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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$01.2

5

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)=.2

7

-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) = .4

6

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 ratior\al 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

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 8713

1

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% 12

3/4%

IVi

14V8

21/4

1

3/4

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 $

10

25

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

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