Question: A homebuyer is considering taking out a
$200,000 loan and is mulling two options. The first option is a 30year FRM at 3.50%.
The second option is a 15year FRM at 2.7%. The 15year FRM comes with a larger monthly
payment. Some homebuyers might choose
the 30year FRM and invest the difference in mortgage payments. What rate of return on financial assets would
make the 30year FRM the preferred choice?
Answer: The first step is to calculate the monthly
mortgage payment for both loans. The
monthly mortgage payment is obtained from the PMT function in Excel.
For the thirtyyear mortgage we get
$898.09=PMT(0.035/12,
360,$200,00)
For the 15year mortgage we get
$1,353.49=PMT(0.027/12,
180, $200,000)
The difference in the 15year versus 30year mortgage
payment is $454.40.
The homebuyer must decide whether to borrow with a 15year
FRM and take the lower interest rate or borrow with the 30year FRM and invest
the additional $454.40 per month.
The relative advantage of the 15year FRM over the 30year
FRM depends upon how long the homebuyer plans to own the property. We solve the problem for holding periods of
5, 10, and 15 years.
The amount of debt paid off at the N^{th} year is
$200,000 minus the future value of the loan.
The future value of the loan is calculated from the FV function in Excel
where FV is a function of interest rate
(0.035/12 or 0.27/12), the number of payments (60, 120, or 180), the
payment ($898.09 or $1352.49) and the loan balance ($200,000).
30year FRM

15year FRM Difference


Interest Rate

0.035

0.027

0.008

Number of Payments

360

180

180

Loan Amount

200000

200000

0

Monthly Payment

($898.09)

($1,352.49)

($454.40)

Paid off Debt after 5
Years

$20,605.85

$57,905.05

$37,299.19

Paid off Debt after 10
Years

$45,146.25

$124,169.45

$79,023.20

Paid off Debt after 15
Years

$74,372.46

$200,000.00

$125,627.54

In order for the homeowner to be better off with the
thirtyyear loan she must earn more than the paid off debt amount by investing
$454.40 per month over the holding period.
This rate of return can be calculated with the RATE function in Excel. Inputs for RATE are the number of payments,
the monthly mortgage payment, the initial value of the annuity, and the future
value of the annuity.
Using values from the fiveyear holding period you can
confirm that
12.2%=12 x
RATE(60,$454.40,0,$37,299.19)
We can confirm that this answer is correct by placing 12.2%/12
into the future value function along with other inputs N=60 and PMT=$454.40 to
get final value $37,299.19.
Rate on annuity

Final Value of annuity


Required Return 5year
Holding Period

12.2%

$37,299.15

Required Return 10year
Holding Period

7.1%

$79,023.12

Required Return 15year
Holding Period

5.4%

$125,627.41

At this time, the advantages of a 15year FRM over the
30year FRM for those who can qualify are compelling. This issue was also discussed at Finance
Memos
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