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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
