe [1982] for a discussion of risk aversion.) For an investment op- portunity, this means accepting more risk, meaning more variability in the re- turn, only if the average or expected
value of the return is higher. For an adjustable-rate loan, this means accepting more risk, meaning greater variability in the obligation, only if the average or ex- pected value of the obligation incurred is lower. In other words, for the risk-averse borrower, the compensation for accepting the greater uncertainty inherent in an ARM is a reduction in the expected value of the mortgage obligation.
The uncertain future behavior of the market interest rates can be quantified us- ing the decision-tree approach. Then, for each possible interest-rate pattern or state of the world, the present value of the mortgage obligation can be calculated given the terms of the mortgage, the ter- mination date, and the discount rate. For each mortgage, a set Z(m,T,r) can be de- fined which is composed of Z{i,m,T,r), where ; represents a distinct interest-rate pattern generated by the decision tree. The number of interest-rate patterns in each set may vary and will depend on the frequency of adjustment and the assumed termination date of the mortgage under consideration.
Each possible adjustment in the decision-tree structure has a probability assigned to it. The probability of each possible interest-rate pattern can be de- rived from these probabilities for high, mid, or low adjustments and appropriate assumptions regarding their indepen- dence. Luna and Reid have assumed that the probability of a high, mid, or low ad- justment at any one adjustment point is independent of the adjustment at any other adjustment point, and the same as- sumption is made here. Using these prob- abilities for various states of the world.
July-August 1988 75
HEIAN, GALE
Year 1 8
mortgage
FRM I- 14.125%-
ARM-5 1- 13.125%- P(2) = 1
20.875%- p(8)=.27
-17.375% — p(9) = .51
-12.375% — p(10)=.22
ARM-3 12.75%- P(3) = 1
-18.875%- p(5)=.27
15.375%- p(6) = .46
25.0%- p(11)=.1215
-23.125% 1 p(12) =.0810
-18.125% 1 p(13)=.O675
-21.5% 1 p(14) =.1656
^
•12% P(7)=.27
18.0%— p(15) = .1426
-14.625% 1 p(16) = .1518
-18.125% 1 p( 17) = .0648
-14.625% 1 p(18)=.1296
11.25% • p(19) = .O756
Figure 1: The Luna and Reid mortgage rate scenario: The mortgage rate is shown for each in- strument for the indicated time periods. The number in each parenthesis is the probability of the sequence of interest rates to that point. For example, p (11) is the probability that the inter- est rates for ARM-3 will be 12.75 percent for years 1, 2, and 3; 18.75 percent for years 4, 5, and 6; and 25.0 percent for years 7, 8, and 9. (For a detailed explanation of the assumptions underly- ing this interest rate scenario, see Luna and Reid [1986].)
INTERFACES 18:4 76
MORTGAGE SELECTION
the expected value and the standard de- viation of each set Z{m,T,r) can be calcu- lated using conventional statistical definitions.
The expected value of Z{m,T,r) is N
E[Z(m,T,r)] = 2 Z(/, m, Z r) • [p ff)]- 1 = 1
The standard deviation of Z{m,T,r) is SD[Z(m,T,r)]
=VE[Z(i,m,T,r)]-E[Z{mXr)Y
where Z(m,T,r) = the set of possible out- comes given the probabilistic description of the market interest rates for mortgage contract tn, terminated at T, and valued using r, and p(i) = the probability of the occurrence of the ith interest-rate pattern or state of the world. Mortgage Selection Comparisons: Time Preference Effects
Luna and Reid show a decision tree for the FRM, ARM-5, and ARM-3, using their “mortgage cost” definition and their assumptions concerning the probabilities of various possible states of the world [1986, Figure 1, p. 74]. This approach con- founds the analysis of the uncertain events (the future mortgage rates) with the analysis of their consequences (the mortgage obligations) and with the analy- sis of borrower’s criteria for comparisons among these consequences.
