Quiz 4: Nominal & Effective Interest Rates

Single amounts with PP ≥ CP

(PP) = Payment Period: length of time between cash flows. For problems involving single amounts, the payment period (PP) is usually longer than the compounding period (CP). For these problems, there are an infinite # of i & n combinations that can be used, with only two restrictions: 1. 'i' must be an effective interest rate 2. The time units on 'n' must be the same as those of 'i' (If 'i' is a rate per quarter, then 'n' is the # of quarters between P & F in the notation F = P(F/P, i, n). For these problems determine an effective interest rate and set n equal to the # of compounding periods between P & F. This usually gives me the % as a whole # which enables me to use the tables in the back of the book, otherwise I would have to use the factor formulas. Note: use r = im

Three ways to express interest rates

1. When no compounding period is given, rate is effective i = 12% per month i = 12% per year 2. When compounding period is given & it is not the same as interest period, it is nominal. i = 10% per year, comp'd semiannually i = 3% per quarter, comp'd monthly 3. When compounding period is given & rate is specified as effective, rate is effective over stated period.

Series with PP < CP

2 Policies: 1. Interperiod cash flows earn no interest. (Assuming no inter period compounding) Here, + cash flows are moved to beginning of the interest period in which they occur and - cash flows are moved to the end of the interest period. Note: This is the only instance in which the actual cash flow is changed 2. Inter period cash flows earn compound interest (Assume inter period compounding is earned) Cash flows are not moved and equivalent P, F, and A values are determined using the effective interest rate per payment period. Here m = CP/PP

Effective annual interest rates

Calculates an effective interest rate per year from any effective rate over a shorter time period. i_a = (1+i)^(m) -1 i_a = effective interest rate per year. i = effective interest rate per compounding period (r/m) m = compounding frequency

Effective Annual Interest rates for any time period

Determines the effective interest rate for any time period (shorter or longer than 1 year). Nominal rates can be converted into effective rates for any time period via the eq: i = (1+r/m)^m -1 i = effective interest rate for any time period r = nominal interest rate per year. m = compounding frequency Note: When using this equation you must find the nominal interest rate per year first. Use the relation: r = i*m Then apply the formula

Series with PP ≥ CP

For series cash flows (A, G = arithmetic, g= geometric). The first step is to determine relationship between PP & CP when PP ≥ CP, the only procedure that can be used is as follows: 1. Find effective 'i' PER PAYMENT PERIOD (PP) (if PP is in quarters, must find effective i/quarter). So determine 'i' as usual r = im then use the equation: i_r = (1+i/m)^m -1 Use this value in your factor 2. Determine n, set n equal to the # of compounding periods within the full period Generally n = CF * # of periods (Quarterly payments for 6 years yields n = 4*6 = 24 and plug it into our factor. Here there is 6 periods and quarterly payments have a frequency of 4 per period so hence : n = 4*6 = 24 Sometimes PP = CP, so in these cases solve for the series as you would as usual except you need to find an appropriate effective interest in order to plug it into your usual equations for those series and make sure your using the appropriate n. Note: use r = im Note: For these problems when you see BEGINNING NOW associated with any annual amount you +1 to the n of what you originally think it is.

Statements

Interest Period: Period of time over which interest is expressed. Ex: 1% per month Compounding period (CP): Shortest time unit over which interest is charged or earned. Ex: 10% per year compounded monthly Compounding Frequency (m): # of times compounding occurs within the interest period t. Ex: At i = 10% per year, compounded monthly, interest would be compounded 12 times during the one year interest period.

There terms 'nominal' and 'effective' enter into consideration when the interest period is less than 1 year.

Nominal Interest Rate (r): Is obtained by multiplying on interest rate that is expressed over a short time period by the # of compounding periods in a longer time period. That is: r = i * m r = nominal interest rate per year i = effective interest rate per compounding period m = compounding period

Varying Rates

When interest rates vary over time, use the interest rates associated with their respective time periods. General Equation when your asked to find the present worth: P = F_1(P/F, i_1, 1) + F_2(P/F, i_2, 1)(P/F,i_1,1) + F_3(P/F,i_3,1)(P/F,i_2,1)(P/F,i_1,1) To find the equivalent uniform series A just take the P found before equal to the RHS just replace all the F with A. Extra: For single amounts just multiply the factors as one, but make sure your using matching effective interest rates These are easier since your just calculating future or present value given the interest rates. Extra: For a series its helpful to draw the cash flow diagram and you need to convert those annual amounts to present values and if the interest changes you need to find the present value of that interest and then bring that value back to the present value similar to last chapter.

Continuous Compounding

When the interest period/compounding period is infinitely small, interest is compounded continuously. Therefor, PP > CP & m increases. Use this equation: i = e^r - 1 Note: Whatever your looking for is the unknown. (You already know). Just make sure that they both have the same periods and solve for the unknown.