optimal leverage strategy: capital structure in real estate investments

Journal of Real Estate Finance and Economics, 13:263-271 (1996) ©Kluwer Academic Publishers

Optimal Leverage Strategy: Capital Structure in Real Estate Investments

ROGER E. CANNADAY Department of Finance, University of Illinois at Urbana-Champaign, 1407 West Gregory Drive, Urbana, 1L 61801

TYLER T. YANG Fannie Mae, 3900 Wisconsin Avenue, N.W., Washington, DC 20016-2899

Abstract

This article examines the optimal leverage strategy for real estate investors who are investing in income-producing properties. Within a discounted cash-flow context, the investment objective for the equity investor is to maximize the contribution to net present value of using mortgage financing. Utilizing more debt decreases the required equity investment and increases the size of the tax shelter. On the other hand, as the loan-to-value ratio increases, the interest rate charged by the lender increases, which indicates a higher cost of debt. This article goes beyond the simple conventional wisdom that debt financing should be used when financial leverage is positive by developing an equation that allows one to determine the optimal level of debt financing to use when positive leverage is possible. The optimal loan-to-value ratio is found to be a function of the investor's characteristics. Several hypotheses about the relationships between such an optimal loan-to-value ratio and the investor's characteristics are derived.

Key Words: leverage; income-producing; discounted cash-flow; debt financing; loan-to-value ratio

This article examines the optimal leverage strategy for real estate investors who are invest-

ing in income-producing properties, rather than owner-occupied, s ingle-family residential properties. Within a discounted cash-flow context, the inves tment objective for the equity

investor is to maximize the contribution to net present value of using mortgage financing. Uti l iz ing more debt decreases the required equity inves tment and increases the size of the

tax shelter. On the other hand, as the loan-to-value ratio increases, the interest rate charged

by the lender increases, which indicates a higher cost of debt.

The convent ional wisdom is that if f inancial leverage is positive, then debt f inancing should be adopted, and vice versa if leverage is negative. The explicit purpose, and ma-

jor new contribution, of this article is to determine what level of debt f inancing, if any, is optimal. The optimal loan-to-value ratio is found to be a function of the investor ' s characteristics. Several hypotheses about the relationships between the optimal loan-to-value ratio and the investor ' s characteristics are derived.

1. The Model

The theoretical model used in this article is s imilar to that of Cannaday and Yang (1995). The net present value (NPV) of a real estate inves tment to the equity investor is derived as

264 CANNADAY AND YANG

follows:

T ATCFt NPV = ~ " (1 + y)t

t = l

ATERT - - + EIo, (1)

(1 + y)T

where ATCFt is the after-tax cash flow from rental income at time t;

ATERT is the after-tax equity reversion from selling the property at time T, the end of the investment period;

EIo is the Equity Invested at time 0;

y is the equity investor's required after-tax rate of return on equity (that is, the after-tax cost of equity); and

T is the holding period of the real estate investment.

By comparing the N P V of using pure equity financing with the N P V of using mortgage financing, the contribution to the net present value of the subject investment of using mortgage financing can be derived. The essence of the derivation is to show the tradeoffbetween: (1) the savings in the initial equity investment; and (2) the reduction in the periodic after-tax cash flows and the after-tax equity reversion; due to mortgage financing. The contribution to NPV for an interest-only (see Cannaday and Yang, 1995, p. 69, n. 3) mortgage is derived as (11) in Cannaday and Yang (1995) and is presented here as (2):

T

ANPV = (1 - k)MVo - t = l

kMVo MVoi(1 - r)

m (1 + y)t

MVo - kMVo m - T'c + MVo~'(1 - ~') m

(1 + y)r

(2)

where Vo is the value of the property at the time of purchase (time 0);

M is the loan-to-value ratio;

k is the discount points as a proportion of the loan amount;

~- is the equity investor's marginal tax rate;

i is the interest rate of the mortgage loan (that is, the before-tax cost of debt);

rn is the years to maturity of the mortgage loan;

7r is the prepayment penalty of the mortgage loan; and other terms are the same as before.

OPTIMAL LEVERAGE STRATEGY 265

Note that this ANPV is independent of the annual net operating income and the net selling price at the end of the holding period. All terms are determined at the time of investment (time 0), and there is no random variable involved. As a result, the investor's optimal leverage strategy at the time of purchase is to choose the mortgage loan contract that maximizes ANPV.