The tree structure in Figure 1, based on the Luna and Reid assumptions implicit in their Figure 1, summarizes the effects of possible market interest rates on the mortgage interest rates for each mortgage contract along with the probability of each possible mortgage rate sequence. Luna and Reid base their assumptions
about the amounts, direction, and proba- bilities of adjustments in the market inter- est rates on an analysis of the historical record for the market interest rates speci- fied in the adjustable rate mortgage con- tracts, the three-year United States Treasury bill rates adjusted for constant maturities for ARM-3, and the five-year Treasury bill rates adjusted for constant maturities for the ARM-5.
As can be seen from Figure 1, Luna and Reid assume that interest rates are much more likely to rise than to fall, tending to make the ARM less attractive. The probability of interest rates on ARM being below that of the FRM after the first adjustment period is 0.22 while the probability of the rate on ARM-3 being below that on the FRM is 0.27 for years 4, 5, and 6 and 0.0756 for years 7, 8, and 9. Even this scenario is more favorable to ARM-5 than a consistent implementation of their methodology would suggest. (From Luna and Reid’s Table 3 and dis- cussion, the low adjustment for ARM-5 should be 4.25-3.5= to 0.75 not -0.75 [1986, p. 75].)
For purposes of comparison, we simu- lated the mortgage obligations, X(0; . .) . . . X(r; . .), for the three alternative mortgage contracts for termination dates up to nine years using the interest-rate patterns shown in Figure L We then val- ued these individual simulated mortgage obligations, Z{i,m,T,r), for discount rates between zero and 20 percent. We applied the minimax decision rule, the minimin decision rule, and the expected-value de- cision rule suggested by Luna and Reid for termination dates five through nine. The result for selected time preference
July-August 1988 77
HEIAN, GALE
Termination in year 5
Discount rate ARM-5 ARM-5 FRM FRM FRM ARM-5 none none none none
(FRM) (FRM) (FRM) (FRM)
none none none none none
Luna and Reid choice
(ARM-5) (FRM) (FRM) (FRM) (FRM) none none none none none (ARM-5) (FRM) (FRM) (FRM) (FRM)
ARM-5 FRM FRM n.a. n.a.
Table 1: Mortgage choice using the minimax decision rule: The most attractive mortgage, if the possibility of not borrowing is excluded, is shown in parenthesis.
Termination in year 5
Discount rate 16% 14%
0%
Luna and Reid choice
ARM-3 ARM-3 ARM-3 ARM-3 ARM-3 ARM-3 ARM-3 ARM-3 ARM-3 ARM-3 none none none none none (ARM-3) (ARM-3) (ARM-3) (ARM-3) (ARM-3) none none none none none (ARM-3) (ARM-3) (ARM:3) (ARM-3) (ARM-3)
ARM-3 ARM-3 ARM-3 n.a. n.a.
Table 2: Mortgage choice using the minimin decision rule: The most attractive mortgage, if the possibility of not borrowing is excluded, is shown in parenthesis.
Termination in year 5 6 7 8 9
Discount rate
12%
Luna and Reid choice
ARM-5 ARM-5 ARM-5 FRM FRM ARM-5 none none none none
(ARM-5) (ARM-5) (FRM) (FRM) none none none none none (ARM-5) (ARM-5) (ARM-5) (FRM) (FRM) none none none none none (ARM-5) (ARM-5) (FRM) (FRM) (FRM)
ARM-5 ARM-5 FRM n.a. n.a.
Table 3: Mortgage choice using the expected value rule: The most attractive mortgage, if the possibility of not borrowing is excluded, is shown in parenthesis.
rates are shown in Tables 1, 2, and 3 along with the Luna and Reid choices.
