Question

4 Balloons You are an important local real estate investor; you just got a $10,000,000 balloon loan to buy a new office building in your home town. The nominal maturity of the loan is 30 years, but the loan has a 10-year balloon payment. In other words, the loan will end at the end of the 10th year, and the outstanding balance will be paid off in a lump sum at that time. The interest on the loan is 6.7% per annum, compounded monthly. 4.1. Assume that the loan has fixed payments. (a) What is the monthly fixed payment that you have to make? (b) Let PV be the present value at time 0 of the fixed payments made over the 10-year life of the loan. What is PV? (c) Can you find a connection between the $10,000,000 principal, PV, and the balloon payment at the end of year 10? Compute, using this connection, the size of the balloon payment at the end of year 10. If you cannot find the connection, a more work-intensive approach is to use an amortization table to compute the answer (use Excel!). 4.2. Assume now that this loan has variable payments. At the end of each month fixed, equal portions of the loan's principal are paid down, such that the principal would be fully amortized at the end of the hypothetical 30-year loan period. What is the size of the balloon payment at the end of year 10 in this case?

Answer 1

4.1 a) Interest rate per month (r) = 6.7%/12 = 0.55833% =0.0055833

Now, let A be the monthly fixed payment , so n = 30*12 = 360 payments in 360 months

From the present value of annuity formula

Present value = A/r * (1- (1/(1+r)^n))

=> 10000000 = A/0.0055833 * (1- (1/1.0055833)^360)

=> 154.9720 *A = 10000000

=> A = $64527.80

So, the monthly fixed payment is $64527.80

b) PV=present value of fixed payments during 10 year period (n=10*12 =120)

= A/r* (1- (1/(1+r)^n))

= 64527.80 * (1-(1/1.0055833)^120)

PV = $5,632,259.40

c) The $10 million principal, PV and balloon payment (X) are related as follows :

Present value of the Balloon payment is equal to the difference in the $10 million principal and  PV

i.e. Present value of the Balloon payment = $10,000,000-$5,632,259.40

=> X/ (1+0.0055833)^120 = 4367740.60

=> X= $ 8,519,710.55

So, the Balloon payment at the end of 10 years is equal to $ 8,519,710.55

4.2 By the end of 10 years, 1/3rd of the principal amount would have been repaid along with accrued interest. So, loan amount remaining at the end of 10th year

=$ 10,000,000 - $10,000,000/3

=$6,666,666.67

If this amount is paid at the end of 10th year, the loan is repaid immediately

Hence, the balloon payment in this case is $6,666,666.67