Re Academic Press Private Real Estate Investment_10 pptx

v ¼ dp þ 1 þ i ð Þt inc pti The balance of the loan, balance n, at the end of any particular year, n, is a function of the interest rate, term, and the initial balance.2 balance nð Þ ¼lo

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imposed maximum payment-to-income ratio, pti, can repay over a full amortization period, t.1

loan ¼

1 þ i

ð Þt

inc pti

The maximum value of the house he can purchase, v, is equal to the amount he can borrow, plus the value of his old residence used as a down payment, dp

v ¼ dp þ

1 þ i

ð Þt

inc pti

The balance of the loan, balance (n), at the end of any particular year, n, is a function of the interest rate, term, and the initial balance.2

balance nð Þ ¼loanð1 þ iÞ

tð1 þ iÞ12n

1 þ i

The sale price, s, at death is the value, v, increased by growth, g, compounded over the life expectancy, le

s ¼ 1 þ g le

dp þ

1 þ i

ð Þt

inc pti i

0 B

@

1 C

The bequest, b, is then merely the remaining equity, the difference between the value at sale and the loan balance

b ¼

dp 1 þ g le

i þ 1

1 þ i

ð Þt 1 þ gle

1

1 þ i

ð Þt

1 þ gle

þð1 þ iÞ12 le

incpti i

ð11-5Þ

1 In the interest of simplicity, we ignore other home ownership operating costs at this stage.

2

Note that this is not the equation for Ellwood Table #5.

Table 11-1 shows three datasets to be used as input values for the examples

in this chapter The second and third datasets are used only in the reverse amortization mortgage section and only differ in life expectancy, growth rate, and loan-to-value ratios Note that the variable for value, v, provided in Equation 11-2 is a computed value, but val in the datasets is a fixed given value

Using data1 we obtain the following values for what we are calling the conventional arrangement, as shown in Table 11-2

The above example ignores the fact that operating costs for the house may increase, but also ignores the fact that retirement income may be indexed In the interest of simplicity, these are assumed to cancel

TABLE 11-1 Three Datasets

data1 data2 data3 Downpayment dp $135,000 $135,000 $135,000

Interest rate i 0.06/12 0.06/12 0.06/12

Operating cost oc 0.04 0.04 0.04 Income inc $3,750 $3,750 $3,750 Payment-to-income ratio pti 0.4 0.4 0.4 Value val $300,000 $300,000 $300,000 Loan-to-value ratio ltv 0.6 0.6 0.4 Payment pmt $1,500 $1,500 $1,500

TABLE 11-2 Values for the Convention Arrangement Purchase price $385,187 Downpayment $135,000

Sale price $487,385 Loan balance at life expectancy $228,666

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THE REVERSE AMORTIZATION MORTGAGE

We now consider a retiree who owns a larger house free of debt and wishes to generate monthly income from his home equity without selling the home The lender will grant the loan based on his life expectancy, le, the value of the house, val, interest rate, i, and payment amount, pmt Ellwood Table #2 handles the way $1 added each period at interest grows The lender sets a maximum loan amount based on the loan-to-value ratio, ltv

hecmbalðnÞ ¼ min ð1 þ iÞ

12n1

i pmt,ltv val 1 þ g

ð11-6Þ

Thus, given data1 and using le for n, the loan balance at life expectancy is

$129,613 As this is less than ltv  val(1 þ g)n, payments occur throughout the full life expectancy of the retiree By incorporating growth into the model,

we assume that the lender is willing to lend against future increases in value (g > 0) Should that not be the case, in data2 where g ¼ 0 and le ¼ 8, the loan reaches its maximum (ltv  initial value) at 94 months and payments stop short of life expectancy

From a lender’s risk perspective, the imposition of a cap is an essential underwriting decision How the cap is computed is also important It can

be based, as above in data1, on a fixed property value and permit a larger initial loan-to-value ratio or it can allow for growth in value but allow a lower loan-to-value ratio as in data3 Clearly, the lender does not want the loan balance to exceed the property value Because the loan documents are a contract, the lender must perform by making payments regardless of the change in value Thus, different assumptions impose different burdens and benefits, respectively, on the lender and borrower When we permit the growth assumption, but reduce the loan-to-value ratio as in data3, the payments stop in 85 months If the dollar amount of appreciation

in house value grows faster than the balance of the loan, it is possible that the house could once again ‘‘afford’’ more payments and payments would resume.3

The sample amounts are not represented to be any sort of standard; they are arbitrary and merely serve as an illustration The plot in Figure 11-1 demonstrates the importance to both parties of estimating life expectancy correctly, obviously not an easy task The type of loan contract most desirable differs depending on how long one expects to need the income

3 From a loan servicing standpoint, this is an unappetizing prospect for the lender.

Using Equation (11-7), one can approach the question from the standpoint

of the maximum payment, mopmt, allowed under the three data scenarios offered in Table 11-1, each requiring one to know life expectancy exactly

1 þ i

ð Þ12n1ltv  val 1 þ g

ð11-7Þ

Table 11-3 shows the maximum payments under the three datasets of Table 11-1

We see in Figure 11-2 that in the choice between a plan with a larger loan-to-value ratio but no growth assumption (data2) and one with a growth assumption but a smaller loan-to-value ratio (data3), the decision changes when one’s life expectancy is ten years or more Not surprisingly, the most

2 4 6 8 10 Years 25000

50000 75000 100000 125000 150000 175000 Balance

No Growth − Hi LTV Growth − Low LTV

FIGURE 11-1 Reverse amortization mortgages under different growth assumptions.

