Posted: February 10th, 2022

Mortgage Selection Using a Decision-Tree

Mortgage Selection Using a Decision-Tree
Approach: An Extension
BETTY C. HEIAN
JAMES R. GALE
School of Business and Engineering Administration
Michigan Technological University
Houghton, Michigan 49931
School of Business and Engineering Administration
Michigan Technological University
Mortgages are available at various interest rates and vary
from traditional fixed-rate contracts to adjustable-rate contracts with a wide range of specific features. A method of
comparison using decision-tree analysis recognizes the borrower’s concern with both the expected value and the variability of possible outcomes. The expected value and the
variance for each of three specific mortgages are calculated
using plausible assumptions regarding time preferences. The
rational choice among mortgages for the risk-averse borrower
depends on the terms or features of the mortgage and the
individual’s expectations and beliefs.
B
orrowers, confronted with several alternative mortgage contracts, seek a
systematic and consistent method for
choosing a mortgage. In a recent article
in Interfaces, Robert E. Luna and Richard
A. Reid suggest a decision-tree approach
to this problem and apply the approach
to a specific case: a choice among a conventional fixed rate mortgage (FRM), an
adjustable rate mortgage for which the
mortgage rate adjusts at three-year intervals (ARM-3), and an adjustable rate
mortgage for which the mortgage rate a(
justs at five-year intervals (ARM-5) [Lun
and Reid 1986]. Their claim is “.. . this
approach provides, at a minimum, a rational framework for what is frequently
decided in a very intuitive or subjective
manner.” As the following discussion w
show, it is neither possible nor desirabl<
Copyright © 1988, The Institute of Management Sciences
0091-2102/88/1804/0072$01.25
This paper was refereed.
DECISION ANALYSIS — RISK
FINANCE
INTERFACES 18: 4 July-August 1988 (pp. 72-83)
MORTGAGE SELECTION
to eliminate from the decision process individual preferences, which are by nature
subjective.
Typically, a decision-tree can be used if
the consequences of each alternative depend on uncertain, discrete future events
that can be described probabilistically. For
the mortgage choice problem, the uncertain future events are the market interest
rates to which the adjustable rate mortgage (ARM) rates adjust. Thus, the objective consequences of each mortgage
choice, which depend on these market interest rates and the terms of the mortgage contract, can also be described
probabilistically. The borrower is assumed
to choose among the available mortgage
contracts or to not borrow using choice
criteria applied to these probabilistic consequences. The operational validity of the
approach depends critically on the probabilistic description of market interest
rates, the simulation of the objective consequences for the borrower, and the
borrower’s choice criteria.
The decision-tree approach, because it
isolates the probabilistic element in the
choice among mortgages, permits careful
consideration of the borrower’s criteria for
choice. Borrower concerns about the timing of payments are defined as time preferences and are reflected in discount rates
used to compute present values. Borrower
concerns about the uncertainty of payments are defined as risk preferences. Because Luna and Reid ignore both the
borrower’s time preferences and preferences with respect to risk in their definition of mortgage cost, they fail to address
two of the central dilemmas facing the
borrower. These are (1) how to choose
between two mortgages for which the sum
of the payments is identical but one of
which has lower payments in early years
and higher payments in later years than
the other; and (2) how to choose between
two mortages, one of which has a higher
expected obligation but less risk or uncertainty than the other. Luna and Reid implicitly make very specific assumptions
regarding time preferences, the central issue in the first problem, and through the
choice of decision rules treat only extreme
cases of preferences with respect to risk.
This suggests the need for, first, a
modified definition of the mortgage obligation, second, a method for valuing the
obligation that incorporates time preferences, and third, a method of defining
risk to permit comparison among
alternative mortgages.
The Mortgage Obligation
A mortgage contract obligates the borrower to make a series of payments over
time that will fully amortize the loan, if it
is held to maturity. These payments will
cover any origination fees and interest
charges, which are generally regarded as
the costs of borrowing, as well as the
principal repayment, which frequently is
not considered a cost of borrowing.
