A mortgage math problem

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.

http://www.freddiemac.com/pmms/

 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+r^N) 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.

http://en.wikipedia.org/wiki/Mortgage_calculator

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.