An exotic mortgage
This post was
motivated by a paper on shared appreciation mortgages that I am scheduled to
present at the 53113 AREUEA meetings in Washington DC.
Situation:
You are purchasing a home worth $200,000. You are given two mortgage options. Option one is a $180,000 30year fixed rate
mortgage with a 5.0% interest rate.
Option two involves a smaller $140,000 30year fixed rate mortgage also
with a 5.0% interest rate. Under option
two, the additional $40,000 is raised from a thirdparty investor. The private investor is promised the return
of $40,000 plus 40% of the capital gain on the home.
The house is sold after seven years. The average annual growth rate in the house
price over the sevenyear period was 4.0%.
Questions:
 What are the monthly mortgage payments under the two options?
 What is the value of the home, the amount owed on the property, and house equity after seven years?
 The reduced mortgage payments obtained from selecting option two over option one are invested in the stock market. What rate of return on stocks would result in an identical return for the two financing options?
Answers:
 The payment on the mortgages is calculated with the PMT function in Excel. For the $180,000 FRM we have PMT(0.05/12,360, $180,000)  $966. For the $140,000 FRM we have PMT(0.05/12,360,$140,000) or $752.
 The house price after 7 years is $200,000 x (1.03)^{7} or $263,186.
 The amount owed from the $180,000 mortgage on the first financing option is obtained from the FV value in Excel. The future value of the mortgage is FV(0.05/12,84,$966, $180,000)=$158,301. There is no SAM so this is total housing debt. The house equity is $263,186$158,301 or $104,885.
 The amount owed from the $140,000 mortgage is $123,123. This is obtained from the FV function FV(0.05/12,84,$752,$140,000). The amount owed on the SAM is $65,275 ($40,0000+.40 x ($263,186$200,000)) The total owed from the mortgage and SAM is $188,398. House equity is $74,789 ($263,186$188,398)
House Equity from Two
Mortgage Arrangements


Option

1

2

Interest rate

0.05

0.05

Term

360

360

Loan

$180,000

$140,000

SAM

$0

$40,000

Holding period (years)

7

7

Monthly Payment

($966)

($752)

Value of Home

$263,186

$263,186

Amount owed on mortgage

($158,301)

($123,123)

Amount owed on SAM

NA

($65,275)

Total Owed

($158,301)

($188,398)

House Equity

$104,885

$74,789

 The mortgage payment for the SAM/FRM combination is $215 lower than the mortgage for the FRM. House equity is $30,097 higher under option 1, the standard FRM with no SAM. In order for a person to invest $215 per month and end up with $30,097 the person must earn an average annual rate of 13.78%. Observe that 12*RATE(84,$215,$0,$30097)=13.78
Check this answer by placing 13.78 into the FV function and
getting a final value of $30,096.
Analysis of returns


Monthly mortgage savings
from SAMS

$215

Future value of annuity
needed to bread even

$30,097

Return needed to break
even

13.78%

Check

($30,096.54)

The primary focus of my paper is on how SAMS affect
household wealth. I find that the gains
to borrowers are small when house prices are low. The loss to homebuyers is large when house
price increases are large. Household
savings is the second most important component of retirement saving. The use of SAMS would erode wealth available
for retirement.
My paper should be available on the AREUEA web site
soon. I will update this post when it is
available.
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