Luna and Reid, for the purposes of
constructing their decision tree, define
mortgage cost per $1,000 borrowed as the
sum of the origination fees and the payments made up to the termination of the
mortgage. Using this definition, mortgage
cost includes origination fees, interest
charges, and payments on the principal
made prior to the termination date but
neglects the balance that must be paid
when the loan is terminated. Mortgage
July-August 1988 73
HEIAN, GALE
obligation is a more appropriate term for
payments that cover fees, interest costs,
and principal repayment. For consistency
with respect to various termination dates,
the balance at the termination date
should be included as part of the mortgage obligation.
Thus the mortgage obligation can be
defined as a series of payments, X(0; i, m)
. . . X(T; i, m) given / and m where
i = a specific state of the world, defined
here as a specific pattern of market
interest rates;
m = a specific mortgage contract in this
example taking values FRM, ARM-5,
and ARM-3;
T = the termination year;
X(0; I, m) = origination fees (paid at
f = 0);
X(l; i,m)…. X(r-1 ; i, m) = annual
payments for years 1 through T-l ,
which may vary for adjustable rate
mortgages; and
X(T; I, m) = the payment for year T plus
the outstanding balance if any.
The mortgage obligation will depend
on the terms of the contract and the state
of the world as indicated by the market
interest rates at each adjustment point. A
specific contract may include a maximum
rate, a limit on the periodic rate increase,
or a payment cap with negative amortization (a provision for increasing the loan
balance rather than increasing the payment by the full amount implied by the
mortgage rate increase). If negative amortization is significant, recognition of the
balance outstanding at the termination of
the mortgage can be important to the borrower. Otherwise, the balance outstanding declines as the period the mortgage is
held increases and becomes less important as the borrower’s time preferences
rise.
Time Preferences
Choosing among alternative mortgages
involves comparing mortgage obligations
that are streams of payments made
through time. In order to be compared,
each stream of payments must be valued
at the same point in time. This can be
done using the present value or possibly
the terminal value. The discount rate
used to compute the present value of the
mortgage obligation for purposes of comparison is a measure of how a borrower
values money available at the beginning
of the period relative to money available
toward the end of the period, in other
words, a measure of the borrower’s time
preferences.
The present value of the mortgage obligation, Z{i,m,T,r), for the ith state of the
world, for mortgage contract m, for termination date T, using discount rate r is
then
Z(,,m,7;r)= S
where Z{i,m,T,r) = the present value of
the mortgage obligation, and r is a discount rate which measures a borrower’s
time preferences.
The value the borrower places on the
mortgage obligation will also depend on
the borrower’s time preferences. Thus,
the present value of the mortgage obligation may take on a different value for
each possible pattern of interest rates, for
each mortgage contract, for each discount
rate, and for each termination date under
consideration.
INTERFACES 18:4 74
MORTGAGE SELECTION
In general, under conditions of certainty, a willingness to borrow implies a
time preference rate at least as great as
the loan rate. Under these assumptions, a
rational individual is assumed to enter
into a mortgage contract only if the value
to the individual of what is received (the
proceeds of the loan) is greater than or
equal to the value to the individual of
what is promised (the obligation undertaken). (For a brief discussion of time
preferences in the context of investment
decisions, see Harris [1981, p. 25].)
Risk Preferences
Adjustable rate mortgages (ARM) are
believed to transfer part of the risk inherent in the variation of market interest
rates from the lender to the borrower. At
It is neither possible nor
desirable to eliminate from
the decision process
individual preferences, which
are by nature subjective.
the same time, borrowers are reluctant to
accept ARM loans because of the uncertainty about the actual payments they will
have to make. They seem to be concerned
about the most likely level of payments
and about the range of possible payments.
In the context of individual investment
decisions, people are considered risk
averse if they will accept additional risk
only in return for additional compensation. (See Radcliffe [1982] for a discussion
of risk aversion.) For an investment opportunity, this means accepting more
risk, meaning more variability in the return, only if the average or expected
value of the return is higher. For an
adjustable-rate loan, this means accepting
more risk, meaning greater variability in
the obligation, only if the average or expected value of the obligation incurred is
lower. In other words, for the risk-averse
borrower, the compensation for accepting
the greater uncertainty inherent in an
ARM is a reduction in the expected value
of the mortgage obligation.