We selected zero percent, 12 percent, 14 percent, and 16 percent time preference rates for presentation. Zero percent is shown for comparison with the Luna and Reid choices. Twelve percent is below the mortgage rate for nearly all the possible outcomes; 14 percent is above that re- quired for borrowing for some of the ARM outcomes; and 16 percent is above that required for borrowing for a fairly wide range of outcomes. The operational significance of the discount rate is clear in Table 1, in which the minimax rule is used. When the value of the mortgage obligations is computed using a 12 per- cent discount rate, even the minimum of the maximum valued obligations is above the loan amount of $1,000 and the ra- tional borrower will not borrow. When the mortgage obligations are valued using 14 percent, the ARM-5 for the five-year termination is acceptable. For termination dates six through nine, the present value of the FRM is below that of the maximum for both ARM-3 and ARM-5 but unac- ceptable because it is above $1,000. Using the 16 percent discount rate lowers the present values for all mortgage obliga- tions. For termination dates of five and six years and a time preference of 16 per- cent, the early low payments on ARM-5 are sufficiently important to the borrower to select ARM-5 using the minimax rule, while for terminations of seven, eight, or nine years the longer period of paying the lower FRM payment dominates. In gen- eral, the higher the discount rate, the more importance the borrower places on relatively low early payments as com-
INTERFACES 18:4 78
MORTGAGE SELECTION
pared to relatively low later payments. The Luna and Reid choice, which in
addition to neglecting the outstanding balance at the termination of the mort- gage, values a dollar paid at the end of five years as equivalent to a dollar paid at the beginning, is also shown in Table 1. ARM-5 is effectively a fixed-rate contract prior to the first adjustment in year six with a rate below the FRM rate and, con- sequently, a payment below the FRM pay- ment in each of the first five years. In such cases the comparison of the mort- gage obligation values will be invariant
The Luna and Reid choice, in addition to neglecting the outstanding balance at the termination of the mortgage, values a dollar paid at the end of five years as equivalent to a dollar paid at the beginning.
with respect to discount rates. If early payments for one mortgage are below and later payments above those of the other mortgage, the discount rate used to com- pute the present value will determine which of the two mortgages has the smaller present value. Luna and Reid do not avoid assuming a time preference, rather they assume a time preference of about 14 percent when they assume bor- rowing will take place and simultaneously a time preference of zero percent in their comparisons among alternative mortgages.
The minimin rule, shown in Table 2, essentially assumes the most rapidly de- clining of the possible interest rate pat-
terns considered will prevail for both adjustable rate mortgages. Thus, ARM-3 which has an initially lower payment than either the FRM or ARM-5 and a lower payment each year, has a lower present value regardless of the time preferences of the borrower. Nonetheless, the rational borrower with a time preference of 12 percent or less will not borrow even as- suming this falling pattern of future interest rates were certain to prevail.
The expected value rule selects the mortgage with the lowest expected pres- ent value, as defined above, for each time preference and termination date. The ex- pected present values of the mortgage ob- ligation for selected discount rates and for termination dates from five to nine years are shown in Table 4. The expected value rule selections are shown in Table 3. The pattern of choices is similar to that of the minimax rule, but because the expected value for ARM-5 is lower than the maxi- mum value, ARM-5 is chosen over FRM for higher discount rates and longer holding periods. Mortgage Selection Comparisons: Risk Preferences Effects
Implicitly Luna and Reid deal with the risk preferences of the borrower through their choice of decision rules. Their mini- max decision criterion assumes that the outcome for each mortgage will be the least favorable (the highest valued mort- gage obligation) under each set of as- sumptions. In effect, the minimax decision rule assumes that the worst case for each mortgage will occur with cer- tainty and selects the mortgage with the minimum value from among these.
The minimin decision rule, in contrast.
