Question

The Johnsons have accumulated a nest egg of $50,000 that they intend to use as a down payment toward the purchase of a new house. Because their present gross income has placed them in a relatively high tax bracket, they have decided to invest a minimum of $2900/month in monthly payments (to take advantage of the tax deduction) toward the purchase of their house. However, because of other financial obligations, their monthly payments should not exceed $3200. If the Johnsons decide to secure a 15-year mortgage, what is the price range of houses that they should consider when the local mortgage rate for this type of loan is 5%/year compounded monthly? (Round your answers to the nearest cent.)

Answer 1

Down payment is $50,000. The minimum monthly payment is $2900 per month whereas the maximum is $3200 per month

Interest rate, r = 5%/year compounded monthly. The monthly rate, i = r/m where m =12 months; so i = 5%/12 = 0.00417

Number of years mortgage, n = 15. Hence, the number of monthly payment periods N = n*m = 15 years * 12 months = 180 months.

The present value of the monthly payments is given by the formula

PV = A * [ (1+i)N - 1] / [ i * (1+i)N ]

where PV is the present value of an annuity of A for N periods at i rate per period.

When minimum payment of $ 2,900 per month is made the present value is:

PV = 2,900 * [ (1+ 0.00417)180 - 1] / [ 0.00417 * (1+ 0.00417)180]   ......equation #1

= 2,900 * [ (1.00417)180 - 1] / [ 0.00417 * (1.00417)180]  

= 2,900 * [ 2.11497 - 1 ] / [ 0.008819 ]

= 2900 * ( 1.11497 / 0.008819 )

= $366,641.68

When maximum payment of $ 3,200 per month is made the present value is:

PV = 3,200 * [ (1+ 0.00417)180 - 1] / [ 0.00417 * (1+ 0.00417)180] which is nearly the same as the earlier equation #1 except that we replaced the annuity amount, A with $3,200

So, PV = 3,200 * ( 1.11497 / 0.008819 )

= $404,570.13

Thus the price range of houses the Johnsons should consider is from $366,641.68 to $404,570.13