The uncertain future behavior of the
market interest rates can be quantified using the decision-tree approach. Then, for
each possible interest-rate pattern or state
of the world, the present value of the
mortgage obligation can be calculated
given the terms of the mortgage, the termination date, and the discount rate. For
each mortgage, a set Z(m,T,r) can be defined which is composed of Z{i,m,T,r),
where ; represents a distinct interest-rate
pattern generated by the decision tree.
The number of interest-rate patterns in
each set may vary and will depend on the
frequency of adjustment and the assumed
termination date of the mortgage under
consideration.
Each possible adjustment in the
decision-tree structure has a probability
assigned to it. The probability of each
possible interest-rate pattern can be derived from these probabilities for high,
mid, or low adjustments and appropriate
assumptions regarding their independence. Luna and Reid have assumed that
the probability of a high, mid, or low adjustment at any one adjustment point is
independent of the adjustment at any
other adjustment point, and the same assumption is made here. Using these probabilities for various states of the world.
July-August 1988 75
HEIAN, GALE
Year 1
8
mortgage
FRM I14.125%-
ARM-5 1- 13.125%-
P(2) = 1
20.875%-
p(8)=.27
-17.375% —
p(9) = .51
-12.375% —
p(10)=.22
ARM-3 12.75%-
P(3) = 1
-18.875%-
p(5)=.27
15.375%-
p(6) = .46
25.0%-
p(11)=.1215
-23.125% 1
p(12) =.0810
-18.125% 1
p(13)=.O675
-21.5% 1
p(14) =.1656
^
•12%
P(7)=.27
18.0%—
p(15) = .1426
-14.625% 1
p(16) = .1518
-18.125% 1
p( 17) = .0648
-14.625% 1
p(18)=.1296
11.25% •
p(19) = .O756
Figure 1: The Luna and Reid mortgage rate scenario: The mortgage rate is shown for each instrument for the indicated time periods. The number in each parenthesis is the probability of
the sequence of interest rates to that point. For example, p (11) is the probability that the interest rates for ARM-3 will be 12.75 percent for years 1, 2, and 3; 18.75 percent for years 4, 5, and
6; and 25.0 percent for years 7, 8, and 9. (For a detailed explanation of the assumptions underlying this interest rate scenario, see Luna and Reid [1986].)
INTERFACES 18:4 76
MORTGAGE SELECTION
the expected value and the standard deviation of each set Z{m,T,r) can be calculated using conventional statistical
definitions.
The expected value of Z{m,T,r) is
N
E[Z(m,T,r)] = 2 Z(/, m, Z r) • [p ff)]-
1 = 1
The standard deviation of Z{m,T,r) is
SD[Z(m,T,r)]
=VE[Z(i,m,T,r)]-E[Z{mXr)Y
where Z(m,T,r) = the set of possible outcomes given the probabilistic description
of the market interest rates for mortgage
contract tn, terminated at T, and valued
using r, and p(i) = the probability of the
occurrence of the ith interest-rate pattern
or state of the world.
Mortgage Selection Comparisons:
Time Preference Effects
Luna and Reid show a decision tree for
the FRM, ARM-5, and ARM-3, using
their “mortgage cost” definition and their
assumptions concerning the probabilities
of various possible states of the world
[1986, Figure 1, p. 74]. This approach confounds the analysis of the uncertain
events (the future mortgage rates) with
the analysis of their consequences (the
mortgage obligations) and with the analysis of borrower’s criteria for comparisons
among these consequences.
The tree structure in Figure 1, based on
the Luna and Reid assumptions implicit
in their Figure 1, summarizes the effects
of possible market interest rates on the
mortgage interest rates for each mortgage
contract along with the probability of
each possible mortgage rate sequence.
Luna and Reid base their assumptions
about the amounts, direction, and probabilities of adjustments in the market interest rates on an analysis of the historical
record for the market interest rates specified in the adjustable rate mortgage contracts, the three-year United States
Treasury bill rates adjusted for constant
maturities for ARM-3, and the five-year
Treasury bill rates adjusted for constant
maturities for the ARM-5.
