The Scenario

An investor is buying a house and the bank offers them a point equal to 1 percent of the loan amount to reduce the APR from 6.5 percent to 6.25 percent on their $400,000 30-year mortgage. This assignment seeks to determine when the investor should and should not buy the point if they:

a) Plan to be in the house for less than five years

A point equal to 1 percent of the loan amount would be 1% of $400,000 = $4,000

Further, we need to convert the APR to the effective annual rate (EAR) at which the bank will charge interest.

Most mortgage loans are compounded monthly. Assuming a monthly compounding, we can obtain the EAR as follows (Berk et al., 2021):

EAR = (1 + APR/k)k where k is the number of compounding periods, which is 12 months

If APR = 6.5%; EAR = (1 + 0.065/12)1/12 = 0.045%

If APR = 6.25%, EAR = (1+ 0.0625/12)1/12 = 0.043%

To determine when the investor should buy the point, we calculate the present value of all future payments, with the monthly repayment rate as an annuity, assuming they pay for the point at the beginning and at some point in the middle, say after year 1 (12 months).

Assuming the investor stays for 5 years (60 months and pays the point at the beginning (month 0);

In this case, they would pay the 4,000 point, and an EAR of 0.045% per month for the remaining $396,000, giving a total balance of $413,820, including $13,820 in interest. The $413,820 would translate to a monthly repayment rate of 6,897 assuming the investor stays for 60 months ($413,820/60 = 6,897). The repayment timeline would be represented by:

Month

0

1

2

3

.60

Amount

4,000

6,897

6,897

6,897

6,897

Using the Excel formula to calculate the present value of an annuity

PV (Rate, NPER, PMT) where the Rate is 0.043%, PMT is 6,897, and NPER is 60 months)

= PV (0.043, 60,6897) = $408,440.66

Assuming, alternatively, the investor pays the point at the end of the first year, they will pay the higher EAR of 0.045% per month for the first 12 months, then the lower EAR of 0.043% on the loan balance for the next 48 months.

In this case, the total loan of $400,000 would attract an interest of $18,000 (0.045% of 400,000), leading to a total repayment of $418,000. The $418,000 would translate to a monthly payment of $6,967 for 60 months. However, at month13, the investor will buy the $4,000 point and thereafter a monthly rate at the lower rate of 0.043% for 48 months. Using this and the amount paid in the first year equal to (($6,967 x 12) + $4,000 point) = $87,604) gives a balance of $330,396. The new monthly rate of $6,883is obtained by dividing the $330,396 balance by 48. Thus, the timeline will be as follows:

Month

1

2

12

13

1

1

48

Amount

6,967

6,967

6.967

4,000

6,883

6,883

6,883

Using the Excel formula to calculate the present value of an annuity

PV (Rate, NPER, PMT) where the Rate is 0.045%, PMT is 6.967, and NPER is 12 months)

= PV (0.045%, 12, 6967) = $83,359.97

The PV for the $4,000 point is = 4,000/1.045 = $3,827

The present value for the next 48 months is given by:

PV (Rate, NPER, PMT) where the Rate is 0.043%, PMT is 6,883 and NPER is 48 months)

= PV (0.043%, 48,6883) = $326,928.21

This gives a total present value of $414,115.18

Conclusion: buying a point upfront for 60 months and paying off the balance at the 0.043% EAR is equivalent to a present value of $408,440.66, while buying it at the end of year 1 (the first 12 months) then only enjoying the 0.043% rate for 48 months is equivalent to a present value of $414,115.18. The cost...…of 0.043% for 348 months. Using this and the amount paid in the first year equal to (($1,186 x 12) + $4,000 point) = $17,932) gives a balance of $400,068. The new monthly rate of $1,150 is obtained by dividing the $400,068 balance by 348. Thus, the timeline will be as follows:

Month

1

2

12

13

1

2

108

Amount

1,161

1,161

1,161

4,000

1,150

1,150

1,150

PV (Rate, NPER, PMT) where the Rate is 0.045%, PMT is 1,161, and NPER is 12 months)

= PV (0.045%, 12, 1161) = $13,891.33

The PV for the $4,000 point is = 4,000/1.043 = $3,835

The present value for the next 348 months is given by:

PV (Rate, NPER, PMT) where the Rate is 0.043%, PMT is 1,150, and NPER is 348 months)

= PV (0.043, 348, 1150) = $370,361.01

This gives a total present value of $388,087.34

Conclusion: buying a point upfront for 360 months and paying off the balance at the 0.45% EAR is equivalent to a present value of $386,139.20, while buying it at the end of year 1 (the first 12 months) then only enjoying the 0.43% rate for 348 months is equivalent to a present value of $388,087.34. The cost is lower when the investor buys the point upfront (in the first month) and enjoys the lower rate for the next 360 months. It would not be advisable to buy the point any other time as this would result in higher costs.

Regardless of how long the investor plans to stay in the house, the most appropriate option is to buy the 1% point immediately or upfront (in month 0) as this results in lower costs than when they buy it any other time. The greatest disadvantage of this kind of present value analysis is that it is not easy to accurately determine the discount rate that represents the true risk premium of the investment (Siegel, 2022). In my view, therefore, the analysis may…