Partial Amortization, or the “Bullet” Loan As we explained earlier, real estate lending usually involves mortgages with a loan period far shorter than the mortgage life. A common example might call for a five-year loan with, say, a 15-year amortization. This means the borrower makes a payment every

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Partial Amortization, or the “Bullet” Loan

As we explained earlier, real estate lending usually involves mortgages with a loan period far shorter than the mortgage life. A common example might call for a five-year loan with, say, a 15-year amortization. This means the borrower makes a payment every month of a fixed amount based on a 15-year amortization. However, after 60 months, the borrower either negotiates a new five-year loan or makes a single, much larger payment called a balloon or bullet to pay off the loan. Balloon payments are common in both commercial and residential mortgages. In either case, because the monthly payments don’t fully pay off the loan, the loan is said to be partially amortized.
Suppose we have a $100,000 commercial mortgage with a 5 percent rate compounded semiannually and a 20-year (240-month) amortization. Further suppose that the mortgage has a five-year balloon. What will the monthly payment be? How big will the balloon payment be?

Accepted Answer

The monthly payment can be calculated based on an ordinary annuity with a present value of $100,000. To find the monthly rate, we first have to find the EAR and then convert it to a quoted monthly rate. To do this, we convert the quoted semiannual rate to an EAR.

EAR = [latex][1 + Quoted rate/m]^{m}[/latex] - 1

= [latex][1 + .05/2]^{2}[/latex] - 1

= [latex]1.0250^{2}[/latex] - 1

= 5.06%

Then, we find the quoted monthly rate used to calculate the payments:

Quoted rate/m = [latex](EAR + 1)^{1/}[/latex]m - 1

Quoted rate/12 = [latex](1.0506)^{1/12 }[/latex]- 1

= 1.0041 - 1 = 0.41%

The quoted monthly rate is 0.41 percent and there are 12 × 20 = 240 payments. To find the payment amount, we use the annuity present value formula.

Annuity present value = $100,000

= C × (1 - Present value factor)/r

$100,000 = C × (1 - 1/[latex]1.0041^{240}[/latex])/.0041

= C × (1 - .3746)/.0041

= C × 152.5366

C = $655.58

Your monthly payments will be $655.58
Now, there is an easy way and a hard way to determine the balloon payment. The hard way is to actually amortize the loan for 60 months to see what the balance is at that time. The easy way is to recognize that after 60 months, we have a 240 - 60 = 180-month loan. The payment is still $655.58 per month, and the interest rate is still .41 percent per month. The loan balance is thus the present value of the remaining payments:

Loan balance = $655.58 × (1 - 1/[latex]1.0041^{180}[/latex])/.0041

= $655.58 × 127.1220

= $83,338.64

The balloon payment is a substantial $83,339. Why is it so large? To get an idea, consider the first payment on the mortgage. The interest in the first month is $100,000 × .0041 = $410. Your payment is $655.58, so the loan balance declines by only $245.58. Since the loan balance declines so slowly, the cumulative pay down over five years is not great[latex].^{7}[/latex]

[latex]^{7}[/latex] To get the precise payment of $657.13 you need to carry 6 decimal places in the interest rate using 0.412392 percent.

Question 6.18: Partial Amortization, or the “Bullet” Loan As we explained e...

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