Nominal interest rate or annual percentage rate (APR) r = the nominal interest rate per year M = the compounding frequency or the number of interest periods per year r/M = interest rate per compounding period Effective interest rate = the rate that truly represents the amount of interest earned in a year or some other time period
i a = (1 + r/M) M – 1 i a = effective annual interest rate
Example If a savings bank pays 1 ½% interest every three months, what are the nominal and effective interest rates per year, Nominal %/year, r = 1 1/2% x 4 = 6% Effective interest rate per year, i a = ( /4) 4 –1 = = 6.1% Notice that when M=1, i a = r
Example A loan shark lends money on the following conditions, Gives you $50 on Monday, you owe $60 the following Monday Calculate nominal interest rate, r, ? Calculate effective interest rate, i a ? If the loan shark started with $50, and stayed in business for one year, how much money would he have in one year?
Example F=P(F/P,i,n) 60=50(F/P,i,1) (F/P,i,1)= 1.2, Therefore, i = 20% per week Nominal interest rate per year = 52 weeks x 0.20 = 10.40, 1040% = r Effective interest rate per year i a = ( /52) 52 –1 = 13,104 = 1,310,400% F = P(1+i) n = 50(1+0.2) 52 = $655,200
Effective interest rate Who said crime doesnt pay? To calculate the effective interest rate for any time duration we have the equation, i a = (1 + r/M) C – 1 i a = (1 + r/CK) C – 1
where M = number of interest periods per year (ie quarterly compounding, M = 4; monthly compounding, M = 12) C = number of interest periods per payment period K = number of payment periods per year (ie weekly payments, K = 52, monthly payments K = 12)
Effective Interest Notice that M = CK or M/K = C Simple case – compounding and payment are the same
Example Borrow $10,000 at yearly nominal rate of 9%. Compounding monthly, payment monthly. You pay on the loan for 6 years. What is your monthly payment? M = 12 (monthly payments), r/M = 0.09/12 = per month, n = 12 months * 6 years = 72 A = P(A/P, i, N) = 10,000 (A/P, , 72) = $180/ month
Example Just using equivalence here. Note that you are really paying. (1.0075) = 9.38% and not really 9% as stated.
Harder - cases when compounding and payment occur at different time periods. Must convert one to the same time period.
Example Invest at yearly nominal of 9%. Compounding monthly, payment quarterly. You will invest for 8 years. If you want to have a fund of $100,000 at the end of the 8 years, how much do you have to invest in each quarter?
Solution M = 12 (monthly compound), K = 4 (quarterly payments). Since we compound more frequently than we pay, we use the CK method. C = number of compound periods per payment period = 3. i per = [1 + r / (CK)] C - 1 = [1 +.09/12] =.022
Solution N = 4 * 8 years = 32 payments. A = F (A/F, i, N) = 100,000 (A/F,.0227, 32) = 2160
Example Invest at yearly nominal 12%. Compounding semi annually, payment quarterly and you will invest for 10 years. If you invest $12,000 per quarter, how much will you have at the end of the 10 th year?
Solution M = 2 (semi-annual), K = 4 (quarterly payments). Two alternate approaches for compounding less frequently that payment. (1)Bank gives us interest on the dollars invested from the point of investment, we use the CK method. This transforms the compound period to the payment period!
Solution Here C = number of compound periods per payment period = ½ i per = [1 + r / (CK)] C - 1 = [ /2] 1/2 - 1 =.0296 compute N = 10 years * 4 payments per year = 40 payments. F = A (F/A, i, N) = 12,000 (F/A, , 40) = 896,654
Solution (2) (2)In the case where the bank does not give interest on middle of period deposits we use the lumping method. Lump all payments in an interest period at the end of the interest period. 2 payments in each semi-annual interest period. Payment is now $24,000 semi-annually. This transforms the payment period to the compound period!
Solution (2) Now, use the r/M formula. r/M = 0.12/2 =.06. N = 10 years * 2 = 20 payments. F = A (F | A, i, N) = 24,000 (F/A,.06, 20) = 882,854 Note that the bank's strategy in the second case has cost you about $14,000!!
Continuous Compounding As an incentive in investment, some institutions offer frequent compounding. Continuous Compounding – as M approaches infinity and r/M approaches zero
When K = 1, to find the effective annual interest of continuous compounding i a = e r – 1
Example $2000 deposited in a bank that pays 5% nominal interest, compounded continuously, how much in two years? i a = e 0.05 – 1 = 5.127% F = 2000( ) 2 = 2210
Now when compounding and payment periods coincide 1. Identify number of compounding periods (M) per year 2. Compute effective interest rate per payment period, i = r/M 3. Determine number of compounding periods, N = M x (number of years)
When compounding and payment periods dont coincide, they must be made uniform before equivalent analysis can continue. 1. Identify M, K, and C. 2. Compute effective interest rate per payment period For discrete compounding, i = (1 + r/M) C – 1 For continuous compounding, i = e r/K - 1
Equivalence 3. Find total number of payment periods, N = K x (number of years) 4. Use i and N with the appropriate interest formula
Example Equal quarterly deposits of $1000, with r = 12% compounded weekly, find the balance after five years M = 52 compounding periods/year K = 4 payment periods per year C = 13 interest periods/payment period
Example i = (1 +.12/52) 13 – 1 =3.042% per quarter N = K x (5) = 4 x 5 = 20 F = A(F/A, 3.042%,20) = $26,985
Example You are deciding whether to invest $20,000 into your home at 6.5% continuously compounding, or the same amount into a CD compounded semi-annually at 7%, which is the wiser investment, assume 10 years?
Home Investment r = 6.5% K = 1 i a = e r/K – 1 = e –1 = 6.7% F = 20,000( ) 10 = $38,254
CD Investment r = 7% M = 2 i a = (1 + r/M) M – 1 = (1 + 7%/2) 2 – 1 = 7.12% F = 20,000( ) 10 = $39,787