TABLE 11-3 Maximum Payment under Different Assumptions

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permissive arrangement (allowance for growth and high loan-to-value ratio)

in the original dataset (data1) provides the highest payment

INTRA-FAMILY ALTERNATIVES

The above examples represent ways to approach the problem using institutional lenders We now turn to intra-family methods where economics only partially control We shall focus on modifications to conventional arrangements That is, we shall assume the reverse annuity mortgage option is not available because the retiree does not own a home of sufficient size to produce the desired results There are two ways to approach such a financing scheme

1 Should someone be willing to purchase a house for our retiree to live in for his lifetime with no right to devise by will, the retiree would have

an additional $1,500 per month discretionary income This, which we will call the Income Viewpoint, considerably enhances his retirement lifestyle

2 Alternatively, the retiree could live in a house he could not otherwise afford if he is unconstrained by the loan qualifying payment-to-income ratio We will call this the Larger House Viewpoint This variation is just

500

1000

1500

2000

Payment

data 3 data 2 data 1

FIGURE 11-2 Payment under different sets of assumptions.

a special case of lifestyle enhancement in which the larger residence is how one elects to apply larger disposable income arising from the life estate arrangement

THE INCOME VIEWPOINT

In the conventional example, our retiree essentially ‘‘purchases’’ the satisfaction of leaving a bequest by incurring the obligation to make loan payments and foregoing the benefits associated with more discretionary income he would have had during his lifetime if he did not have loan payments to make The income viewpoint amounts to ‘‘selling’’ that satisfac-tion in return for the enhanced present income The interesting quessatisfac-tion is: How much of one is the other worth?

The tradeoff is between leaving a bequest, b, and current income, inc.4A rational retiree chooses based on his calculation of the greater of these two Such a calculation involves assumptions that can, at times, be uncomfort-able to make Using Equation (11-5), the value of the bequest in Tuncomfort-able 11-2 for data1 circumstances is $258,719

To make a fair comparison we need to know the present value of the income foregone in order that a bequest may be left If our retiree is able to live in a house without paying loan payments, he enjoys that income for the remainder of his life The present value of this income is computed via Equation (11-8)

pv ¼

1 þ i

ð Þ12 le

inc pti

If we value that income at the same interest rate as the bank and accurately predict life expectancy (recall we said some uncomfortable assumptions would be necessary), using data1 the present value of those payments is

$90,509 As the $258,719 bequest is larger than the present value of the foregone income, if one takes the simple (too simple!) position that the investor chooses the largest of these, he buys a house, makes payments, and leaves a bequest

Why is this too simple? It is naive to equate the nominal value of money left to someone else in the future with the present value of dollars one may

4 This is popularized by the bumper sticker adorning many recreational vehicles that reports,

‘‘We’re spending our children’s inheritance!.’’

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personally consume Merely incorporating the time value of money and using the same rate as the bank, a present value calculation performed

on the bequest seems at least reasonable Thus, the decision rule becomes Equation (11-9)

1 þ i

ð Þ12 le,pv

ð11-9Þ

But under data1, at bank interest rates the discounted value of the bequest,

$180,664, is larger than the $90,509 present value of the foregone income, so this retiree still buys a house and leaves a bequest

Present value may imperfectly adjust for the difference between the value our retiree places on his own consumption and the value he places on financing the future consumption of others One way to deal with this is to increase the discount rate on the bequest Suppose we arbitrarily value bequest dollars considerably less than present consumption dollars by making the discount rate thrice the interest rate Now, for data1 the present value of the bequest, $88,567, is below the present value of the foregone income Under these conditions our retiree opts to have someone else buy him a house, someone who will receive the house at his death.5

So for the Income Viewpoint and given data1, the decision turns on how dollars the retiree may consume are valued versus how he values dollars he leaves behind This means the retiree carefully selects a discount rate that adjusts future dollars others receive to equal the value of dollars he may otherwise consume

THE LARGER HOUSE VIEWPOINT

One point illustrates how this may, indeed, be creative financing An institutional lender evaluates risk based on the probability of repayment taking place over the investor’s lifetime As there is a cap on his dollar return (all interest payments plus the principal), the lender makes a loan governed

by the realities of (a) the income the retiree has during his lifetime to make payments and/or (b) the liquidation value of the property needed to retire any balance remaining at the retiree’s death The Remainderman as lender has a different perspective Since he captures the entire (uncertain) value of the property at death, the Remainderman’s payoff prospects are different Also, it

is possible that an older relative’s care of a larger property for the

5

We are reminded that we assumed the ‘‘someone’’ who buys the retiree a house is not his heir If this were not the case the retiree would be, in a sense, merely deciding the form of the bequest.

