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Chapter 4

Chapter 4. Future Value, Present Value and Interest Rates. Future Value, Present Value and Interest Rates: The Big Questions. How can we compare payments at different dates? What is an interest rate? What is a bond? What is the relationship between interest rates and inflation?.

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Chapter 4

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  1. Chapter 4 Future Value, Present Value and Interest Rates

  2. Future Value, Present Value and Interest Rates:The Big Questions • How can we compare payments at different dates? • What is an interest rate? • What is a bond? • What is the relationship between interest rates and inflation?

  3. Future Value, Present Value and Interest Rates:Roadmap • Future Value • Present Value • Internal Rate of Return • Bond Basics • Real vs. Nominal Interest Rates

  4. A Brief History of Lending • Lenders despised throughout history. • Credit is so basic that we evidence of loans going back 5 thousand years. • Hard to imagine an economy without it. • Yet, people still take a dim view of lenders because they charge interest

  5. Lending and Interest • Why do lenders charge interest? • The existence of alternatives means that lenders face an opportunity cost. • Borrowers rent resources from lenders. Interest is the rent.

  6. Valuing Monetary Payments Now and in the Future • Fundamental to studying financial instruments is the ability to value payments made at different times. • Tools:Future value and Present Value

  7. Future Value:Definition The value on a future date of an investment made today. If you invest $100 today at 5 percent interest per year, in one year you will have $105.

  8. Future Value:One Year Future Value = Present Value + Interest FV = PV + PVxi $105 = $100 + $100x(0.05)

  9. Future Value:One Year FV = PV + PVxi = PVx(1+i) Future Value in one year =Present Value x (one plus interest rate)

  10. Future Value:Two Years $100+$100(0.05)+$100(0.05) + $5(0.05) =$110.25 Present Value of the Initial Investment + Interest on the initial investment in the 1st Yr + Interest on the initial investment in the 2nd Yr+ Interest on the Interest from the 1st Yr in the 2nd Yr = Future Value in Two Years

  11. Future Value: General Formula Future value of an investment of PV in n years at interest rate i FVn = PVx(1+i)n (Remember: The interest rate is measured is a decimal so if 5%, i = .05)

  12. Future Value:$100 Investment at 5% Annual Interest After 10 years, $100 as grown to $162.89 – that’s the initial investment of $100 plus interest of $62.89. Ignoring compounding, you would have just multiplied 5 percent times 10 years and gotten $50. The difference of $12.69 comes from compounding.

  13. Future Value: Caution Time (n) & interest rate (i) must be in same time units If i is at annual rate, then n must be in years.Future Value of $100 in 18 months at 5% annual interest rate is FV = 100 x (1+.05)1.5

  14. Invest $100 at 5% annual interest • How until you have $200? • The Rule of 72: • Divide the annual interest rate into 72 • So 72/5=14.4 years. • 1.0514.4 = 2.02

  15. Present Value: Definition Present Value (PV) is the value today (in the present) of a payment that is promised to be made in the future. • At a 5 percent interest rate, the present value of $105 one year from now is $100. • Reverses the future value calculation

  16. Present Value:One Year Solve the Future Value Formula for PV: FV = PV x (1+i) so Present Value = Future Value divided by one plus interest rate

  17. Present Value:One Year Example $100 received in one year, i=5% Note: FV = PVx(1+i) = $95.24x(1.05) = $100

  18. Present Value:General Formula Present Value of payment received n years in the future:

  19. Present Value:Example Present Value of $100 received in 2½ yrs at interest rate of 8%. Note: FV = PVx(1+i)n=$82.50x (1.08)2.5 = $100

  20. Present Value:Important Properties Present Value is higher: 1. The higher future value of the payment.(FV bigger) 2. The shorter time period until payment.(n smaller) 3. The lower the interest rate. (i smaller)

  21. Present Value:$100 at 5% interest rate Note rate of decline of Present Value. At a 5% interest rate, a $100 payment made in 14.4 years has a PV=$50.

  22. Present Valueof $100 Payment • As the interest rate rises from 1% to 5%, a payment due • 1 year falls by $3.77 • 10 years falls by $29.14

  23. Divine law of Islamic religion (Shari’a) forbids paying interest • Banks developed alternatives. • Liabilities • Deposit accounts: No interest • Investment accounts: Share in bank’s profits or losses • Assets • Profit share in exchange for loan

  24. Investment grows 0.5% per month • What is the compound annual rate?FVn=PV(1+i)n = 100x(1.005)12=106.17 Compound annual rate = 6.17% (Note: 6.17 > 12x0.05=6.0)

  25. To decide you need to compare • The value of the extra savings you will accumulate from waiting that allows you to purchase a more expensive care • The value of having the new care sooner.

  26. Internal Rate of Return:Definition The interest rate that equates the present value of an investment with its cost.

  27. Internal Rate of Return:Example You run a sports equipment factory. Should you purchase new tennis racquet machine? • Cost: $1 million • Produces 3000 racquets per year • Sell racquets for $50 apiece • The machine lasts 10 years and collapses with no resale value. • Should you buy the machine?

  28. Internal Rate of Return:Example • Balance the cost of the machine against the revenue • $1 million today vs. $150,000 a year for ten years. • Is the $150,000 revenue enough to make payments on a $1 million loan?

  29. Internal Rate of Return:Example Solve for i:$1,000,000 Solving for i, i=.0814 or 8.14%

  30. Can you retire when you’re 40? • Assume • Live to 85 • Interest rate = 4% • Want to have $100,000 per year • You will need

  31. Bond Basics • Bond: A promise to make a series of payments on specific future date • Bond Price = Present Value of payments

  32. Coupon Bond $1000 Face Value50-yr, 3½% coupon bond issued on May 1, 1945. Coupons

  33. Coupon Bond • A type of loan: • Interest paid during the life of the loan • Loan repaid at maturity • Coupon Rate: the annual interest the borrower pays (ic) • Maturity Date: when the payments stop and the loan is repaid (n) • Principal: the final payment (F)

  34. Coupon Bond:Valuing the Principal Present value of Bond Principal = Payment divided by one plus the interest rate raised to n

  35. Coupon Bond:Valuing the Coupon Payments Value of Coupon Payments = Present value of the sequence Note that C= icx F

  36. Price of Coupon Bond:Principal + Coupons Price of Coupon Bond (PCB) = Present value of Coupon Payments (PCP) + Present Value of the Principal (PBP)

  37. Bond Pricing:Important Property The price of a bond (PCB) and the interest rate (i) are inversely related: i  PCB 

  38. Credit cards are useful. • But lenders charge high interest rates. • Pay off your balance as fast as you can.

  39. Real and Nominal Interest Rates • Borrowers care about the resources required to repay. • Lenders care about the purchasing power of the payments they received. • Neither cares solely about the number of dollars, they care about what the dollars buy.

  40. Real and Nominal Interest Rates Nominal Interest Rates (i) Interest Rates expressed in current dollar terms. Real Interest Rates (r) Nominal Interest Rate adjusted for inflation.

  41. Real and Nominal Interest Rates Nominal interest rate = Real Interest Rate + Expected Inflation i = r + e (This is called the “Fisher Equation”)

  42. Nominal Interest Rate, Inflation Rate and Real Interest Rate Nominal Interest Rate = Real Interest Rate + Expected Inflation

  43. Real and Nominal Interest Rates Countries with high nominal interest rates have high inflation:   i

  44. Chapter 4 End of Chapter

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