July-August 1988 79
HEIAN,
Mortgage
FRM
ARM-5
ARM-3
Mortgage
FRM
ARM-5
ARM-3
Mortgage
FRM
ARM-5
ARM-3
Mortgage
FRM
ARM-5
ARM-3
GALE
Termination in year
E(Z) SD(Z)
E(Z) SD(Z)
E(Z) SD(Z)
Termination in year
E(Z) SD(Z)
E(Z) SD(Z)
E(Z) SD(Z)
Termination in year
E(Z) SD(Z)
E(Z) SD(Z)
E(Z) SD(Z)
Termination in year
E(Z) SD(Z)
E(Z) SD(Z)
E(Z) SD(Z)
5
$ 1719.30 0
1673.77 0
1707.48 50.04
5
$ 1093.68 0
1062.78 0
1080.91 30.00
5
$ 1021.77 0
992.66 0
1009.04 2770
5
$ 956.43 0
928.97 0
943.77 25.62
0% discount –
6
$ 185799 0
1838.09 29.52
1858.61 75.27
12% discount •
6
$ 1104.25 0
1086.55 14.96
1097.88 42.59
14% discount
6
$ 1022.33 0
1004.97 13.43
1015.37 38.90
16% discount
6
$ 948.88 0
932.16 12.12
941.43 35.77
7
$ 1995.94 0
2001.81 59.23
2036.98 10720
7
$ 1113.64 0
110771 28.29
1125.61 56.54
7
$ 1022.82 0
1016.02 25.21
1032.07 51.25
7
$ 942.40 0
934.92 22.52
949.31 46.56
8
$ 2133.62 0
2164.85 89.11
2214.74 142.95
8
$ 1121.97 0
1126.54 40.17
1150.31 70.34
8
$ 1023.24 0
1025.57 35.5
1046.69 63.13
8
$ 936.85 0
93731 31.44
956.11 56.81
9
$ 2269.14 0
232709 119.19
2391.78 180.41
9
$ 1129.35 0
1143.29 50.75
1172.29 83.18
9
$ 1023.61 0
1033.92 44.48
1059.49 73.99
9
$ 932.10 0
939.38 39.09
961.00 66.02
Table 4: Expected value and standard deviation for selected mortgage obligations: E(Z) is the expected value and SD(Z) is the standard deviation of Z{m,T,r). The mortgage obligations were simulated using the interest rate index scenario suggested by Luna and Reid.
INTERFACES 18:4 80
MORTGAGE SELECTION
assumes the “best” lowest-valued mort- gage obligation will occur with certainty and selects the mortgage with the mini- mum lowest-valued obligation. In neither of these cases is there any attempt to deal with the possibility that extreme interest rate patterns will occur with a very low probability. Thus, the decisions are hard to reconcile with intuition about borrower preferences. For example, if the worst case (maximum value of the mortgage ob- ligation for all considered possibilities) for one adjustable-rate contract has a low probability while an alternative fixed-rate mortgage obligation is certain and, given the discount rate, has a value just slightly lower than the ARM, the minimax rule will select the fixed-rate instrument. This assures the borrower of a certain mort- gage obligation with a value nearly as high as the worst possible outcome for the adjustable rate instrument. In con- trast, if the minimin rule is applied, the adjustable rate instrument will be selected even if the probability of the ARM value being below the value of the FRM is very small. If borrowers are risk averse, they are most likely to prefer the FRM if its value is close to the minimum possible ARM value. They are likely to accept some variability in their mortgage obligation if they perceive a low probability for the ARM value being higher than the FRM value.
The discussion of risk aversion can be formalized by postulating the existence of a preference map for the risk-averse bor- rower over the expected present value of the mortgage obligation and its standard deviation as defined above. If it is as- sumed that given the expected value of the mortgage obligation, a smaller stand-
Expected Value Z(m,T,R)
A Is preferred to points In this
region.
Points In region are preferred to point A.
Standard Deviation Z(m,T,r)
Figure 2: Preference space for expected value and standard deviation of mortgage contracts: Expected value, standard deviation pairs be- tow and to the teft of A are unambiguously preferred to A by risk-averse borrowers. Point A is unambiguously preferred to points above and to the right of it. Points in the shaded re- gions can be compared if the borrower’s pref- erences for risk relative to obligation are known.
ard deviation is preferred to a greater standard deviation, and given the stand- ard deviation, a smaller expected value of the mortgage obligation is preferred to greater expected value, then the limits to the borrowers’s preference map can be shown as in Figure 2. Risk-averse borrow- ers necessarily prefer situations to the left and below point A to point A, and prefer point A to points to the right and above it. Borrowers may prefer, be indifferent between, or not prefer points to the left and above or to the right and below point A (the shaded area in Figure 2) depend- ing on the individual’s willingness to trade lower expected values of the mort- gage obligation for greater uncertainty.