Usually, the interest rate (i) of a mortgage loan is positively related to the desired loan-to- value ratio (M). This relationship is primarily induced by the level of default risk carried by the lender. As the loan-to-value ratio increases, the likelihood of the investor defaulting on the mortgage loan increases. In other words, the probability that the value of the underlying property becomes lower than the book value of the loan becomes greater. Given any loan-to- value ratio, we can always find a minimum interest rate that will be offered by all lenders. This set of minimum interest rates represents a constraint to the investor. That is, the interest rate on the mortgage loan must be greater than or equal to the minimum interest rate at the given loan-to-value ratio. Give this constraint, the leverage strategy of the equity investor can be written as:

MaxMANPV

s.t. i >- I(M)

M>~O,

(3)

where I(M) is the minimum interest rate offered by lenders for the subject investment for the given M. Since ANPV monotonically decreases as the interest rate increases, the lowest possible interest rate should always be chosen by the borrower for a given M. Therefore, the maximization problem can be simplified to be:

kMVo T MVoI(M)(1 - .r)

MaxMANPV = (1 - k)MVo - Z (1 + m y)t t = l

m - T MVo - kMVo z - MVoTr(1 - 7)

t n

(1 + y)r

(4)

The first-order condition for the maximization of (4) isl:

3ANPV f [ ~ ] ( 1 + Y ) r - 1 3M - Vo l - k - I * ( l - ' r ) - y ~ y ~

- I;4M*Vo(1 - r)(ly(1 y)T+ y)r--1

ANPV I~M*Vo(1 .(1 + y)T _ 1 - M* - r ) y ~ l ~ y 7

1 - km m ~f-+-y~r T'r - z r ( 1 - "r)}

(5)

= 0 .


Solving (5) for ANPV yields:

1 + y)T _ 1 ANPV* = I~uM*2Vo(1- r ) ( ) ( i T y ~ ' (6)

where M* is the optimal loan-to-value ratio, and I~ is the slope of I(M) at the optimal M*. Note that this ANPV needs to be compared with the trivial solution where the investor uses zero debt, in which case, zero ANPV is realized. If the solution in (6) is negative, then the situation is referred to as "negative leverage"; the investor would be better off using pure equity financing (assuming that the NPV in (1) is positive). If this ANPV* is greater than zero, "positive leverage," the investor's optimal strategy is borrowing a loan amount of M*Vo and achieving the highest NPV from the investment project. Thus, the leverage strategy is only relevant in the positive leverage situation, i.e., when ANPV* > 0.

Rearranging (6), we find that the I(M) must be an increasing function of M in the relevant range such that an interior solution may exist. That is,

ANPV*y(1 + y)r : IM = M*2Vo(1 - ~')[(1 + y ) T _ 1] > 0. (7)

The second-order necessary condition is checked to make sure that the solution to (6) is a maximum. The second-order condition is:

02ANPV

3M 2 1 + y)r _ 1

- --2IMV0(1 -- ~')( ( l y 1 + y)r _ 1

IMMMVo(1- "r)( ( l y < o. (8)

Rearranging (8), we obtain

21;, I~t M > - M~. (9)

Combining the first-order condition and the second-order condition, we find the necessary and sufficient conditions for an interior solution to exist: I(M) is increasing and cannot be too concave with respect to M. This is consistent with Hendershott and Shilling (1989, p. 107), who hypothesize that I(M) is increasing and convex with respect to M, and then present empirical evidence to support that hypothesis.

Suppose that the constraint I(M) satisfies the necessary and sufficient conditions, then the optimal loan-to-value ratio can be computed as:

M* =

1-- y(1 + - y ~ 5 q--Y~ y(1 + y)r

l~t(1 - ~-)[(1 + y)T -- 1]

f , ANPV*y(1 + y)r > 0. (lO)


Several comparative statics can be derived from (10). First of all, taking the partial derivative of M* with respect to r yields:

f OM* _ / ANPV*y(1 + y)r

Or V4I~t(1 - r)3Vo[(1 + y)T _ 1] > 0 . (11)

The optimal loan-to-value ratio increases as the marginal tax rate of the investor increases. Secondly, taking the partial derivative of M* with respect to T yields:

OM*_A[1 l + y ] c~T (1 + y)r _ 1

> < 0 if and only if T < > l n ( 2 + y ) ln(1 +y)'

(12)

where

A = / ANPV*y(1 + y)rln(1 + y)

~/2 I~ (-----1 7 r )--Vo[ (-1 + y )--~ -- 1-]

= M*~ fln(12 + y ) M > 0 .

