a typical down payment of 20 to 30 % of the purchase price, which can vary significantly ... 'time value of money' effect are present value and terminal value.
OMEGA, The Int. J1 of Mgrat Sci., Vol. 4, No. 1, 1976. Pergamon Press. Printed in Great Britain
A Deterministic Model for the
Housing Decision RICHARD MOJENA G G E O F F R E Y BOOTH JAMES P B O W M A N (Received March 1975; in revised form July 1975)
The decision to rent or buy housing, or to choose among various housing acquisition alternatives, represents an important and complex financial problem, which confronts virtually all individuals. This paper develops a comprehensive model (within the institutional and legislative context of the United States) which provides theoretically appropriate criteria for evaluating personal housing alternatives. Empirical results clearly indicate the economic consequences of housing alternatives, and both substantiate some and contradict other "wisdoms" concerning the housing decision.
INTRODUCTION THIS PAPER is concerned with the economic decision of renting vs owning housing for personal use. Within the institutional and legislative context of the United States, a mathematical model is presented which provides the decision maker with a tool for evaluating housing alternatives. The importance of such a model is underscored by noting that over the past two decades in the United States, the average cost of housing amounts to approximately 22 ~o of personal income [8, pp. 322-3]. Furthermore, a commitment to buying housing requires a typical down payment of 20 to 30 % of the purchase price, which can vary significantly among regions. For example according to a survey in 1973 [1], median purchase prices were $57,500 for New York City, $34,400 for Chicago, and $53,200 for Los Angeles. Monthly rentals also vary substantially: $398 in New York City, $302 in Chicago, and $240 in Los Angeles. For the most part, the literature treats the personal housing decision in a descriptive manner, as typified in Refs. [2], [3] and [5]. Existing quantitative approaches lack rigor, fail to explicate relevant variables, ignore the time value of money, or rely on questionable assumptions, as in Refs. [6], [7] and [9]. In 85
Mojena, Booth, Bowman--The Housing Decision contrast, leasing investment models are well established and documented [4], but inappropriate to the personal housing decision because of differences in the treatment of taxes, depreciation, and capital gains. The proposed quantitative model explicitly defines theoretically appropriate costs of rearing and buying based on a comprehensive set of variables and parameters. By distinguishing between rent-buy aspects, the model may be used to evaluate rent versus buy alternatives, different rentals, or different purchasing options. Because the pattern of cash flows normally varies among alternatives, differential cash positions result over time, each of which is subject to foregone investment opportunities. Two common methods used to account for this 'time value of money' effect are present value and terminal value. This model employs terminal value because, from an interpretive standpoint, it indicates the total cost of renting or buying up to a given horizon, which includes not only actual costs but the investment income foregone by having to pay for shelter. From another perspective, the terminal value of cost over a given time indicates the amount of funds that an individual would accumulate at the end of the time, including investment income, if payment for housing could be avoided. In addition, the cost criterion is defined on an after-tax basis, as the incidence of taxes and subsidies varies across housing alternatives. In the United States, for example home owners must pay annual property taxes (typically I-5 ~o of assessed value) to the local municipality; property taxes and interest payments on the mortgage are deductible for income tax purposes; capital gains tax must be paid to the federal government if a profit is realized on the sale of a house (unless a more expensive purchase is effected within one year); and front-end purchasing subsidies (if any) can be deducted from down payments. Since the model excludes qualitative criteria such as quality of life, security of tenancy and other intangible factors, it is appropriately regarded as one of the inputs to the decision-making process. Should it be used as the sole criterion, it must be assumed that qualitative criteria are equal for all alternatives or that alternatives exhibit qualitative balance through tradeoffs.
