You need to borrow $2,500,000 for the purchase of a retail development. You expe

Question

You need to borrow $2,500,000 for the purchase of a retail development. You expect to sell the property after 5 years and have the following mortgage options:

a. A 30-year fixed rate mortgage at 8% with no points or prepayment penalties

b. A 30-year fixed rate mortgage at 7% with 3 points and a 1% prepayment penalty if repaid within the first 7 years

c. A 30-year adjustable rate mortgage with no points or prepayment penalty. Mortgage contract rates for loans of this size and risk are currently 6.75%. Rates are expected to rise to 7% in year 2, 7.25% in year 3, 7.75% in year 4, and 8% in year 5.

Which loan option gives you the lowest effective borrowing cost?

Explanation / Answer

Amount borrowed P = $2,500,000

Time of repayment is after 5 years.

Here we need to floow the following procedure:

1) Find equated monthly intallments for 30 years

2) Find the future value of the loan, we would have paid in 30 years.

3) Calculate the present value at the 5th year of this 30 year future value.

4) The above value will the amount that you will pay in the 5th year, plus the prepayment penalty if any.

5) Compare this with the principal amount or the discounted principal, if we have discount points. Calculate the interest that you will pay after 5 years.

a)

Here P = $ 2,500,000

i = 8% per annum

r = 8/12 = 0.67 per month

n = 30 years = 360 months , N= 5 years = 60 months

t = no.of months in a year = 12

EMI = (P x i) x ((1 + r)n)/ (t x ((1 + r)n)- 1) = (2,500,000 x 0.08) x ((1 + 0.0067)360)/ (12 x ((1 + 0.0067)360)- 1)

=> EMI= R = $16,712.78

Future value of the loan = A = R((1+r)n - 1)/r = 16712.78((1+0.0067)360-1)/0.0067 = $25,111,247.54

After the years the present value of A = A/(1+r)n-N =   $25,111,247.54/(1+0.0067)360-60 =$25,047,717.80/(1+0.0067)300 = $3,387,253.54

So we will pay $3,387,253.54 after 5 years or 60 months for the loan. So the new interest rate r' is

A = P(1+r)N => r = (A/P)1/N-1 = ($3,387,253.54 /$2,500,000)1/60 - 1

r = 0.0051= 0.51%

EAR = (1+0.0051)12 - 1 = 0.0626= 6.26%

b)

Here P = $ 2,500,000

i = 7% per annum

r = 7/12 = 0.58 per month

n = 30 years = 360 months , N= 5 years = 60 months

t = no.of months in a year = 12

EMI = (P x i) x ((1 + r)n)/ (t x ((1 + r)n)- 1) = (2,500,000 x 0.07) x ((1 + 0.0058)360)/ (12 x ((1 + 0.0058)360)- 1)

=> EMI= R = $14,623.72

Future value of the loan = A = R((1+r)n - 1)/r = 14,623.72((1+0.0058)360-1)/0.0067 = $17,700,344.88

After the years the present value of A = A/(1+r)n-N =   $17,700,344.88/(1+0.0058)360-60 =$17,700,344.88/(1+0.0058)300 = $3,122,427.36

The policy has a prepayment clause of 1%, which means 1% of the remaining balance need to be paid as penalty.

Remaining balance = $3,122,427.36 - [$14,623.72*((1+0.0058)60-1)/0.0058 = $2076556.04

1% pre-payment panalty = 0.01*$2076556.04 = $20,765.56

So we will pay $3,122,427.36+$20,765.56 = $3,143,192.92 after 5 years or 60 months for the loan.

Also we have 3 points discount rate, so our principal now becomes P' = (1-0.03)P = 0.97*2,500,000= $2,425,000

So the new interest rate r' is

A = P'(1+r)N => r = (A/P')1/N-1 = ($3,143,192.92/$2,425,000)1/60 - 1

r = 0.0043 = 0.43%

EAR = (1+0.0043)12 - 1 = 0.0533 = 5.33%

c)

Here P = $ 2,500,000

i1 = 6.75% per annum, i2? = 7% per annum, i3 = 7.25% per annum, i4 = 7.75% per annum, i5 = 8% per annum  

i1 = 0.5625% per month, i2? =0.5833% per month, i3 = 0.6042% per month, i4 = 0.6458% per month, i5 = 0.6667% per month  

n = 30 years = 360 months , N= 5 years = 60 months

t = no.of months in a year = 12

Since this is a floating rate loan, we cannot calculate the exact future value after 30 years so we will use the future value of the loan after 5 years, which will be nearly equal to the exact value.

Future value of the loan = A = P(1+i1)(1+i2)(1+i3)(1+i4)(1+i5) = 2,500,000*(1.0675)*(1.07)*(1.0725)*(1.0775)*(1.08)

= $ 3,563,936.89

So the repayment amount for 5 years is  $ 3,563,936.89

So the new interest rate r' is

A = P'(1+r)N => r = (A/P')1/N-1 = ( $ 3,563,936.89/$2,425,000)1/60 - 1

r = 0.0059 = 0.59%

EAR = (1+0.0059)12 - 1 = 0.0735 = 7.35%

Thus from the above, we can see that the option b, has the lowest borrowing cost as its interest rate is the least. Thus we should go with option b.

Option Loan amount to be repaid in 5 years Interest rate EAR a $3,387,253.54 0.51% 6.26% b $3,143,192.92 0.43% 5.33% c $3,563,936.89 0.59% 7.35%

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