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Mortgage Selection Using a Decision-Tree

Approach: An Extension

BETTY C. HEIAN

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 contracts with a wide range of specific features. A method of

comparison using decision-tree analysis recognizes the borrower’s concern with both the expected value and the variability 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.

B

orrowers, confronted with several alternative 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 conventional fixed rate mortgage (FRM), an

adjustable rate mortgage for which the

mortgage rate adjusts at three-year intervals (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 rational 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 individual preferences, which are by nature

subjective.

Typically, a decision-tree can be used if

the consequences of each alternative depend on uncertain, discrete future events

that can be described probabilistically. For

the mortgage choice problem, the uncertain future events are the market interest

rates to which the adjustable rate mortgage (ARM) rates adjust. Thus, the objective consequences of each mortgage

choice, which depend on these market interest rates and the terms of the mortgage 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 consequences. The operational validity of the

approach depends critically on the probabilistic description of market interest

rates, the simulation of the objective consequences 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 timing of payments are defined as time preferences and are reflected in discount rates

used to compute present values. Borrower

concerns about the uncertainty of payments are defined as risk preferences. Because Luna and Reid ignore both the

borrower’s time preferences and preferences with respect to risk in their definition 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 uncertainty than the other. Luna and Reid implicitly make very specific assumptions

regarding time preferences, the central issue 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 obligation, second, a method for valuing the

obligation that incorporates time preferences, and third, a method of defining

risk to permit comparison among

alternative mortgages.

The Mortgage Obligation

A mortgage contract obligates the borrower 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 payments 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 mortgage 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 amortization (a provision for increasing the loan

balance rather than increasing the payment by the full amount implied by the

mortgage rate increase). If negative amortization is significant, recognition of the

balance outstanding at the termination of

the mortgage can be important to the borrower. Otherwise, the balance outstanding declines as the period the mortgage is

held increases and becomes less important 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 comparison 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 obligation, Z{i,m,T,r), for the ith state of the

world, for mortgage contract m, for termination 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 discount 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 obligation 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 certainty, 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 undertaken). (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 inherent 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 uncertainty 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 compensation. (See Radcliffe [1982] for a discussion

of risk aversion.) For an investment opportunity, this means accepting more

risk, meaning more variability in the return, 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 expected 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 using 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 termination date, and the discount rate. For

each mortgage, a set Z(m,T,r) can be defined 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 derived from these probabilities for high,

mid, or low adjustments and appropriate

assumptions regarding their independence. Luna and Reid have assumed that

the probability of a high, mid, or low adjustment at any one adjustment point is

independent of the adjustment at any

other adjustment point, and the same assumption is made here. Using these probabilities for various states of the world.

July-August 1988 75

HEIAN, GALE

Year 1

8

mortgage

FRM I14.125%-

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 instrument 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 interest 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 underlying this interest rate scenario, see Luna and Reid [1986].)

INTERFACES 18:4 76

MORTGAGE SELECTION

the expected value and the standard deviation of each set Z{m,T,r) can be calculated 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 outcomes 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 confounds the analysis of the uncertain

events (the future mortgage rates) with

the analysis of their consequences (the

mortgage obligations) and with the analysis 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 probabilities of adjustments in the market interest rates on an analysis of the historical

record for the market interest rates specified in the adjustable rate mortgage contracts, 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 discussion, 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 simulated 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 valued 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 decision 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 required 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 percent discount rate, even the minimum of

the maximum valued obligations is above

the loan amount of $1,000 and the rational 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 unacceptable because it is above $1,000. Using

the 16 percent discount rate lowers the

present values for all mortgage obligations. For termination dates of five and

six years and a time preference of 16 percent, 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 general, the higher the discount rate, the

more importance the borrower places on

relatively low early payments as comINTERFACES 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 mortgage, 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, consequently, a payment below the FRM payment in each of the first five years. In

such cases the comparison of the mortgage 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 compute 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 borrowing 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 declining of the possible interest rate patterns 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 assuming this falling pattern of future

interest rates were certain to prevail.

The expected value rule selects the

mortgage with the lowest expected present value, as defined above, for each time

preference and termination date. The expected present values of the mortgage obligation 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 maximum 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 minimax decision criterion assumes that the

outcome for each mortgage will be the

least favorable (the highest valued mortgage obligation) under each set of assumptions. In effect, the minimax

decision rule assumes that the worst case

for each mortgage will occur with certainty 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 mortgage obligation will occur with certainty

and selects the mortgage with the minimum 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 obligation 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 mortgage obligation with a value nearly as

high as the worst possible outcome for

the adjustable rate instrument. In contrast, 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 borrower over the expected present value of

the mortgage obligation and its standard

deviation as defined above. If it is assumed that given the expected value of

the mortgage obligation, a smaller standExpected 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 betow 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 regions can be compared if the borrower’s preferences for risk relative to obligation are known.

ard deviation is preferred to a greater

standard deviation, and given the standard 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 borrowers 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) depending on the individual’s willingness to

trade lower expected values of the mortgage obligation for greater uncertainty.

The minimize-the-maximum-regret criteria also ignores the uncertainty of possible 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, however, it implies the borrower would

choose the mortgage with the lower

expected or mean value of the mortgage

obligation regardless of the standard deviation of the outcomes.

In Table 4, expected values and standard deviations of the mortgage obligation for selected termination dates and selected discount rates are reported. These

results illustrate the importance and the

feasibility, given the decision-tree approach to analyzing the behavior of market 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 alternatives 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 comparison between the FRM with an expected 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 present 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 possible consequences of each mortgage contract. Luna and Reid go a step in that

direction. The decision-tree approach allows the development of interest-rate

scenarios that incorporate both the direction and magnitude of changes in the underlying 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 marketclearing prices for risk and return. The

borrower, facing several alternative mortgage contracts is assumed to make a partial 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 borrowing under the contract terms or terminate

the loan at will. This is likely to be particularly 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 (MayJune), pp. 73-81.

Radcliffe, Robert C. 1982, Investment Concepts,

Analysis, and Strategy, Scott, Foresman and

Company, Glenview, Illinois.

July-August 1988 83

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