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## Chapter 9

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**Chapter 9**Debt Instruments Quantitative Issues**Pricing a Bond**where P0 = price of bond today T = maturity of the bond Y = appropriate discount rate PAR = par or face value of the bond**Bond Prices with Semiannual Payments**• Divide coupon payment by two • Multiply maturity of bond by two. • Divide discount rate by two**Bond Yields & Rates**• Coupon rate (nominal yield) • Current yield (coupon / price) • Yield to maturity (YTM = IRR) • Realized compound yield to maturity (RCYTM) • Yield to First (earliest) Call • Realized return**ABC Example**• Coupon: $40 per year • Par Value: $1,000 • Maturity: 6 years • Callable: in 3 years @ $1040 • Price: $950**Coupon Rate**• Stated dollar return of fixed-income investment • Equals annual interest payments divided by par value**Current Yield**• Bond’s coupon rate divided by current market price OR • Stock’s indicated dividend rate divided by per-share price**Yield to Maturity**• Measure of bond yield that takes into account capital gain or loss, as well as coupon payments • Discount rate that would make present value of bond’s cash flows (payments plus face value at maturity) equal purchase price of bond where C= the coupon payment**Yield to Call**where Tc = time to earliest call Yc = yield to first call • Almost identical to YTM, except • Call price replaces par value • Time to call replaces term to maturity**Realized Rate (Yield)**• Ex post rate of return or yield from investment (internal rate of return) where TH = holding period YH = realized rate of return**Bond Price Volatility**• Bond prices and interest rates inversely related • Maturity effect: longer a bond’s term to maturity, greater percentage change in price for given change in interest rates • Coupon effect:lower a bond’s coupon rate, greater percentage change in price for given change in interest rates • Yield-to-maturity effect: For given change in interest rates, bonds with lower YTMs have greater percentage price changes than bonds with higher YTMs – all other things equal**Which Bond’s Price Is Most Volatile?**• Bond X: 25 years to maturity, 10% coupon rate, and a 6% YTM • Bond Y: 10 years to maturity, 2% coupon rate, and a 6% YTM • Bond Z: 17.5 years to maturity, 6% coupon rate, and a 4% YTM**Answer**• Based on maturity effect, it would be X • Based on coupon effect, it would be Y • Based on yield-to-maturity effect, it would be Z**Duration**• Weighted average amount of time until present value of bond’s purchase price repaid to the investor • Based on time-weighted present value of bond’s principal and interest payments divided by the bond’s price • Used as measure of bond’s sensitivity to interest rate changes**Formula for Duration**Where P0 = price of the bond today Y = yield to maturity Ct = cash flow in period t (coupon, principal or both) T = term to maturity**Equation 9-6**• Insert Equation • Where Y = yield to maturity C = coupon rate T = term to maturity**Uses of Duration**• Price volatility index • Larger duration statistic, more volatile price of bond • Immunization • Interest rate risk minimized on bond portfolio by maintaining portfolio with duration equal to investor’s planning horizon**Major Characteristics of Duration**• Duration of zero-coupon bond equal to term to maturity • Duration of coupon bond always less than term to maturity • Inverse relationship between coupon rate and duration (continued)**Major Characteristics of Duration (continued)**• Inverse relationship between yield to maturity and duration • Direct relationship between maturity and duration**Modified Duration**• Adjusted measure of duration used to estimate a bond’s interest rate sensitivityD* = D (1 + YTM)% Chg in price of bond = –D x % Chg in YTM% Chg in price of bond = – D* x [Chg in YTM]**Portfolio Duration**• Market value weighted average of durations of individual securities in the portfolio**Components of InterestRate Risk**• Price Risk • Reinvestment Rate Risk**Price Risk**• Risk of existing bond’s price changing in response to unknown future interest rate changes • If rates increase, bond’s price decreases • If rates decrease, bond’s price increases**Reinvestment Rate Risk**• Risk associated with reinvesting coupon payments at unknown future interest rates • If rates increase, coupons are reinvested at higher rates than previously expected • If rates decrease, coupons are reinvested at lower rates than previously expected**Immunizing a Portfolio**• If a single time horizon goal, purchasing zero-coupon bond whose maturity corresponds with planning horizon • If multiple goals, purchasing series of zero-coupon bonds whose maturities correspond with multiple planning horizons (continued)**Immunizing a Portfolio (continued)**• Assembling and managing bond portfolio whose duration is kept equal to planning horizon Note: this strategy involves regular adjustment of portfolio because duration of portfolio will change at SLOWER rate than will time itself**Bond Swaps**• Technique for managing bond portfolio by selling some bonds and buying others • Possible benefits achieved: • tax treatment • yields • maturity structure • trading profits**Types of Swaps**• Substitution swap • Tax swap • Intermarket spread swap • Pure-yield pick-up swap • Rate anticipation swap**Strategies for Managing a Bond Portfolio**• Bullet Portfolio – Entire portfolio is placed in one maturity • Bond ladders • Equally distributed dollar allocations over time • Barbells • Majority of dollar allocations in shortest-term and longest-term holdings**Yield Curve or Term Structure**• Vertical axis: yield to maturity • Horizontal axis: term to maturity • Bonds of like quality • Always based on Treasuries**Shapes of Yield Curve**• Rising: Most common (used to be only one observed) • Falling: Next most common • Humped • Flat: Rare**Theories of the Yield Curve**• Unbiased expectations • Long-term rates reflect market’s expectation of current and future short-term rates. • Preferred habitat • Significantly more attractive rates can induce investors and borrowers out of their preferred maturity structures (continued)**Theories of Yield Curve(continued)**• Market Segmentation: • Yields reflect supply and demand for each maturity class. • Liquidity Preference: • Borrowers are risk averse and demand premium for buying long-term securities • Yield curves tend to be upward sloping. (continued)**Theories of the Yield Curve (continued)**• Preferred habitat • Significantly more attractive rates can induce investors and borrowers out of their preferred maturity structures • Unbiased expectations • Long-term rates reflect market’s expectation of current and future short-term rates.**Factors Affecting Bond Yields**• General credit conditions: Credit conditions affect all yields to one degree or another. • Default risk: Riskier issues require higher promised yields. • Term structure: Yields vary with maturity • Duration: Weighted average amount of time until present value of purchase price is recouped. • Coupon effect: Low-coupon issues offer yields that are partially taxed as capital gains. (continued)**Factors Affecting Bond Yields (continued)**• Seasonings: Newly issued bonds may sell at slight discount to otherwise-equivalent established issues. • Marketability: Actively traded issues tend to be worth more than similar issues less actively traded. • Call protection: Protection from early call tends to enhance bond’s value. • Sinking fund provisions: Sinking funds reduce probability of default, thereby tending to enhance bond’s value. • Me-first rules: Bonds protected from diluting effect of additional borrowings are generally worth more than otherwise-equivalent unprotected issues.