Question: A person buying a house is considering two
different financing options. The first
financing option entails a $200,000 30-year mortgage. The
second financing option entails a larger down payment and a smaller 15-year
mortgage. The monthly mortgage payment
is identical in both financing options.
The interest rate on the 30-year FRM is 3.43%. The interest rate on the 15-year FRM is
2.65%. These are the rates that
actually exist in the week of April 11, 2013.
Based on these facts:
·
What is the monthly mortgage payment for the two
financing options?
·
What is the loan balance on the 15-year FRM
under the second financing option?
·
How much liquid assets must the homebuyer use to
reduce the mortgage under financing option two.
·
What is the outstanding balance on the 30-year
FRM after 15 years?
·
What rate of return on liquid assets must a
person who takes financial package one earn to have the same wealth as a person
who takes financing package two?
Background: The
answer to this problem relies on mortgage payment and annuity formulas. The formula for the monthly mortgage payment
is
C=r x P/ (1-(1+r)-N)
Where c is the monthly payment on the mortgage, r is the
monthly interest rate, N is the maturity of the loan in months, and P is the
initial balance of the loan.
The formula for the outstanding loan balance after N
payments is
AO=(1+rN) x
P – ((1+r)N-1)/(r)) x C
Where AO is the amount outstanding on the loan, r, N, P and
c are as defined above.
Wikipedia has an excellent discussion of the derivation of
these formulas.
The values for the mortgage payments and the outstanding
balance equation can also be calculated with pre-programmed finance functions
in Excel. The Excel function for the
mortgage payment is PMT. The Excel
function for the outstanding balance value is FV.
Answer:
Mortgage payments:
Plugging r=0.0343/12, N=360, and P=$200,000 into the formula
for monthly payment we get c=$890.32.
The loan balance on a
15-year loan with a mortgage payment of $890.32:
Plugging r=0.0265/12 N=180 and P=$200,000 we find the
mortgage payment of a 15-year mortgage on a $200,000 loan is $1,347.7. In order to reduce the payment to $890.32 the
loan balance must be reduced to $132,116 (890.29/1347.7) x $200,000. See the formula for c for the intuition
behind this adjustment.
Reduction in liquid assets:
In order to reduce the loan from $200,000 to $132,116 the
person must reduce liquid assets by $67,884 ($200,000-$132,116.)
Mortgage balances
after 15 years:
The loan balance for the 15-year mortgage after 15 years of
payment is $0.
The loan balance for the 30-year mortgage after 15 years can
be obtained by plugging r=0.0343/12, N=180, PMT=890.29, and PV=$200,000 into
the FV function in Excel. (You can also
use the formula for AO above.) The
30-year loan balance after 15 years is $125,137.
Breakeven point:
The person using financial package number one initially has
$67,884 more in liquid assets than the person using financial package two. (Under investment strategy two the mortgage
is reduced by this amount.) The rate of
return needed for $67,884 to grow to $125,137 in 15 years is:
Rate=100*((125,137/67,884)(1/15)-1)=4.16%
Some concluding remarks: Under current market conditions, a person
with some cash should take financing package two. Use the cash to pay down the mortgage and
take out the 15-year FRM. The 15-year
FRM has a lower interest rate. CD rates
are much lower than 4.16%.
True mortgage interest is tax deductible. However, interest income, dividends and
capital gains are also taxed and investments in equities are risky.
Individuals with higher cost student loans or credit card
debts would be better off with a 30-year FRM, or perhaps renting until they can
afford a 15-year FRM.
The riskless opportunity cost for people who do not have
high-interest debt and have a lot of cash is the CD rate, which is very low
right now. (Ten-year CDs are around
1.0%.) Homebuyers with cash and little
debt should take the 15-year mortgage.
Most homebuyers do not have a ton of cash so the selection
of a 15-year FRM will probably involve higher mortgage payments.
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