Equation (12) indicates that when the investment period is short, the optimal loan-to-value ratio increases with the holding period of the underlying property; when the holding period is relatively long, the optimal loan-to-value ratio decreases with the holding period, This implies that there is a particular holding period in which the optimal loan-to-value ratio

ln(2+y) is maximum. The investor with the holding period equal to ~ will utilize the highest loan-to-value ratio.

Thirdly, taking the partial derivative of M* with respect to y yields:

OM* - B{[(1 + y ) T _ 1](1 + y ) - - Ty} Oy (13)

> 0 2,

where

ANPV*(1 + y)T-2 B = 2I~(1 - r)Voy[(1 + y)r _ 113

> 0 .

This indicates that the higher the required rate of return on equity the more the investor will borrow. Note that because of the heterogeneity of investors (due to different risk preferences, different marginal costs of equity, etc.), at any point in time, different investors may have different required rates of return for the same potential investment. For example, for a given


level of risk involved in a given investment, the more risk-averse the investor is, the higher the rate of return that will be required to compensate for the level of risk. That is, the required rate of return is increasing with the level of risk aversion of the investor.

2. Geometrical Analysis

Equation (2) can be analyzed in a different way. If we fix the ANPV and solve for the relationship between the interest rate and the loan-to-value ratio, a set of iso-ANPV curves can be derived for a particular investor. When the investor must choose among different interest-rate and loan-to-value ratio combinations, he/she will try to choose the combination that falls on the highest magnitude ANPV curve. This can be analyzed on an interest-rate and loan-to-value ratio plane as in Figure 1. The iso-ANPV curves take the following form:

D 1 = C - ~ , (14)

where

k~[(1 + y ) r _ 1] 1 kr(m-T) 7r(1 - r ) ~ y ( l + y ) r l - k + _ m

C = (1 - r ) [ ( 1 +y)r_ 1] my(1 +y)r (1 +y)r

and

D = ANPV(y)(1 + y)r > O. (1 - ~ ) [ (1 + y ) r _ l l V o

Maximizing the ANPV is equivalent to choosing the lowest possible iso-ANPV curve in Figure 1. When different ANPVs are plugged into (14) we obtain different iso-ANPV curves on the interest-rate and loan-to-value ratio plane as in Figure 1. The fact that the zero iso- ANPV curve is a horizontal line in the figure conforms with the conventional wisdom that there exists a break-even interest rate, C, for every investor. When the borrowing rate is above this break-even rate, the ANPV is negative regardless of the loan-to-value ratio. The investment project has negative leverage. The borrower should avoid borrowing if possible. When the interest rate is lower than this break-even rate, positive leverage is available and the ANPV is positive regardless of the loan-to-value ratio. 3 At this particular break- even rate, the effective borrowing yield is equal to the investor's required rate of return and therefore does not affect the profitability to the investor. This break-even rate is independent of the size of the investment (Vo). It is determined by the investor's characteristics and other terms involved in the mortgage contract.

Comparative statics also can be used to analyze the break-even rates among different investors. For example, the break-even interest rate is higher when the investor's marginal tax rate is higher. This is indicated by the positive sign of the partial derivative of C with respect to ~-.


Ir

0.3

0.25

0.2

0.15

0.1

0.05

0 I I,, ' ' ' I I I . . . . . . . . . . . . . . II l l l l l l l l l l l l l l

Q ~ O ~ Q

Loan To Value Ratio

Figure 1. Iso-~NPV curves.

3C k + my(1 - k) k y T - + > O. ( 1 5 )

3~- m(1 - "0 2 m(1 - "r)2[(1 + y):r _ 1]

When the zero iso-ANPV curve moves up in Figure 1, the investor with a higher tax rate finds it easier to realize positive leverage. As a result, these investors are the ones that are most likely to use debt financing.

Meanwhile, given the same interest-rate and loan-to-value ratio combination, the slope of the relevant iso-ANPV curve is steeper for the higher tax rate investors. Hence, the optimal loan-to-value ratio, which is the point where the iso-ANPV curve is tangent to the market interest-rate constraint I (M) , will be higher. This situation is proved geometrically by Figure 2.

