Title
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 87131
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
egularity. This result provided confidence that the potential
enefits associated with a more structured approach could be
ealized. Using concepts from classical decision theory and a
easonable range of alternative future scenarios, a rational
choice for financing a personal investment was identified.
y \ ma
ied 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 cu
ent 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
XXXXXXXXXX/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% 123/4%
IVi
14V8
21/4
13/4
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 cu
ent index rate for this
type of mortgage is 10% percent.
(3) A conventional fixed rate mortgage
which cu
ently 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 bo
owers use cu
ent interest
ates to assess the attractiveness of a con-
ventional mortgage, evaluating ARMs re-
quires that they estimate future interest
ates. 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 bo
ower'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
Co
elation 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 .03
0.2745
XXXXXXXXXX
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 — Fe
uary 1983; five-year T-bill interest rates were ob-
tained from the Statistical Abstract of the United States: 1984 for years XXXXXXXXXX; and earlie
XXXXXXXXXXfive-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, =
XXXXXXXXXX«) for the three-year T-bill in-
terest rates over time had a high co
ela-
tion coefficient XXXXXXXXXXThe regression {u,
= XXXXXXXXXX30n) for the five-year T-bill
interest rate data showed an even greate
co
elation coefficient XXXXXXXXXXIn 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-yea
time intervals indicated relatively little
co
elation XXXXXXXXXXand -0.10, respec-
tively) with time. Moreover, the fitted
egression lines possessed slopes which
were very small XXXXXXXXXXand XXXXXXXXXX,
espectively) in comparisor\ with the mag-
nitudes of the average fractional differ-
ences XXXXXXXXXXand 0.3993, respectively).
These combined results provided a sound
ationale 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-
ived 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-Yea
ARM
10'/4%
0.2595
0.3359
2.66%
3.44%
- V4%
+ 2%%
+ 6'/B%
5-Yea
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
ate values are rounded to the nearest one-
eighth percent. High and low index rate val-
ues co
espond to the mean value plus and
minus one standard deviation, respectively.
May-June XXXXXXXXXX
LUNA, REID
PLANNING HORIZON
©I =P(High hterest Rates)
©2= P( Average Interest Rates)
©3=P{Low Interest Rates)
$737 $881 $1025
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-
ing and costs associated with these events are recorded respectively on appropriate tree
anches 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
y 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
ounded 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-
le outcomes. Figure 1 shows a decision
tree that identifies the relative costs asso-
ciated with the different mortgage ar-
angements in terms of $1,000 investment
increments. This basic unit of bo
owing
permits the results to be generalized fo
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,
eflects 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
88
$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 XXXXXXXXXX
LUNA, REID
produce the highest bo
owing costs are
assumed to prevail. The decision make
then selects the mortgage type which
minimizes total costs. In this case, the
five-year ARM is prefe
ed for a planning
horizon of five years. However, if the con-
dominium is held for six or seven years,
the conventional mortgage has