Investment Course - 2005 Day Three: Fixed-Income Analysis and Portfolio Strategies
The Role of Fixed-Income Securities in the Financial Markets and Portfolio Management
Par vs. Spot Yield Curves (Cont.) • A par value yield curve summarizes the yields for coupon-bearing instruments where the coupon rate is equal to the yield-to-maturity. Assuming that the above example is based a collection of Eurobonds (i.e., bonds that pay an annual coupon), the 10% yield for the three-year instrument can be interpreted as the average annual return that the investor can expect if he or she: (i) Holds the bond until maturity, (ii) Reinvests all intermediate cash flows (i.e., the first two coupons) at the same 10% rate for the remaining time until maturity. • A spot, or zero coupon, yield curve summarizes the yields for non-coupon-bearing instruments (i.e., pure discount bonds). These yields can therefore be interpreted as more of a "pure" return since there is no concern about having to reinvest intermediate coupon cash flows. For example, if the above yield curve corresponded to zero coupon securities, the 10%, three-year yield would represent the average annual price appreciation in the bond if it were held to maturity.
Uses for Implied Forward Rates • Predictions of Future Spot Rates: This assumes that investors set yield curves with unbiased expectations, which is seldom true. Generally, implied forward rates are upward-biased predictions of future spot rates because of liquidity premiums attached to yields of longer-term maturity bonds relative to shorter-term instruments • Maturity Choice Decisions: Helps fixed-income investors decide on appropriate maturity structure for a bond portfolio by quantifying the reinvestment rate embedded in longer-term securities compared to shorter-term ones • Pricing Interest Rate Derivatives: Sets the arbitrage boundaries for the rates attached to actual forward agreements (e.g., bond futures, interest rate swaps)
Basics of Bond Valuation • Bonds are simply loans from bondholder to issuer (e.g., firm or government). Just like loans, bonds require interest payments and repayment of principal (also called face value or par) at a pre-specified future date. Interest payments are called coupon payments and bond principal repayments are usually non-amortizing (i.e., paid all at once at maturity) • The current market value of a fixed-income bond is the present value of its future coupon and principal cash flows. In theory, the interest rates used to discount those future cash flows are the zero-coupon (or pure discount)rates corresponding to the dates of each cash flow.
Basics of Bond Valuation (cont.) • Consider a five-year, 9% (annual coupon payment) Eurobond. The market value of the bond, 103.99 (% of par value), can be obtained by calculating the present value of each scheduled cash flow using a sequence of zero-coupon rates commensurate with the riskiness of the bond.
Basics of Bond Valuation (cont.) • The yield to maturity (y) of the bond is the constant interest rate per period that solves the following equation: • The yield-to-maturity is the internal rate of return of all cash flows. It is the rate such that the present values of the cash flows, each discounted by that same rate, exactly equal the market value of the bond. The yield to maturity of this bond turns out to be 8.00%.
Basics of Bond Valuation (cont.) • The yield to maturity is a statistic about the rate of return on the bond that includes both the coupon cash flows as well as any inevitable capital gain or loss if the bond is held to maturity (a gain if the bond is purchased at a discount below par value, a loss if the bond is purchased at a premium above par value). • Therefore, it contains more information than the current yield, which is simply the coupon rate divided by the current price, e.g., 9 103.99 = .0865 . The current yield of 8.65% overstates the investor’s rate of return since it neglects the capital loss.
Basics of Bond Valuation (cont.) • Notice that the yield to maturity can be interpreted as a "weighted average" of the sequence of zero-coupon rates, with most of the weight placed on the last cash flow since that is when the principal is redeemed, in that both deliver the same present value: • Clearly, the yield to maturity must lie within the range of the zero-coupon rates.
Current Coupon, Premium, and Discount Bonds • A Current Coupon (or Par-Value) Bond is one for which the current market price equals the face value. In that case, the coupon rate (C/F) will equal the current yield (C/P), which will equal the yield-to-maturity (y). P = F <===> C/F = C/P = y The bond is priced at par value since its coupon rate is "fair" in that it equals the current market interest rate as represented by the yield-to-maturity. • A Premium Bond has a current market price that exceeds the face value. In this case, the coupon rate will be higher than the current yield, which in turn will be higher than the yield-to-maturity. P > F <===> C/F > C/P > y The bond is priced at a premium above par value since its coupon rate is "high" given current market rates. A par-value, current coupon bond would have a lower coupon rate, so the premium represents the value of the "excessive" coupon cash flows. In fact, the amount of the premium is the present value of the annuity represented by the difference between the coupon rate and the bond's yield, discounted at that yield. • A Discount Bond has a current market price that is less than the face value. The coupon rate will be less than the current yield, which will be less than the yield-to-maturity. P < F <===> C/F < C/P < y The bond is priced at a discount below par value since its coupon rate is "low" given current market rates. The amount of the discount is the present value of the annuity represented by the difference between the yield and the coupon rate. For example, a zero-coupon bond will usually be at a deep discount to par value.
