RPN-38 — 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)

Keys: Display: Description:
6 NOM 12 n 12.00 Stores the known nominal percentage rate and the compounding periods.
EFF 6.17 Calculates the effective annual rate on the loan.
4 n 4.00 You wish to know the interest rate that will yield the same effective rate with quarterly compounding .
NOM 6.03 Calculates the equivalent rate for quarterly compounding.
4 ÷ i 1.51 Stores this newly converted rate as the quarterly interest rate .
(Slide switch)   Set mode switch to M.DY/END.
3000 PV 0 FV 0.00 Stores the known values for the student loan.
8 ENTER 4 × n 32.00 Stores the number of compounding periods.
PMT -118.87 Calculates your quarterly payment.

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)

Keys: Display: Description:
(Slide switch)   Set mode switch to M.DY/END.
10 EFF 10.00 Stores annual effective interest rate.
4 n 4.00 Stores number of compounding periods per year for nominal rate.
NOM 9.65 Calculates annual nominal rate with quarterly compounding.
4 ÷ i 2.41 Stores quarterly nominal rate.
24 n 24 Enter number of quarterly payments.
800 PMT 800.00 Stores quarterly cash flow.
0 FV 0.00 Clears FV register.
PV -14,449.09 Calculates present value.
If you can purchase the investment for $13,000, what is the yield expressed as an annual effective
interest rate? Solve this by calculating the yield with quarterly compounding and converting it to the
equivalent annual effective interest rate.
13000 CHS PV -13,000.00 Stores purchase price of investment.
i 4 × 13.55 Calculates IRR% as a yearly nominal rate with quarterly compounding.
NOM 4 n EFF 14.25 Enters nominal rate and calculates annual effective rate of return.



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 cash-flow 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?

Keys: Display: Description:
Step 1   Set mode switch to END.
9 g 12÷ 0.75 Interest per compounding period (month).
20 g 12× 240.00 Stores the number of months.
100000 CHS PV -100,000.00 Stores the original amount of the mortgage.
0 FV 0.00 The loan will be completely paid off in 20 years.
PMT 899.73 Calculates the regular payment.
Step 2    
f RND PMT 899.73 Rounds the payment to 2 decimal places for accuracy.
5 g 12× 60.00 Stores the number of months to the balloon.
FV 88,706.74 Calculates the balloon payment.
Step 3    
RCL n 42 - n 18.00 Stores remaining number of payments.
79000 CHS PV -79,000.00 Stores the price of the mortgage.
i 1.73 Calculates the monthly return on this discounted mortgage.
RCL g 12÷ 20.72 Retrieves the yearly return.


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 3-year loan with an annual interest rate of 3.5%.
2. A 3-year 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?

Keys: Display: Description:
    Set mode switch to END.
36 n 12500 PV 0 FV 0.00 Store the known values.
3.5 g 12÷ 0.29 Interest per month.
PMT -366.28 Calculates payment for the first financing option.
RCL n × -13,185.94 Calculates the total amount you will pay in interest and principal for the first option.
11500 PV 11,500.00 Stores the loan amount with the rebate.
9.5 g 12÷ 0.79 Stores the second interest rate.
PMT -368.38 Calculates the payment for the second option.
RCL n × -13,261.64 Calculates the total amount you will pay in interest and principal for the second option.


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.

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 cash-flow diagram starting with an interest rate conversion because of the monthly compounding:

Keys: Display: Description:
    Set mode switch to BEGIN.
9 NOM 9.00 Stores the nominal annual rate.
12 n 12.00 Stores the number of compounding periods used with this nominal rate.
EFF 9.38 Calculates the effective annual rate.
i 9.38 Stores the effective rate as the annual rate.
15000 PMT 15,000.00 Stores the annual withdrawal.
4 n 4.00 Stores number of withdrawals.
0 FV 0.00 Stores the balance at the end of four years.
PV -52,713.28 Calculates the amount required when your daughter starts college.

Then use that PV as the FV on the following cash-flow diagram, and calculate the PMT.

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?

Keys: Display: Description:
    Set mode switch to BEGIN.
35 n 35.00 Stores the number of payment periods until retirement.
8.175 ENTER 1 ENTER .28 - × 5.89 Calculates the interest rate diminished by the tax rate.
i 5.89 Stores the interest rate.
0 PV 0.00 Stores the amount you are starting with.
FV 345,505.61 Calculates the amount in the account at retirement.
8 i 0 PMT PV -23,368.11 Calculates present value purchasing power of the above FV assuming an 8% inflation rate.

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