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Lecture 6

Lecture 6. Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test. Review. Review Problems 2.28, 2.30(b) (reviewed other problems, took the entire class). 2.28 Compute P for:. Problem 2.30. Lecture 7. Due Tuesday Read Chapter 3 115-136 Problems 3.1, 3.2, 3.5, 3.6, 3.7.

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Lecture 6

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  1. Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.

  2. Review • Review Problems 2.28, 2.30(b) (reviewed other problems, took the entire class)

  3. 2.28 Compute P for:

  4. Problem 2.30

  5. Lecture 7 Due Tuesday Read Chapter 3 115-136 Problems 3.1, 3.2, 3.5, 3.6, 3.7

  6. Chapter 3

  7. 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

  8. ia = (1 + r/M)M – 1 • ia = effective annual interest rate

  9. 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, • ia = ( 1 + 0.06/4)4 –1 = 0.061 = 6.1%  • Notice that when M=1, ia = r

  10. 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, ia? • If the loan shark started with $50, and stayed in business for one year, how much money would he have in one year?

  11. 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 ia = ( 1+ 10.40/52)52 –1 = 13,104 • = 1,310,400% • F = P(1+i)n = 50(1+0.2)52 = $655,200

  12. Effective interest rate • Who said crime doesn’t pay? • To calculate the effective interest rate for any time duration we have the equation, • ia = (1 + r/M)C – 1 • ia = (1 + r/CK)C – 1

  13. 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)

  14. Effective Interest • Notice that M = CK or M/K = C • Simple case – compounding and payment are the same

  15. 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 = 0.0075 per month, n = 12 months * 6 years = 72 • A = P(A/P, i, N) = 10,000 (A/P, 0.0075, 72) = $180/ month

  16. Example • Just using equivalence here. • Note that you are really paying. • (1.0075)12 - 1 = 9.38% and not really 9% as stated.

  17. Harder - cases when compounding and payment occur at different time periods. • Must convert one to the same time period.

  18. 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?

  19. 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. • iper = [1 + r / (CK)]C - 1 = • [1 + .09/12]3 - 1 = .022

  20. Solution • N = 4 * 8 years = 32 payments. • A = F (A/F, i, N) • = 100,000 (A/F, .0227, 32) • = 2160

  21. 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 10th year?

  22. 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!

  23. Solution • Here C = number of compound periods per payment period = ½ • iper = [1 + r / (CK)]C - 1 = [1 + 0.12/2]1/2 - 1 = .0296 • compute N = 10 years * 4 payments per year = 40 payments. • F = A (F/A, i, N) = 12,000 (F/A, 0.0296, 40) = 896,654

  24. 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!

  25. 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!!

  26. Continuous Compounding • As an incentive in investment, some institutions offer frequent compounding.Continuous Compounding – as M approaches infinity and r/M approaches zero

  27. Continuous Compounding

  28. Continuous Compounding • When K = 1, to find the effective annual interest of continuous compounding • ia = er – 1

  29. Example • $2000 deposited in a bank that pays 5% nominal interest, compounded continuously, how much in two years? • ia = e0.05 – 1 = 5.127% • F = 2000(1 +0.05127)2 = 2210

  30. Now when compounding and payment periods coincide • Identify number of compounding periods (M) per year • Compute effective interest rate per payment period, i = r/M • Determine number of compounding periods, N = M x (number of years)

  31. When compounding and payment periods don’t coincide, they must be made uniform before equivalent analysis can continue. • Identify M, K, and C. • Compute effective interest rate per payment period For discrete compounding, i = (1 + r/M)C – 1 For continuous compounding, i = er/K - 1

  32. Equivalence • Find total number of payment periods, N = K x (number of years) • Use i and N with the appropriate interest formula

  33. 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

  34. 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

  35. 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?

  36. Home Investment r = 6.5% • K = 1 • ia = er/K – 1 = e0.065 –1 = 6.7% • F = 20,000(1+0.067)10 = $38,254

  37. CD Investment • r = 7% • M = 2 • ia = (1 + r/M)M – 1 • = (1 + 7%/2)2 – 1 = 7.12% • F = 20,000(1+0.0712)10 = $39,787

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