Current Coupon, Premium, and Discount Bonds (cont.) • Example: Calculate the yield-to-maturity statistic on a seven-year, 6-3/4% Treasury note priced at 98.125. Assume that a semi-annual coupon payment has just been made so that exactly 14 periods remain until the principal is refunded at maturity. • Algebraically, the yield is the solution “y” to the following equation: • Solving for the periodic yield (i.e., y/2) on a financial calculator (such as the HP 12C) obtains 3.5472 [100 FV, 14 n, 3.375 PMT, -98.125 PV, i …. 3.5472]. • The annualized yield-to-maturity would then be reported as y = 7.0944% (i.e., 3.5472 x 2).
Sources of Bond Risk • Primary: • Default: Will the borrower honor its promise to repay? • Interest Rate: How will changing market conditions affect the value of the bond? • Price risk component • Reinvestment risk component • Secondary: • Call: Will the borrower refinance the loan under conditions that are disadvantageous to investor? • Liquidity: How easily can bond be bought or sold? • Tax: Will changes in the tax code affect bond values?
Bond Yields, Pricing, and Volatility • Theorem #1: Bond prices are inversely related to bond yields. Implication: When market rates fall, bond prices rise, and vice versa. • Theorem #2: Generally, for a given coupon rate, the longer is the term to maturity, the greater is the percentage price change for a given shift in yields. (The maturity effect) Implication: Long-term bonds are riskier than short-term bonds for a given shift in yields, but also have more potential for gain if rates fall. • Theorem #3: For a given maturity, the lower is the coupon rate, the greater is the percentage price change for a given shift in yields. (The coupon effect) Implication: Low-coupon bonds are riskier than high-coupon bonds given the same maturity, but also have more potential for gain if rates fall. • Theorem #4: For a given coupon rate and maturity, the price increase from a given reduction in yield will always exceed the price decrease from an equivalent increase in yield. (The convexity effect)
Bond Yields, Pricing, and Volatility (cont.) Implication: There are potential gains from structuring a portfolio to be more convex (for a given yield and market value) since it will outperform a less convex portfolio in both a falling yield market as well as a rising yield Price Convex Price-Yield Curve Yield
Bond Yields, Pricing, and Volatility: Example • Consider the following bonds: • Initial Prices:
Bond Yields, Pricing, and Volatility: Example (cont.) • Prices after yields increase by 50 bp: • Percentage price changes: Bond A: (906.43 - 924.18) / (924.18) = -1.92% (least) Bond B: (668.78 - 705.46) / (705.46) = -5.20% (most) Bond C: (952.68 - 1000.00) / (1000.00) = -4.73% (middle)
Bond Yields, Pricing, and Volatility: Example (cont.) • Question: Where would Bond D, which has a coupon rate of 6% and a maturity of 19 years, fit into this price sensitivity spectrum? (Assume its initial yield is also 8%.) Initial: After: So, percentage change: Bond D: (768.31 - 807.93) / (807.93) = -4.90%
Calculating the Duration Statistic • The duration of a bond is a weighted average of the payment dates, using the present value of the relative cash payments as the weights: • This statistic is the Macaulay duration, named after Frederick Macaulay who first developed it, and can be interpreted as the point in the life of the bond when the average cash flow is paid.
Calculating the Duration Statistic: Example • Consider a five-year, 12% annual payment bond having a face value of $1,000. Suppose that the bond is priced at a premium to yield 10% (p.a.). The price of the bond is $1,075.82 and the Macaulay duration is 4.074: or:
Duration as a Measure of Price Volatility • Basic Price-Yield Elasticity Relationship: • Convert to “Volatility Prediction” Equation: • Prediction Equation in Modified Form (% price change):
Duration as a Measure of Price Volatility (cont.) • Convert to dollar (or cash) sensitivity: DMV ~ -(Mod D)( Dy)(MV) • Sensitivity to a one bp yield change (i.e., Dy = 0.0001): DMV ~ -(Mod D)(0.0001)(MV) = Basis Point Value = BPV
Duration and Price Volatility: Example • Consider again the five-year, 12% coupon bond with a yield to maturity of 10%: • Macaulay D: 4.074 • Modified D: 3.704 (= 4.074 / 1.1) • This means that an increase in yields of 100 bp will change the bond’s price by approximately 3.7% in opposite direction • Basis Point Value: $0.0398 [= (3.704)(.0001)(107.582)] • This means that a one bp change in yields will cause the bond’s price to move by about 4 cents per $100 of par value (which would correspond to a 40 cent movement for a bond with a par value of $1000)