As can be seen from Figure 1, Luna
and Reid assume that interest rates are
much more likely to rise than to fall,
tending to make the ARM less attractive.
The probability of interest rates on ARM
being below that of the FRM after the
first adjustment period is 0.22 while the
probability of the rate on ARM-3 being
below that on the FRM is 0.27 for years 4,
5, and 6 and 0.0756 for years 7, 8, and 9.
Even this scenario is more favorable to
ARM-5 than a consistent implementation
of their methodology would suggest.
(From Luna and Reid’s Table 3 and discussion, the low adjustment for ARM-5
should be 4.25-3.5= to 0.75 not -0.75
[1986, p. 75].)
For purposes of comparison, we simulated the mortgage obligations, X(0; . .)
. . . X(r; . .), for the three alternative
mortgage contracts for termination dates
up to nine years using the interest-rate
patterns shown in Figure L We then valued these individual simulated mortgage
obligations, Z{i,m,T,r), for discount rates
between zero and 20 percent. We applied
the minimax decision rule, the minimin
decision rule, and the expected-value decision rule suggested by Luna and Reid
for termination dates five through nine.
The result for selected time preference
July-August 1988 77
HEIAN, GALE
Termination
in year 5
Discount rate
ARM-5 ARM-5 FRM FRM FRM
ARM-5 none none none none
(FRM) (FRM) (FRM) (FRM)
none none none none none
Luna and
Reid choice
(ARM-5) (FRM) (FRM) (FRM) (FRM)
none none none none none
(ARM-5) (FRM) (FRM) (FRM) (FRM)
ARM-5 FRM FRM n.a. n.a.
Table 1: Mortgage choice using the minimax
decision rule: The most attractive mortgage, if
the possibility of not borrowing is excluded,
is shown in parenthesis.
Termination
in year 5
Discount rate
16%
14%
0%
Luna and
Reid choice
ARM-3 ARM-3 ARM-3 ARM-3 ARM-3
ARM-3 ARM-3 ARM-3 ARM-3 ARM-3
none none none none none
(ARM-3) (ARM-3) (ARM-3) (ARM-3) (ARM-3)
none none none none none
(ARM-3) (ARM-3) (ARM:3) (ARM-3) (ARM-3)
ARM-3 ARM-3 ARM-3 n.a. n.a.
Table 2: Mortgage choice using the minimin
decision rule: The most attractive mortgage, if
the possibility of not borrowing is excluded,
is shown in parenthesis.
Termination
in year 5 6 7 8 9
Discount rate
12%
Luna and
Reid choice
ARM-5 ARM-5 ARM-5 FRM FRM
ARM-5 none none none none
(ARM-5) (ARM-5) (FRM) (FRM)
none none none none none
(ARM-5) (ARM-5) (ARM-5) (FRM) (FRM)
none none none none none
(ARM-5) (ARM-5) (FRM) (FRM) (FRM)
ARM-5 ARM-5 FRM n.a. n.a.
Table 3: Mortgage choice using the expected
value rule: The most attractive mortgage, if the
possibility of not borrowing is excluded, is
shown in parenthesis.
rates are shown in Tables 1, 2, and 3
along with the Luna and Reid choices.
We selected zero percent, 12 percent, 14
percent, and 16 percent time preference
rates for presentation. Zero percent is
shown for comparison with the Luna and
Reid choices. Twelve percent is below the
mortgage rate for nearly all the possible
outcomes; 14 percent is above that required for borrowing for some of the
ARM outcomes; and 16 percent is above
that required for borrowing for a fairly
wide range of outcomes. The operational
significance of the discount rate is clear in
Table 1, in which the minimax rule is
used. When the value of the mortgage
obligations is computed using a 12 percent discount rate, even the minimum of
the maximum valued obligations is above
the loan amount of $1,000 and the rational borrower will not borrow. When
the mortgage obligations are valued using
14 percent, the ARM-5 for the five-year
termination is acceptable. For termination
dates six through nine, the present value
of the FRM is below that of the maximum
for both ARM-3 and ARM-5 but unacceptable because it is above $1,000. Using
the 16 percent discount rate lowers the
present values for all mortgage obligations. For termination dates of five and
six years and a time preference of 16 percent, the early low payments on ARM-5
are sufficiently important to the borrower
to select ARM-5 using the minimax rule,
while for terminations of seven, eight, or
nine years the longer period of paying the
lower FRM payment dominates. In general, the higher the discount rate, the
more importance the borrower places on
relatively low early payments as comINTERFACES 18:4 78
MORTGAGE SELECTION
pared to relatively low later payments.
