Understanding Interest Rate, Present Value, and Future Value

2. The Interest Rate, Present Value, and Future Value • Time Value of Money: Money today is worth more than money in the future. • The interest rate should include: • Compensation for inflation: preserve purchasing power. • Compensation for default risk: the borrower might default on the loan. • Compensation for the opportunity cost of waiting to spend the money. • Simple interest is paid on principle only • FV = PV (1 + I*N) • Compounded interest is paid on both principle and interest (moves $ forward) • FV = PV (1 + I)N FVFN,I = (1 + I)N • Discounting (moves $ backward) • PV = FV (1 + I)-N PVFN,I = (1 + I)-N = 1 / FVFN,I

3. Comparing Investments • Suppose you are considering investing $1,000 in one of the following bank CDs: • CD A pays 4% interest rate per year for 3 years = 1,000 (1 + 0.04)3 = 1,124.86 • CD B pays 10% the 1st and 1% the 2nd and 3rd year = 1,000(1.1)(1.01)2 = 1,122.11 • How Do You Value a College Education? • These are the additional earnings of college over high school graduates by age: Age 22: $7,200; Age 23: $7,200; Age 24: $7,300; Age 25: $7,300 • What is PV of a college education from ages 22 to 25 at age 21 with 5% interest ? • Suppose you are 18 years old, explain how you calculate the present value of a college education in order to make a decision whether to take a job immediately after graduating from high school or to attend college and then work at age 22. • Find PV of a college education through the normal retirement at 67 versus • cost of college explicit (tuition, books) and implicit or opportunity (cannot work)

4. Loans, Bonds, and the Timing of Payments • Price of any asset (fin and real) should not exceed PV of future cash flows. • Four basic categories of debt instruments (assume 10% interest): • Simple loans: borrower agrees to repay the principal plus interest at maturity. • Borrow $10,000 and repay $10,000 + ($10,000 × 0.10) = $10,000(1.1) = $11,000. • Discount bonds: similar to simple loan with current price being discount of face.Borrow $9,901 = $10,000 / 1.1 and repay $10,000 • Coupon bonds: pays interest (coupon C = 11%) over life & principle (par or face value FV = $1,000) at maturity (N = 6 years). • P = ∑C(1+I)-n + FV(1+I)-N = PVAFN,I C + PVFN,I FV = [(1- PVFN,I)/I]C + PVFN,I FV or Fin Calculator: N = 6, I = 10, PMT = 110, FV = 1,000 solve for PV = -1,043.55 • 4. Fixed-payment loans: both principle and interest paid through life in equal PMTs. • Borrow $10,000 to buy car for 5 years with monthly payments • Loan = PMT * PVAFN,I (I in PVAF can only be solved by trial and error) • Fin Calculator: N = 5*12, I = 10/12, PV = 10,000 solve for PMT = 212.47 • Payment on student loans starts upon graduation but adds interest to loan. Extending maturity lowers each payment but increases total interest paid.

5. Yield to Maturity YTM is discount rate that equates PV of asset’s payments with asset’s current price. Current yield is the coupon as percentage of current price. Write the equation to find YTM for each of the following situations a) A simple loan or discount bond for $5,000 with $7,000 payment in 4 years: $7,000 = $5,000(1 + I)4 or Fin Calc: N = 4, PV = 5000, FV = -7000 solve for I = 8.78 b) A bond with 10% coupon, 5 years maturity selling for $975: Formula: Trial & error or Fin Calc: N = 5, PV = -975, PMT = 100, FV = 1000, I = 10.67 Coupon Rate >/=/< YTM => Price >/=/< Face => Premium/Par/Discount c) Perpetuity paying $25 coupon for price of $500: P = C / YTM => YTM = C / P = 25 / 500 = 0.05 or 5%. • A 25 years $40,000 mortgage with $2,000 annual payments: Formula: Trial & error or Fin Calc: N = 25, PV = 40000, PMT = -2000, I = 1.8

6. Banks Take a Bath on Mortgage-Backed Bonds • Banks reduce lending significantly during the financial crisis of 2007-2008. • For mortgage-backed securities, borrowers began to default on their payments, so buyers required much higher yields to compensate for more default risk. • By 2008, the prices of many mortgage-backed securities had declined by 50% or more. Higher yields on these securities meant lower prices. • By early 2009, U.S. commercial banks had suffered losses of about $1 trillion on their investments. Bond Prices and Yields to Maturity Move in Opposite Directions • Yields to maturity and bond prices move in opposite directions. • If interest rates on newly issued bonds rise, the prices of existing bonds will fall, and vice versa. • Reason: If interest rates rise, existing bonds issued with lower interest rates become less desirable to investors, and their prices fall. • This relationship should also hold for other debt instruments.

7. Reading the Bond Tables in the Wall Street Journal Treasury Bonds and Notes Price quoted as decimal % of par like corporate bonds. (Old quotes in x/32, with “:”.) Fin Intermediary buys at bid price and sells at asked price (to make $ buy low sell high), so customers buy from asked and sell to bid. Treasury Bills Bonds quote prices, T-bills quote discounts relative to Face. Bank discount rate = ((Face - Price)/Face)*(360/N) => P = F(1 – discount*N/360). NYSECorp Bonds Bond’s rating shows the likelihood that the firm will default on the bond.

8. Interest Rates and Rates of Return $ Return = Cash In – Cash Out and % Return = $ Return / Cash Out Bond % Ret = (Sell P + C – Buy P) / Buy P = C / Buy P + (Sell – Buy P) / Buy P Current Yield + Capital Gains Yield Both YTM and current yield do not include Capital Gains Yield and can mislead. Stock % Ret = (Sell P + D – Buy P) / Buy P = D / Buy P + (Sell – Buy P) / Buy P Dividend Yield + Capital Gains Yield Interest Rates Risk and Bond Price Price risk: if need to sell before maturity and YTM goes up (affects more bonds with longer maturity and lower coupon) Reinvestment Risk: if YTM goes down realized return lower than when bond was purchased (affects more bonds with longer maturity and higher coupon) Nominal Interest Rates Versus Real Interest Rates Fisher’s Effect: (1 + R) = (1 + r)(1 + π) R = r + π + r π ≈ r + π (for small π)

9. Will Investors Lose Their Shirts in the Market for Treasury Bonds? • Treasury bonds have little default risk as the U.S. government is almost certain to make payments on its bonds. • In September 2012, many financial advisors warned investors not to buy Treasury bonds due to their interest rate risk. • The Fed had responded to the weak U.S. economy by increasing the money supply that would lead to high inflation in the long run. • The expectation of high future inflation would lower the prices of bonds as a result of higher interest rates on those bonds.

10. TIPS as a Percentage of All Treasury Securities Figure 3.2 Since 1997, the U.S. Treasury has issued inflation indexed bonds called TIPS (Treasury Inflation Protection Securities). To protect bond holders from inflation par is increased by inflation rate at every coupon payment. TIPS were an increasing percentage of all U.S. Treasury securities until 2009.

11. Secondary Markets, Arbitrage, and the Law of One Price • A trader buys and sells securities to profit from small differences in prices. • During the period before bond prices fully adjust to changes in interest rates, there is an opportunity for arbitrage. • The prices of financial securities at any given moment allow little opportunity for arbitrage profits, so that investors receive the same yields on comparable securities. • This rationale follows the principle of the law of one price: identical products should sell for the same price everywhere. Financial arbitrageis the process of buying and selling securities to profit from price changes over a brief period of time.