Understanding Loan Amortization and Calculating EAR
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Loan Amortization and EAR: You want to buy a car, and a local bank will lend you $40,000. The loan will be fully amortized over 5 years (60 months), and the nominal interest rate will be 8% with interest paid monthly. What will be the monthly loan payment? What will be the loan’s EAR?
Using a financial calculator: N=60, I/YR=8/12=0.6667, PV=-40,000, FV=0. Solve for PMT: $811.06.
To calculate the EAR: EAR = (1 + nominal rate/m)^m - 1. In this case, EAR = (1 + 0.08/12)^12 - 1.0 = (1.00667)^12 - 1.0 = 8.30%.
Loan Amortization Example: Jan sold her house on December 31 and took a $10,000 mortgage as part of the payment. The 10-year mortgage has a 10% nominal interest rate, with semiannual payments beginning next June 30. Next year, Jan must report on Schedule B of her IRS Form 1040 the amount of interest included in the two payments she received during the year.
What is the dollar amount of each semiannual payment?
N = 10 years * 2 payments/year = 20. I/YR = 10%/2 = 5%. PV = -10,000, FV = 0. Solve for PMT: $802.43.
How much interest was included in the first payment? How much was repayment of principal? How do these values change for the second payment?
Here is an amortization schedule for the first two payments:
| Period | Beginning Balance | Payment | Interest | Principal Repayment | Ending Balance |
|---|---|---|---|---|---|
| 1 | $10,000.00 | $802.43 | $500.00 | $302.43 | $9,697.57 |
| 2 | $9,697.57 | $802.43 | $484.88 | $317.55 | $9,380.02 |
Because the mortgage balance declines with each payment, the portion of the payment applied to interest declines, while the portion applied to principal increases. The total payment remains constant over the life of the mortgage.
Jan must report $984.88 in interest income on Schedule B for the first year. Her interest income will decline in each successive year for the reason explained in part b.
Interest is calculated on the beginning balance for each period. This is the amount the lender has loaned and the borrower has borrowed. As the loan is amortized (paid off), the beginning balance, and hence the interest charge, declines, and the repayment of principal increases.
Expected Return Calculation: A stock’s returns have the following distribution based on demand for the company’s products:
| Demand | Probability | Rate of Return |
|---|---|---|
| Weak | 0.1 | (30%) |
| Below Average | 0.1 | 14% |
| Average | 0.3 | 11% |
| Above Average | 0.3 | 20% |
| Strong | 0.2 | 45% |
| Total | 1.0 |