The Luna and Reid choice, which in
addition to neglecting the outstanding
balance at the termination of the mortgage, values a dollar paid at the end of
five years as equivalent to a dollar paid at
the beginning, is also shown in Table 1.
ARM-5 is effectively a fixed-rate contract
prior to the first adjustment in year six
with a rate below the FRM rate and, consequently, a payment below the FRM payment in each of the first five years. In
such cases the comparison of the mortgage obligation values will be invariant
The Luna and Reid choice, in
addition to neglecting the
outstanding balance at the
termination of the mortgage,
values a dollar paid at the
end of five years as
equivalent to a dollar paid at
the beginning.
with respect to discount rates. If early
payments for one mortgage are below and
later payments above those of the other
mortgage, the discount rate used to compute the present value will determine
which of the two mortgages has the
smaller present value. Luna and Reid do
not avoid assuming a time preference,
rather they assume a time preference of
about 14 percent when they assume borrowing will take place and simultaneously
a time preference of zero percent in their
comparisons among alternative mortgages.
The minimin rule, shown in Table 2,
essentially assumes the most rapidly declining of the possible interest rate patterns considered will prevail for both
adjustable rate mortgages. Thus, ARM-3
which has an initially lower payment than
either the FRM or ARM-5 and a lower
payment each year, has a lower present
value regardless of the time preferences
of the borrower. Nonetheless, the rational
borrower with a time preference of 12
percent or less will not borrow even assuming this falling pattern of future
interest rates were certain to prevail.
The expected value rule selects the
mortgage with the lowest expected present value, as defined above, for each time
preference and termination date. The expected present values of the mortgage obligation for selected discount rates and for
termination dates from five to nine years
are shown in Table 4. The expected value
rule selections are shown in Table 3. The
pattern of choices is similar to that of the
minimax rule, but because the expected
value for ARM-5 is lower than the maximum value, ARM-5 is chosen over FRM
for higher discount rates and longer
holding periods.
Mortgage Selection Comparisons: Risk
Preferences Effects
Implicitly Luna and Reid deal with the
risk preferences of the borrower through
their choice of decision rules. Their minimax decision criterion assumes that the
outcome for each mortgage will be the
least favorable (the highest valued mortgage obligation) under each set of assumptions. In effect, the minimax
decision rule assumes that the worst case
for each mortgage will occur with certainty and selects the mortgage with the
minimum value from among these.
The minimin decision rule, in contrast.
