RPN38 — More Examples
Interest Rate Conversions
When you are approaching a financial problem, it is important to recognize that the interest rate or rate of return can be described in at least three different ways:
Annual interest rates are typically stated in the United States as nominal rates. The periodic interest rate per compounding period equals the annual rate divided by the number of compounding periods per year. Occasionally, discount rates or required investment rates are stated as annual effective rates, while the cash flows occur more than once a year. To compare investments with differing compounding periods, the annual effective interest rate must be converted to the equivalent annual nominal interest rate. For example, if the periodic rate is ½% (per month), this periodic rate would usually be quoted as a nominal annual rate, which is 6% (½% x 12). This same periodic rate could be quoted as an effective annual rate, which considers compounding. The balance after 12 months of compounding would be $1,061.68, which means the effective annual interest rate is 6.168%. The Interest Conversion application (NOM, EFF, available as key shortcuts above the display) converts between nominal and effective annual interest rates. Example 1: Monthly Compounding, Quarterly Payments You are taking out a student loan for $3,000. The loan will be amortized over the first 8 years after graduation. The annual interest rate will be 6% compounded monthly, but you will make quarterly payments (at the end of each quarter). What will your payments be? (Make sure Top Bar Operations are enabled in Settings)
Example 2: Present Value of Cash Flows You are evaluating an investment that earns $800 at the end of each quarter for 6 years. Assuming you must earn a 10% annual effective interest rate, what is the present value of the cash flows? (Make sure Top Bar Operations are enabled in Settings)
interest rate? Solve this by calculating the yield with quarterly compounding and converting it to the equivalent annual effective interest rate.

Yield of a Discounted (or Premium) Mortgage
The annual yield of a mortgage bought at a discount or premium can be calculated given the original mortgage amount (PV), interest rate (I/YR), periodic payment (PMT), balloon payment amount (if any) (FV), and the price paid for the mortgage (new PV). Remember the cashflow sign convention: money paid out is negative; money received is positive. Example: Discounted Mortgage. An investor wishes to purchase a $100,000 mortgage taken out at 9% for 20 years. Since the mortgage was issued, 42 monthly payments have been made. The loan is to be paid in full (a balloon payment) at the end of its fifth year. What is the yield to the purchaser if the price of the mortgage is $79,000?


Automobile Loan
You are buying a new $14,000.00 sedan. Your down payment is $1,500 and you are going to finance the remaining $12,500. The car dealer is offering two choices for financing: 1. A 3year loan with an annual interest rate of 3.5%. 2. A 3year loan with an annual interest rate of 9.5% and a $1,000.00 rebate. With which of the above two choices do you pay less for the car?


Saving for College Costs
Suppose you want to start saving now to accommodate a future series of cash outflows. An example of this is saving money for college. To determine how much you need to save each period, you must know when you'll need the money, how much you'll need, and at what interest rate you can invest your deposits. Example: Your daughter will be going to college in 12 years and you are starting a fund for her education. She will need $15,000 at the beginning of each year for four years. The fund earns 9% annual interest, compounded monthly, and you plan to make monthly deposits, starting at the end of the current month. The deposits cease when she begins college. How much do you need to deposit each month? This problem needs to be solved in two steps. First calculate the PV of this cashflow diagram starting with an interest rate conversion because of the monthly compounding:

Keys:  Display:  Description: 
Set the mode switch to END.  
CHS FV_{}  52,713.28  Stores the amount you will need. 
0 PV  0.00  Stores the amount you are starting with. 
12 g 12×  144.00  Stores the number of deposits. 
9 g 12÷  0.75  Stores the interest rate per compounding period. 
PMT  204.54  Calculates the monthly payment required now to meet the future demands of college. 
Value of a Taxable Retirement Account
This problem uses the TVM application to calculate the future value of a taxable retirement account that receives regular, annual payments beginning today (Begin mode). The annual tax on the interest is paid out of the account. (Assume the deposits have been taxed already.) Example: Taxable Retirement Account. If you invest $3,000 each year for 35 years, with dividends taxed as ordinary income, how much will you have in the account at retirement? Assume an annual dividend rate of 8.175% and a tax rate of 28%, and that payments begin today. What will be the purchasing power of that amount in today's dollars, assuming 8% inflation?

