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  2. Time Value of Money • A dollar today is worth more than a dollar received at some future date. • Money may be spent on consumption or saved by investing in real capital assets (machinery) or by buying financial assets (deposits or stock). • Investing means giving up consumption.

  3. Time Value of Money (concluded) • With a positive time preference for consumption, investment means giving up consumption (opportunity cost). • The opportunity cost of giving up consumption is known as the time value of money. It is the minimum rate of return required on a risk-free investment.

  4. Future Value or Compound Value • The future value (FV) of a sum (PV) is FV = PV (1+i)n. • (1+i)n is referred to as the Future Value Interest Factor. • Multiply by the dollar amount involved to calculate the FV of an investment. • Interest factor formulas are included in financial calculators.

  5. Present Value • The value today (at present) of a sum received at a future date discounted at the required rate of return. • Given the time value of money, one is indifferent between the present value today or the future value received in the future.

  6. Present Value (concluded) • With risk present, a premium return may be added to the risk-free time value of money. The higher the risk or higher the interest rate, the lower the present value.

  7. Valuing a Financial Asset • There are two necessary ingredients for valuing financial assets. • Estimates of future cash flows. • The estimates include the timing and size of each cash flow. • An appropriate discount rate. • The discount rate must reflect the risk of the asset.

  8. The Mechanics of Bond Pricing • A fixed-rate bond is a contract detailing the par value, the coupon rate, and maturity date. • The coupon rate is close to the market rate of interest on similar bonds at the time of issuance. • In a fixed-rate bond, the interest income remains fixed throughout the term (to maturity).

  9. The Mechanics of Bond Pricing (concluded) • The value of a bond is the present value of future contractual cash flows discounted at the market rate of interest. • Ci is the coupon payment and Fn is the face value of the bond. • Cash flows are assumed to flow at the end of the period and are assumed to be reinvested at i. Bonds typically pay interest semiannually. • Increasing i decreases the price of the bond (PB).

  10. Basic Bond Pricing Formula • The stream of coupon payments on a fixed rate bond is an annuity which allows the pricing of a bond with the following formula:

  11. Pricing Zero Coupon Bonds • Bonds that do not pay periodic interest payments are called zero-coupon bonds. • Zero coupon bonds trade at a discount. • The value of the "zero" bond is • There is no reinvestment of coupon payments with zeros and thus, no reinvestment risk. The yield to maturity, i, is the actual yield received if held to maturity.

  12. Bond Yields • Bond yields are related to several risks. • Credit or default risk is the chance that some part or all of the interest or principal payments will be delayed or not paid. • Reinvestment risk is the potential variability of market interest rates affecting the reinvestment rate of the periodic interest received resulting in an actual, realized rate different from the expected yield to maturity. • Price risk relates to the potential variability of the market price of the bond caused by a change in market interest rates.

  13. Bond Yields (continued) • Bond yields are market rates of return which equate the market price of the bond with the discounted expected cash flows of the bond. • A bond yield measure should reflect all three cash flows from the bond and their timing: • Coupon payments. • Interest income from reinvestment of coupon interest. • Any capital gain or loss.

  14. Bond Yields (continued) • The yield to maturity is the investor's expected or promised yield if the bond is held to maturity and the cash flows are reinvested at the yield to maturity. • Bond yields-to-maturity vary inversely with bond prices. • If the market price of the bond increases, i, or the yield to maturity declines.

  15. Bond Yields (continued) • If the market price of the bond decreases, the yield to maturity increases. • When the bond is selling at par, the coupon rate approximates the market rate of interest. • Bond prices above par are priced at a premium; below par, at a discount.

  16. Bond Yields (continued) • The realized yield is the ex-post, actual rate of return, given the cash flows actually received and their timing. Realized yields may differ from the promised yield to maturity due to: • A change in the amount and timing of the promised cash flows. • A change in market interest rates since the purchase of the bond, thus affecting the reinvestment rate of the coupons. • The bond may be sold before maturity at a market price varying from par.

  17. Bond Yields (concluded) • The expected, ex-ante yield, assuming a realized price and future interest rate levels, are forecasted rates of return.

  18. Bond Theorems • Bond yields vary inversely with changes in bond prices. • Bond price volatility increases as maturity increases. • Bond price volatility decreases as coupon rates increase.

  19. Bond Price Volatility • The percentage change in bond price for a given change in yield is bond price volatility. • %PB= the percentage change in price. • Pt = the new price in period t. • Pt-1 = the price one period earlier.

  20. Relationship Between Price, Maturity, Market Yield, and Price Volatility

  21. Relationship Between Price, Coupon Rate, Market Yield, and Price Volatility

  22. Interest Rate Risk • Reinvestment risk--variability in realized yield caused by changing market rates for coupon reinvestment. • Price risk--variability in realized return caused by capital gains/losses or that the price realized may differ from par. • Price risk and reinvestment risk offset one another, depending upon maturity and coupon rates.

  23. Duration • Duration is a measure of interest rate risk that considers both coupon rate and term to maturity. • Duration is the ratio of the sum of the time-weighted discounted cash flows divided by the current price of the bond. • Duration equals maturity for zero coupon securities.

  24. Duration Calculations • D = duration. • CFt = interest or principal at time t. • t = time period in which cash flow is received. • n = number of periods to maturity. • i = the yield to maturity (interest rate).

  25. Duration Calculations (concluded) • Calculate duration of a bond with 3 years to maturity, an 8 percent coupon rate paid annually, and a yield to maturity of 10%.

  26. Duration for Bonds Yielding 10% (Annual Compounding)

  27. Properties of Duration • The greater the duration, the greater is price volatility. • Bonds with higher coupon rates have shorter durations. • Generally, bonds with longer maturities have longer durations.

  28. Properties of Duration (concluded) • Except for bonds with a single payment, duration is less than maturity. For bonds with a single payment duration equals maturity. • The higher the yield to maturity, the shorter is duration.

  29. Using Duration to Estimate the Percent Change in Bond Prices • The formula for estimating the percent change in price for a given change in the market rate of interest using duration is:

  30. Convexity • The formula for estimating the percent change in a bond’s price using duration works well for small changes in interest rates, but not for large changes in interest rates. • The formula can be modified to work well for large interest changes and the modification is an adjustment for convexity.

  31. Calculating Convexity • The formula for convexity is:

  32. Using Duration and Convexity to Estimate the Percent change in a Bond’s Price • The formula for using duration and convexity to estimate the percent change in a bond’s price is:

  33. Managing Interest Rate Risk with Duration • Zero-coupon bonds have no reinvestment risk. The duration of a zero equals its maturity. Buy a zero with the desired holding period and lock in the yield to maturity. • To assure that the promised yield to maturity is realized, investors select bonds with durations matching their desired holding periods. (duration-matching approach).

  34. Managing Interest Rate Risk with Duration (concluded) • Selecting a bond maturity equal to the desired holding period (maturity-matching approach) eliminates the price risk, but not the reinvestment risk.