July-August 1988 79
HEIAN,
Mortgage
FRM
ARM-5
ARM-3
Mortgage
FRM
ARM-5
ARM-3
Mortgage
FRM
ARM-5
ARM-3
Mortgage
FRM
ARM-5
ARM-3
GALE
Termination
in year
E(Z)
SD(Z)
E(Z)
SD(Z)
E(Z)
SD(Z)
Termination
in year
E(Z)
SD(Z)
E(Z)
SD(Z)
E(Z)
SD(Z)
Termination
in year
E(Z)
SD(Z)
E(Z)
SD(Z)
E(Z)
SD(Z)
Termination
in year
E(Z)
SD(Z)
E(Z)
SD(Z)
E(Z)
SD(Z)
5
$ 1719.30
0
1673.77
0
1707.48
50.04
5
$ 1093.68
0
1062.78
0
1080.91
30.00
5
$ 1021.77
0
992.66
0
1009.04
2770
5
$ 956.43
0
928.97
0
943.77
25.62
0% discount –
6
$ 185799
0
1838.09
29.52
1858.61
75.27
12% discount •
6
$ 1104.25
0
1086.55
14.96
1097.88
42.59
14% discount
6
$ 1022.33
0
1004.97
13.43
1015.37
38.90
16% discount
6
$ 948.88
0
932.16
12.12
941.43
35.77
7
$ 1995.94
0
2001.81
59.23
2036.98
10720
7
$ 1113.64
0
110771
28.29
1125.61
56.54
7
$ 1022.82
0
1016.02
25.21
1032.07
51.25
7
$ 942.40
0
934.92
22.52
949.31
46.56
8
$ 2133.62
0
2164.85
89.11
2214.74
142.95
8
$ 1121.97
0
1126.54
40.17
1150.31
70.34
8
$ 1023.24
0
1025.57
35.5
1046.69
63.13
8
$ 936.85
0
93731
31.44
956.11
56.81
9
$ 2269.14
0
232709
119.19
2391.78
180.41
9
$ 1129.35
0
1143.29
50.75
1172.29
83.18
9
$ 1023.61
0
1033.92
44.48
1059.49
73.99
9
$ 932.10
0
939.38
39.09
961.00
66.02
Table 4: Expected value and standard deviation for selected mortgage obligations: E(Z) is the
expected value and SD(Z) is the standard deviation of Z{m,T,r). The mortgage obligations were
simulated using the interest rate index scenario suggested by Luna and Reid.
INTERFACES 18:4 80
MORTGAGE SELECTION
assumes the “best” lowest-valued mortgage obligation will occur with certainty
and selects the mortgage with the minimum lowest-valued obligation. In neither
of these cases is there any attempt to deal
with the possibility that extreme interest
rate patterns will occur with a very low
probability. Thus, the decisions are hard
to reconcile with intuition about borrower
preferences. For example, if the worst
case (maximum value of the mortgage obligation for all considered possibilities) for
one adjustable-rate contract has a low
probability while an alternative fixed-rate
mortgage obligation is certain and, given
the discount rate, has a value just slightly
lower than the ARM, the minimax rule
will select the fixed-rate instrument. This
assures the borrower of a certain mortgage obligation with a value nearly as
high as the worst possible outcome for
the adjustable rate instrument. In contrast, if the minimin rule is applied, the
adjustable rate instrument will be selected
even if the probability of the ARM value
being below the value of the FRM is very
small. If borrowers are risk averse, they are
most likely to prefer the FRM if its value is
close to the minimum possible ARM value.
They are likely to accept some variability in
their mortgage obligation if they perceive a
low probability for the ARM value being
higher than the FRM value.
The discussion of risk aversion can be
formalized by postulating the existence of
a preference map for the risk-averse borrower over the expected present value of
the mortgage obligation and its standard
deviation as defined above. If it is assumed that given the expected value of
the mortgage obligation, a smaller standExpected Value
Z(m,T,R)
A Is preferred
to points In this
region.
Points In
region are
preferred
to point A.
Standard Deviation
Z(m,T,r)
Figure 2: Preference space for expected value
and standard deviation of mortgage contracts:
Expected value, standard deviation pairs betow and to the teft of A are unambiguously
preferred to A by risk-averse borrowers. Point
A is unambiguously preferred to points above
and to the right of it. Points in the shaded regions can be compared if the borrower’s preferences for risk relative to obligation are known.
ard deviation is preferred to a greater
standard deviation, and given the standard deviation, a smaller expected value of
the mortgage obligation is preferred to
greater expected value, then the limits to
the borrowers’s preference map can be
shown as in Figure 2. Risk-averse borrowers necessarily prefer situations to the left
and below point A to point A, and prefer
point A to points to the right and above
it. Borrowers may prefer, be indifferent
between, or not prefer points to the left
and above or to the right and below point
A (the shaded area in Figure 2) depending on the individual’s willingness to
trade lower expected values of the mortgage obligation for greater uncertainty.