Remainderman can produce positive results for the Remainderman that are not included in these computations

Let us begin by noting how the retiree will approach the possibility of a larger house Remember that ‘‘larger’’ is just a metaphor for ‘‘better’’ in some tangible way The house may be better located, newer, have a better view, be larger, or otherwise in some sense be more desirable than the house the retiree might purchase We assume that all of these desirable attributes will be captured in a higher price, making possible the measurement of larger or better

Suppose that the retiree’s self-imposed limit on the portion of his income

he will spend on housing is the same fraction a lender will allow That is, he wishes to have the most house he can support, paying in operating costs, oc, the same amount as his loan payment would have been had he purchased the property The point is that our retiree has a housing budget that is a self-imposed constraint on the size of house he is willing to ‘‘support,’’ whether that support is in the form of loan payments, upkeep, or some combination of the two Clearly, ‘‘bigger’’ or better is more feasible without loan payments We will suppose that annual operating costs on an expensive residence run 4% of its purchase price Thus, he can ‘‘carry’’ a house the value of which is equal to the ratio of his annual housing budget to operating costs Using Equation (11-10) and data1, our retiree acquires a house valued at $300,000

lg hse ¼12inc pti

If we assume, naively, that the utility of different houses is represented

by the difference in their values, using Equation (11-11), the retiree chooses the greater of this difference or the bequest, again requiring

an ‘‘appropriate’’ discount, which we have again set three times bank interest rate

Max lg hse  v, b

1 þ:18 12

2 6 6

3 7

Under data1 conditions, the larger of these alternatives, $88,567, is the bequest

Setting the two equations in Equation (11-11) equal and solving for payment-to-income ratio, we can find an indifference point based on the

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portion of the retiree’s income he is willing to devote to housing Using data1 inputs we find that, if all else is equal and the retiree uses only 17.73% of his income for housing rather than the 40% the lender would allow, he is indifferent between the large house and the bequest This provides planning flexibility in that under these circumstances the retiree may choose to use an additional 22.27% of his income either for housing or for other retirement comforts

The qualifier ‘‘if all else is equal’’ is important Combining the variables using different values provides an infinite number of permutations For instance, leaving the discount rate at the bank interest rate, i, moves the indifference point of the payment-to-income ratio to 28.57%, again making the choice of discount rate critical The case shown here is a template for further reflection following some simulation using the Excel workbook that accompanies this chapter

THE REMAINDERMAN’S POSITION

The Remainderman’s position is conceptually much simpler He may be viewed as buying a zero coupon bond with an uncertain payoff date and amount We assume that the Remainderman buys the house for its value, v, and concurrently sells a life estate to the retiree for the amount the retiree realizes from the sale of his old residence, dp In that way the Remainderman really is providing financing, creative or not, for he takes the place of the lender His net investment is the amount of the loan The payoff is the sale price of the property, an unknown amount, at the death of the retiree, on an unknown date

THE INCOME CASE

Given data1, the Remainderman’s investment would be a loan of $250,187 on which he computes an annual return of 11.11% using Equation (11-12)

retInc ¼Log s=loan½ 

Figure 11-3 shows that, as one might expect, the return is negatively related to life expectancy and positively related to growth Because higher returns occur in the early years, the choice of which relative to stand in as lender is critical One does not want to create a perverse incentive in such

an arrangement Measuring the utility our Remainderman gains from his

relations’ longevity (or lack of it!) is at best an unsavory task that even an economist would not relish

THE LARGER HOUSE CASE

The larger house alternative may be less attractive for the younger family member One reason is that in our example the retiree’s purchase price for the life estate is limited to the value of his former residence So even though the growth takes place on a bigger number, unless the larger house comes with a larger growth rate, because of the larger investment this alternative yields less, 9.87% per annum using data1, to the junior member of the family

retLghse ¼

Log lg hse 1 þ g

 le

lg hse  dp

The longer the arrangement continues, the lower the yield At le ¼ 20 years, the yield drops to 5.71%

The return is again negatively related to life expectancy Figure 11-4 shows that if a larger house comes with higher growth, the return is respectable across the likely range of the investment time horizon

5

10

15

20 0.04

0.05 0.06 0.07

Growth 0.1

0.2 0.3 0.4 0.5

Return

Life Expectancy

FIGURE 11-3 Return based on growth and life expectancy.