Paying to the bank problem

[tuanl]

(Problem 4.37) On September 1, 1998, Susan Chao bought a motorcycle for $10,000. She paid $1,000 down and financed the balance with a five-year loan at a stated annual interest rate of 9.6 percent, compounded monthly. She started the monthly payment exactly one month after the purchase, i.e., October, 1998. In the middle of October, 2000, she got a new job and decided to pay off the loan. If the bank charges her 1 percent prepayment penalty based on the loan balance, how much should she pay the bank on November 1, 2000?
This is a problem from Steven Ross's wonderful book(Corporate finance). I'm not sure about my interpretation and solution, so I hope someone can check it for me

My solution: Since she paid $1000 right away, she owes 10000-1000=$9000 on Sep 1,1998. Now, the future value of this loan on October 1,2000= FV(0.096/12,25,0,-9000)= $10983.88 (n=25 because it's from September 1,1998 to October 1, 2000). In addition, monthly payment= PMT(0.096/12,60,-9000)= $189.46. Thus, the future value of Chao's payment on October 1, 2000= FV(0.096/12,24,-189.46)=$4990.89.
The remaining loan (on October 1,2000) =10983.88 - 4990.89=$5992.99. Now, since the bank charges her 1 percent prepayment penalty when she pays off the remaining loan on mid. October 2000, she will owes 5992.99*0.01=$59.9299. Thus, on November 1,2000, she needs to pay 59.9299*(1+0.096/24)= $60.16972

 

[LochWulf]

You need to use 25 periods in the FV-of-annuity formula, since the Oct 1 loan payment is the 25th of the contract. The fact that there are only 24 compounding periods involved (i.e., the Oct 1 contribution to the hypothetical annuity compounds for zero periods, since the formula returns the annuity's value immediately following the final contribution) is already baked into the formula.

With that change, you have that the outstanding loan balance immediately following the Oct 1 loan payment is 5,763.54 rather than 5,992.99 (quickly verified with an amortization schedule).

To pay off early, then, li'l Suzie "Chopper" Chao would owe 5,763.54 * 1.01 + per diem interest on the 5,763.54; the latter amount depending on exactly when she motors down to the bank. You've interpreted that to be in mid-Oct, when she landed the new gig.

But since the question assumes that something remains to be paid on Nov 1, I think a reasonable alternate interpretation is that in mid-Oct, Suzie simply arrives at the prepay decision, with the actual payoff occurring on Nov 1 in lieu of her next regularly-scheduled loan payment. On Nov 1 she pays the outstanding bal, the 1% thereof, and accrued interest for the month of November.

 

[tuanl]

You need to use 25 periods in the FV-of-annuity formula, since the Oct 1 loan payment is the 25th of the contract. The fact that there are only 24 compounding periods involved (i.e., the Oct 1 contribution to the hypothetical annuity compounds for zero periods, since the formula returns the annuity's value immediately following the final contribution) is already baked into the formula.

With that change, you have that the outstanding loan balance immediately following the Oct 1 loan payment is 5,763.54 rather than 5,992.99 (quickly verified with an amortization schedule).

But since the question assumes that something remains to be paid on Nov 1, I think a reasonable alternate interpretation is that in mid-Oct, Suzie simply arrives at the prepay decision, with the actual payoff occurring on Nov 1 in lieu of her next regularly-scheduled loan payment. On Nov 1 she pays the outstanding bal, the 1% thereof, and accrued interest for the month of November.

Thanks for your correctness! I was too careless( I think your interpretation is reasonable.