RENT MODEL The total cost of renting is defined to consist of the cash flows associated with downpayments, rent payments, and maintenance expenses. The cost of the downpayment, including the impact of the opportunity cost, may be expressed as~ n--I
D,
/7
[1 + kt (1 -- 7",)1 -- D,,
i=0
86
(1.1)
Omega, Vol. 4, No. I where D, is the initial dollar deposit; n is the number of periods (months) in the planning horizon; ki is the before-tax opportunity cost expressed as a ratio at the end of the i tn period; and Ti is the income tax rate in the i" period. This specification represents the total after-tax income (assuming interest income is taxed at rate Tt) foregone over n periods by paying a rental deposit. This is to say that, if the individual did not have to pay the rental deposit, it could have been invested for n periods at various rates for a net monetary value at the end of the horizon given by (1.1). Similarly, the cost of rent payments is given by:
Z'
P,~
j=l
H
(1.2)
[1 + k~ (1 -- T~)I
i=j--I
where P,j is the rent payment in dollars at the beginning of t h e f h period and the remaining notation is as before. The final category of costs are those associated with maintenance. These include such items as utilities, insurance, and minor repairs not covered by the rental. Additionally, if the geographic locations of housing alternatives vary significantly, then transportation expenses can be included under maintenance. Expressing these costs and their opportunity impact yields: n
S j=l
[1 -k k, (1 -- T,)I ,
(1.3)
J
where U,j represents maintenance and related costs in dollars at the end of the jth period and the other symbols are as previously defined. Summing these three types of costs and combining certain terms yield the total cost of renting:
C,=O,
~
li+k,(I--7,)1-
+
~r j-- l
ki=O +
~' j=l
U,~ /-/
}
[1-k- kt ( 1 - - T,)] ,
i=j
where Cr is the total cost of renting. 87
p,~
[l+~,(1--r,)l i=j--1
(1)
Mojena, Booth, Bowman--The Housing Decision
BUY MODEL The buy model, although similar to the rent model, is somewhat more complex because of additional variables that need to be considered. The total cost of buying is defined to consist of the cash flows associated with the downpayment, principal and interest payments, maintenance expenses, property taxes, and sale of the house. The time-value cost of the downpayment and related closing costs is given by: (D~+K)
n--I /-/ [I + k , ( 1 --T,)], i-----0
(2.1)
where Db is the downpayment in dollars (net of any subsidies); and K represents closing costs in dollars. The various periodic costs may be expressed as: ~' j=l
[eoj+ Usj + ( 1 --T~)(Ij+Lj)I /"/ i=j
[1 q-k,(1 -- T0]
(2.2)
where P~j is the principal repayment in dollars at the end of the j,h period; U~j represents maintenance and related expenses in dollars at the end of the j,h period; la is the interest charge in dollars at the end of t h e j th period; and Lj indicates property taxes in dollars at the end o f t h e j th period. This specification reflects the tax deductibility of interest and property taxes from state and federal income taxes. It should be pointed out that the incidence of the tax deductibility is simplified in this model since it normally does not occur monthly. It should also be noted that the principal repayment and the interest payment are determined from standard annuity formulae based on the amount borrowed, the life of the mortgage, and the mortgage rate. The last major element in the buy model is that of divesture. The costs associated with this action are: - sn (I - b) -{- IS, (1
-
b) -- So -- K] G -- R,,
(2.3)
where Sn a n d
b G Rn
So
represent the market value of the dwelling in dollars at the end of periods n and o, respectively; is the brokerage fee in the form of a ratio; is the capital gains tax rate; and is the remaining principal in dollars. 88
Omega, Vol. 4, No. 1 The first term represents a cash inflow from selling the house in period n. This amounts to the market value less brokerage fees. The second term is used to calculate the amount of tax owed when the house is sold. This terms should be ignored if it is negative or if another, more expensive, purchase is contemplated within one year. The third term, R,, represents the amount of the mortgage which has not yet been paid. It is determined easily as a function of the amount borrowed, the mortgage life, the mortgage rate and the length of time the house is owned. Combining (2.1), (2.2), and (2.3) and simplifying selected terms yield: C6 = (D~+K)
+
Z' j=l
n-1 1"]" [1 + k , ( l -- T , ) I - S , ( 1 - b ) ( l -- G) - (So -b K) G -- R, i----0 [P~ + Ubj + (1 - Tj) (lj + Lj)I /7
i=j
[1 + k, (1 - r,)l ,
(2)
where
Cu is the total cost of buying.