The minimize-the-maximum-regret criteria also ignores the uncertainty of possible outcomes. The fourth decision rule,
choose the mortgage with the minimum
expected present value for the mortgage
obligation, recognizes that the possible
outcomes for each alternative mortgage
should be thought of as occurring with
some specific probability. By looking
July-August 1988 81
HEIAN, GALE
exclusively at the expected value, however, it implies the borrower would
choose the mortgage with the lower
expected or mean value of the mortgage
obligation regardless of the standard deviation of the outcomes.
In Table 4, expected values and standard deviations of the mortgage obligation for selected termination dates and selected discount rates are reported. These
results illustrate the importance and the
feasibility, given the decision-tree approach to analyzing the behavior of market interest rates, of considering both the
expected present value and the standard
deviation of the mortgage obligation. For
example, for the six-year termination date
valued using a 16 percent discount rate,
the expected present value for all three alternatives is below the loan amount
($1,000). Using the expected present
value, standard deviation criteria ARM-5
is clearly preferred over ARM-3, because
it has both a lower expected present value
and a lower standard deviation. The comparison between the FRM with an expected present value of $948.88 and zero
standard deviation and ARM-5 with a
lower expected present value of $932.16
and a higher standard deviation of 12.12
is ambiguous. In this case, the borrower
may, ir principle, be indifferent between
the two choices, but probably will prefer
one mortgage to the other depending on
his or her preferences regarding risk and
obligations.
The mortgage selections based on the
expected value, standard deviation rule
are shown in Table 5. For the 16 percent
discount rate, there is a clear choice for
termination in years five, eight, and nine
Termination
year
Discount rate
16%
14%
12%
0 %
5
ARM-5
ARM-5
none
(ARM-5)
none
6
***
none
***
none
***
none
7
*»*
none
***
none
»*»
none
8
FRM
none
(FRM)
none
(FRM)
none
9
FRM
none
(FRM)
none
(FRM)
none
(ARM-5) ” ‘ (FRM) (FRM) (FRM)
Table 5: Mortgage choice using the expected
value-standard deviation rule: *** indicates
there is no clear choice. ARM-5 has a lower
expected value and a larger standard deviation
than FRM. The most attractive mortgage, if the
possibility of not borrowing is excluded, is
shown in parenthesis.
— all cases in which the mortgage with
the lowest expected value has a zero
standard deviation. For termination in
years six and sever, the ARM-5 has a
lower expected present value but a higher
standard deviation. The borrower must
choose between a higher expected present value with a lower standard deviation
and a lower expected present value with
a higher standard deviation of the
mortgage obligation.
Conclusion
In general, the ratioral borrower would
like to know the distribution of the possible consequences of each mortgage contract. Luna and Reid go a step in that
direction. The decision-tree approach allows the development of interest-rate
scenarios that incorporate both the direction and magnitude of changes in the underlying index rates and the probabilities
associated with these changes. However,
they have failed to utilize the full power
of their innovation.
The decision-tree approach suggested
by Luna and Reid and extended in this
paper treats the ARM as a risky liability
INTERFACES 18:4 82
MORTGAGE SELECTION
and assumes implicitly that the terms of
the mortgage contracts reflect marketclearing prices for risk and return. The
borrower, facing several alternative mortgage contracts is assumed to make a partial equilibrium choice, the mortgage that
best fits the borrower’s risk and outlay
preferences given time preferences. An
alternative approach is to treat the ARM
as an option written by the lender in
which the borrower may continue borrowing under the contract terms or terminate
the loan at will. This is likely to be particularly fruitful if the question of ARM
pricing is addressed from the lender’s
perspective. Our approach, focusing as it
does on the borrower’s risk and time
preferences in a partial equilibrium
framework, extends our understanding of
the possible benefits of ARM to the
individual.
References
Harris, Laurence 1981, Monetary Theory,
McGraw-Hill, New York.
Luna, Robert E. and Reid, Richard A. 1986,
“Mortgage selection using a decision tree
approach,” Interfaces, Vol. 16, No. 3 (MayJune), pp. 73-81.
Radcliffe, Robert C. 1982, Investment Concepts,
Analysis, and Strategy, Scott, Foresman and
Company, Glenview, Illinois.
July-August 1988 83

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