Interest Rate

c.

c~

iso-ANPV H

~176 ..,.-" iso-A NPVL

i / : : I ~"

i ! ~ . f Loan to Value

Figure 2. Lower vs. higher marginal tax rate.


In Figure 2, the dashed curves are the iso-ANPV curves for L, an investor with lower marginal tax rate; the dotted curves are the iso-ANPV curves for H, an investor with higher marginal tax rate; and the thick solid curve is the I(M) for the particular market conditions. Point L* is the optimal interest-rate and loan-to-value ratio combination for investor L. At L*, an iso-ANPVL is tangent to I(M), and both of them have the same slope. From Figure 2, the iso-ANPVH that goes through L* has a steeper slope. As a result, the investor H can achieve a higher magnitude ANPV by moving along the I(M) curve in the upper right direction. H* represents the point where I(M) is the tangent to an iso-ANPVH, and is the optimal interest-rate and loan-to-value ratio combination for investor H.

3. Conclusion

This article goes beyond the simple conventional wisdom that debt financing should be used when financial leverage is positive by developing an equation that allows one to determine the optimal level of debt financing to use when positive leverage is possible. It is possible that some investments will have a positive NPV only if debt financing is used. Therefore, one should first identify the optimal loan-to-value ratio which maximizes NPV. Then, an investment decision could be made on the basis of whether this maximized NPV is positive or not.

Several hypotheses about the relationship between the optimal loan-to-value ratio and the equity investor's characteristics have been derived. These are as follows: (1) the optimal loan-to-value ratio increases as the marginal tax rate of the investor increases; (2) for relatively short holding periods, the optimal loan-to-value ratio increases as the holding period increases; (3) for relatively long holding periods, the optimal loan-to-value ratio decreases as the holding period increases; and (4) the optimal loan-to-value ratio increases as the required rate of return on equity for the investor increases.

Empirical tests of these hypotheses are proposed for future research. The results of these tests will be compared with the results of relevant earlier studies such as those by Gau and Wang (1990), Jaffe (1991), and Maris and Elayan (1990). This article is already differenti- ated from those earlier studies by the more rigorous manner in which the hypotheses to be tested are derived.

Acknowledgment

We wish to thank an anonymous referee for substantive comments that helped us to improve the final product.

Notes

1. For this analysis, it is assumed that the cost of equity (y) is a constant. The rationale for this assumption is that the analysis is for only one project and that separate non-recourse financing is available for each project. Note that this analysis is for real estate projects and is not necessarily generalizable to non-real estate projects. However, while generally accepted corporate finance theory says that the cost of equity of the firm generally


increases as the firm raises more and more equity, this is not a continuous increase. Rather, the cost of equity increases in discrete steps. For example, at some point, the finn may have to issue new stock instead of using retained earnings. As long as the required equity for the project being analyzed does not reach the point where the next step-up in cost occurs, a fiat cost of equity should be a reasonable assumption.

2. [(1 + y)r _ 1](1 + y) - Ty

> [(1 + y)r _ 1] - Ty (Since y > 0)

= (1 q- y)r _ (1 + Ty)

> 0 (Since compound interest is more than simple interest) 3. However, exactly which loan-to-value ratio should be adopted will depend on other parameters of the ANPV

function and on the shape of the interest-rate-loan-to-value ratio tradeoff function (I(M)) available on the market. The optimal loan-to-value ratio is the one at which the ANPV is maximized; that is, the one which gives the investor the highest NPV.

References

Cannaday, R. E., and T. L. T. Yang. (1995). "Optimal Interest Rate-Discount Points Combination: Strategy for Mortgage Contract Terms," Real Estate Economics (Spring), 65-83.

Gau, G. W., and K. Wang. (1990). "Capital Structure Decisions in Real Estate Investment," Journal of the Amer- ican Real Estate & Urban Economics Association (Winter), 501-521.

Hendershott, E H., and J. D. Shilling. (1989). "The Impact of the Agencies on Conventional Fixed-Rate Mortgage Yields," Journal of Real Estate Finance and Economics (June), 101 - 115.

Jaffe, J. E (1991). "Taxes and the Capital Structure of Partnerships, REITs, and Related Entities," Journal of Finance (March), 401-407.

Maris, B. A., and F. A. Elayan. (1990). "Capital Structure and the Cost of Capital for Untaxed Firms: The Case of REITs," Journal of the American Real Estate & Urban Economics Association (Spring), 22-39.

optimal leverage strategy: capital structure in real estate investments

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