CRITERIA Four criteria of general interest may now be defined: (a) cost of renting for a given planning horizon, (b) cost of buying for a given planning horizon, (c) breakeven period, and (d) cost of buying-renting at breakeven. The first two are determined by the straight-forward calculation of equations (1) and (2), respectively. Note that by defining
Ac= c,-
cb,
it follows that AC> 0 for a given horizon implies a decision in favor of buying, all other things equal, and AC < 0 implies a decision to rent. The breakeven period, n*, is defined as the first value of n for which A C > 0 , i.e. n* is the first value of n for which the cost of buying is less than, or equal to, the cost of renting. If the decision maker's expected length of stay is less than the breakeven period, the appropriate financial decision is to rent. The cost of buying-renting at breakeven is obtained by evaluating equation (2) at n*. As previously expressed, the use of these quantitative criteria must be tempered by qualitative criteria; that is, financial aspects represent only one facet of the housing decision. DISCUSSION As might be expected, the rent-buy decision is sensitive to variations in several important variables. The sensitivity of breakeven period is illustrated 89
Mojena, Booth, Bowman--The Housing Decision TABLE 1. VARIATIONSIN BREAKEVEN PERIOD (MONTHS)
Initial Monthly Rental Payments ($) Annual house appreciation rate
225
250
275
300
325
350
0-00 0-03 0-06
387 214 38
273 108 28
185 59 22
128 40 18
84 30 15
59 24 14
Note: Dowapayment set to 25 y. of $40,000 market price; annual inflation, mortgage, and opportunity cost rates fixed at 0-06, 0"08, and 0.07, respectively; rental deposit assumed equal to initial rent; initial and final income tax rates were 0"20 and 0"25, respectively; U,, = S30/month, Ub, = $100/month, K = $300, b = 0'06, t = 300 months, Lo = $90/month, and G = 0.125; dynamic behaviors of P,j, U,j, Ubj, and Lj specified by compounding at rate of inflation. in Table 1 for variations in the house appreciation rate and the rental payment. The large values of some of the entries clearly dispel the commonly-held belief that 'it is always best to buy housing because equity accumulates.' The decision to rent is favored when the opportunity cost is high relative to the rate of appreciation. Additionally, the effect of lower rental payments is dramatic at low rates of housing appreciation. To illustrate costs, assume that the alternatives are characterized by a 5 year planning horizon, a 6 ~o annual rate of house appreciation, a monthly rental of $325 per month, and values for other variables and parameters as noted in Table 1. Employing these data, the model indicates that the cost of renting for the 5 year period is $29,512 and the cost of buying for the same period is $18,249. The breakeven period is 15 months and the cost of buying for this period is $5,766. In this case the decision maker should rent if the expected length of stay is less than 15 months. The cost of renting has a unique interpretation: if it were possible to avoid the cost of shelter altogether and cash flows associated with renting were invested at the opportunity cost, the individual would be better off by $29,512 at the end of five years. Policy implications based on regression analyses of 750 cases include the following: the importance of accurately estimating rates of housing appreciation; high inflation rates favor the purchasing of housing, all other things equal; renting is a favorable financial alternative under a variety of circumstances, particularly for transients and when the opportunity cost is high relative to property appreciation; and the effects of the size of downpayments on the decision to buy or rent and on the costs of buying are minimal, all other things equal. In concluding, we suggest that this model provides a valid procedure for assessing the financial impact of housing alternatives. As such, it should prove 90
Omega, Vol. 4, No. 1
useful as one of the inputs to the common and economically important housing decision. In particular, this type of modelling can be used by organizations which provide services related to housing decisions (e.g. realtors and banks) and by personnel departments of organizations which relocate employees (e.g. armed services and large corporations).
REFERENCES 1. B~LABON MB (1974) Inter-city differences in rent vs purchase decisions in housing. Geography Conference, West Point, New York. 2. COHEN JB and I'I~NSEN AW (1972) Personal Finance: Principles and Case Problems. Richard D Irwin, Homewood, IL. 3. DONAJ-DSONEP and Pr~HL JK (1966) Personal Finance. Ronald Press, New York. 4. FI~a~t~Y MC III (1974) Financial lease evaluation: survey and synthesis. Fin. Rev. 1-15. 5. HAS~NGS P and MiE'rus N (1972) Personal Finance. McGraw-Hill, New York. 6. MUMEYGA (1972) Personal Economic Planning. Holt, Rhinehart & Winston, New York. 7. SHEL'rONJP (1968) The cost of renting vs owning a home. Land Econ. 44, 59-72. 8. U.S. Bureau of the Census (1973) Stat. Abs. U.S. Washington, D.C. 9. WEST DA and WOOD GS (1972) Personal Financial Management. Houghton Miflin, Boston.
Professor R Mojena, College of Business Administration, Department of Mgmt Science, University of Rhode Island, Kingston RI 02881, USA.
ADDRESS FOR CORRESPONDENCE:
9t