The minimize-the-maximum-regret cri- teria also ignores the uncertainty of possi- ble outcomes. The fourth decision rule, choose the mortgage with the minimum expected present value for the mortgage obligation, recognizes that the possible outcomes for each alternative mortgage should be thought of as occurring with some specific probability. By looking
July-August 1988 81
HEIAN, GALE
exclusively at the expected value, how- ever, it implies the borrower would choose the mortgage with the lower expected or mean value of the mortgage obligation regardless of the standard de- viation of the outcomes.
In Table 4, expected values and stan- dard deviations of the mortgage obliga- tion for selected termination dates and se- lected discount rates are reported. These results illustrate the importance and the feasibility, given the decision-tree ap- proach to analyzing the behavior of mar- ket interest rates, of considering both the expected present value and the standard deviation of the mortgage obligation. For example, for the six-year termination date valued using a 16 percent discount rate, the expected present value for all three al- ternatives is below the loan amount ($1,000). Using the expected present value, standard deviation criteria ARM-5 is clearly preferred over ARM-3, because it has both a lower expected present value and a lower standard deviation. The com- parison between the FRM with an ex- pected present value of $948.88 and zero standard deviation and ARM-5 with a lower expected present value of $932.16 and a higher standard deviation of 12.12 is ambiguous. In this case, the borrower may, ir principle, be indifferent between the two choices, but probably will prefer one mortgage to the other depending on his or her preferences regarding risk and obligations.
The mortgage selections based on the expected value, standard deviation rule are shown in Table 5. For the 16 percent discount rate, there is a clear choice for termination in years five, eight, and nine
Termination
year
Discount rate
16%
14%
12%
0 %
5
ARM-5
ARM-5
none
(ARM-5)
none
6
***
none ***
none ***
none
7
*»*
none ***
none »*»
none
8
FRM
none
(FRM)
none
(FRM)
none
9
FRM
none
(FRM)
none
(FRM)
none
(ARM-5) ” ‘ (FRM) (FRM) (FRM)
Table 5: Mortgage choice using the expected value-standard deviation rule: *** indicates there is no clear choice. ARM-5 has a lower expected value and a larger standard deviation than FRM. The most attractive mortgage, if the possibility of not borrowing is excluded, is shown in parenthesis.
— all cases in which the mortgage with the lowest expected value has a zero standard deviation. For termination in years six and sever, the ARM-5 has a lower expected present value but a higher standard deviation. The borrower must choose between a higher expected pres- ent value with a lower standard deviation and a lower expected present value with a higher standard deviation of the mortgage obligation. Conclusion
In general, the ratioral borrower would like to know the distribution of the possi- ble consequences of each mortgage con- tract. Luna and Reid go a step in that direction. The decision-tree approach al- lows the development of interest-rate scenarios that incorporate both the direc- tion and magnitude of changes in the un- derlying index rates and the probabilities associated with these changes. However, they have failed to utilize the full power of their innovation.
The decision-tree approach suggested by Luna and Reid and extended in this paper treats the ARM as a risky liability
INTERFACES 18:4 82
MORTGAGE SELECTION
and assumes implicitly that the terms of the mortgage contracts reflect market- clearing prices for risk and return. The borrower, facing several alternative mort- gage contracts is assumed to make a par- tial equilibrium choice, the mortgage that best fits the borrower’s risk and outlay preferences given time preferences. An alternative approach is to treat the ARM as an option written by the lender in which the borrower may continue borrow- ing under the contract terms or terminate the loan at will. This is likely to be partic- ularly fruitful if the question of ARM pricing is addressed from the lender’s perspective. Our approach, focusing as it does on the borrower’s risk and time preferences in a partial equilibrium framework, extends our understanding of the possible benefits of ARM to the individual. References Harris, Laurence 1981, Monetary Theory,
McGraw-Hill, New York. Luna, Robert E. and Reid, Richard A. 1986,
“Mortgage selection using a decision tree approach,” Interfaces, Vol. 16, No. 3 (May- June), pp. 73-81.
Radcliffe, Robert C. 1982, Investment Concepts, Analysis, and Strategy, Scott, Foresman and Company, Glenview, Illinois.
July-August 1988 83
