If could solve part A through part C. Thank you! the arbitrage argument says tha

Question

If could solve part A through part C. Thank you!

the arbitrage argument says that the price of a house should equal the present discounted value of the amount that the house could be rented for, adjusting for leverage.

7. Housing prices: Suppose a condominium can be rented for S1,000 a month, it depreciates at 10 percent per year, and the annual interest rate is 5 percent. Let the down-payment rate and the annual growth rate of condominium prices be given by the table below Growth rate of condo prices (percent) Down-payment rate, X (percent) 20 20 20 20 100 10 Price of the condo 10 (a) For each case, compute the value of the housing price according to the (b) Based on your results, discuss the sensitivity of condo prices to the expected (c) Based on your results, discuss the sensitivity of condo prices to the down- simple theory developed in the chapter. capital gain. payment rate

Explanation / Answer

(a)

A simple theory to value house can be as if you are valuing a share. The formula is Po = Div1 + P1 / 1 + K.

In case of house or real estate, Div 1 can be taken as Net operating income

as Rent - Depreciation (taken as maintenance cost) - interest cost (the cost of leverage).

K as the cost of capital will be the interest rate, although a better approximation will be to take weighted average cost of capital with interest as cost of debt (or leverage) and cost of equity (or of own funds ) calculated through a model like CAPM.

Monthly rental is 1,000 and hence yearly rental is 12,000. Depreciation is taken as 10% of initial cost i.e., Po. The leverage depends upon down payment. Hence if down payment is 20%, 80% funds are borrowed with a cost of 80%*Po*.05, because interest cost is 5%.

In our case

Div1 or Net operating income from the condominium is (12000 - .10*Po - .80*Po*.05), if down payment is 20%. P1 will be 1.02Po if the growth rate of prices is 2%

Case 1 Growth rate – 0%; Down payment – 20%

Po = Div1 + P1 / 1+k

= (12,000 - .10*Po - .80*Po*.05) + 1*Po(because growth rate is zero and hence Po will be equal to P1) / 1+.05

Po = (12,000 - .10*Po - .80*Po*.05) + 1*Po / 1.05

1.05Po = 12000 - .14Po + Po

.19Po = 12000

Po = 12000 / .19 = $63157

Case 2 Growth rate – 2%; Down payment – 20%

Po = (12,000 - .10*Po - .80*Po*.05) + 1.02*Po / 1.05

1.05Po = 12000 - .14Po + 1.02Po

.17Po = 12000

Po = 12000 / .17 = $ 70588

Case 3 Growth rate – 5%; Down payment – 20%

Po = (12,000 - .10*Po - .80*Po*.05) + 1.05*Po / 1.05

1.05Po = 12000 - .14Po + 1.05Po

.14Po = 12000

Po = 12000 / .14 = $ 85714

Case 4 Growth rate – 10%; Down payment – 20%

Po = (12,000 - .10*Po - .80*Po*.05) + 1.10*Po / 1.05

1.05Po = 12000 - .14Po + 1.10Po

.09Po = 12000

Po = 12000 / .09 = $ 133,333

Case 5 Growth rate – 5%; Down payment – 100%

Po = (12,000 - .10*Po ) + 1.05*Po / 1.05

1.05Po = 12000 - .10Po + 1.05Po

.10Po = 12000

Po = 12000 / .10 = $ 120,000

Case 6 Growth rate – 5%; Down payment – 10%

Po = (12,000 - .10*Po - .90*Po*.05) + 1.05*Po / 1.05

1.05Po = 12000 - .145Po + 1.05Po

.145Po = 12000

Po = 12000 / .145 = $ 82,758

Case 7 Growth rate – 5%; Down payment – 5%

Po = (12,000 - .10*Po - .95*Po*.05) + 1.05*Po / 1.05

1.05Po = 12000 - .1475Po + 1.05Po

.1475Po = 12000

Po = 12000 / .1475 = $ 81,355

(b) Compare case 1 to 4, the sensitivity of the condo price to capital gain is subatantial and range is from $63,000 to $133,000.

(c ) Compare case 3,5,6, and 7, with a constant growth rate and changing down payment. The sensitivity is much less

as compared to change in the growth rate

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