W. Sean Cleary Charles P. Jones

INVESTMENTS:

Analysis and Management

Second Canadian Edition

Chapter 11

Bond Yields and Prices

Learning Objectives

• Calculate the price of a bond.

• Explain the bond valuation process.

• Calculate major bond yield measures, including yield to maturity, yield to call, and horizon return.

• Account for changes in bond prices.

• Explain and apply the concept of duration.

Bond Valuation Principle

• Intrinsic value  Is an estimated value   Present value of the expected future cash flows Required to compute intrinsic value • Expected future cash flows • Timing of expected cash flows • Discount rate, or required rate of return by investors

Bond Valuation

• Value of a coupon bond with semi-annual payments:

V



t 2n

 

1 (1 C



t /2 r/2) t



(1



MV r /2) 2n

• Biggest problem is determining the discount rate or required yield • Required yield is the current market rate earned on comparable bonds with same maturity and credit risk

Interest Rates

• • Rates and basis points  100 basis points are equal to one percentage point Short-term riskless rate    Provides foundation for other rates Approximated by rate on Treasury Bills Other rates differ because of • Maturity differentials • Security risk premiums

Interest Rates

• • Maturity differentials  Term structure of interest rates • Accounts for the relationship between time and yield for bonds the same in every other respect Risk premium   Yield spread or yield differential Associated with issuer’s particular situation

Determinants of Interest Rates

• • Real rate of interest   Rate that must be offered to persuade individuals to save rather than consume Rate at which real capital physically reproduces itself Nominal interest rate  Function of the real rate of interest and expected inflation premium

Determinants of Interest Rates

• Market interest rates on riskless debt (nominal rate)  real rate + expected inflation  Fisher Hypothesis • Real rate estimates obtained by subtracting the expected inflation rate from the observed nominal rate

Current Yield

• Defined as the ratio of the coupon interest to the current market price • Uses the current market price instead of the face amount of a bond ($1,000) • Not a true measure of the return – does not account for the difference between bond’s purchase piece and eventual redemption at par value

Yield to Maturity

• Yield to maturity (YTM)  Rate of return on bonds most often quoted for investors  Promised compound rate of return received from a bond purchased at the current market price and held to maturity  Equates the present value of the expected future cash flows to the initial investment • Similar to internal rate of return

Yield to Maturity

• Solve for YTM (semi-annual coupons):

P



t 2n

 

1 (1



C t /2 YTM/2) t



MV (1



YTM/2) 2t

• Investors earn the YTM if the bond is held to maturity and all coupons are reinvested at YTM

Yield to Call

• Yield to a specified call date and call price • Substitute number of periods until first call date for and call price for face value (semi annual coupons) • Applies to callable bonds

P



t 2c

 

1 (1



C t /2 YTC/2) t



(1



CP YTC/2) 2c

Reinvestment Risk

For: (1) longer-term bonds (2) bonds with higher coupon rates (i.e., have more money to reinvest)

NO

reinvestment risk for “Zeroes”

Realized Yield

• • Rate of return actually earned on a bond given the reinvestment of the coupons at varying rates • Can only be calculated after investment period is over Horizon return analysis  Bond returns based on assumptions about reinvestment rates

Bond Price Changes

• Over time, bond prices that differ from face value must change • Bond prices move inversely to market yields • The change in bond prices due to a yield change is directly related to time to maturity and inversely related to coupon rate

Bond Price Changes

Market yield

• Holding maturity constant, a rate decrease will raise prices a greater percent than a corresponding increase in rates will lower prices

Measuring Bond Price Volatility: Duration

• Important considerations    Different effects of yield changes on the prices and rates of return for different bonds Maturity inadequate measure of a bond’s economic lifetime A measure is needed that accounts for both size and timing of cash flows

Duration

• A measure of a bond’s lifetime, stated in years, that accounts for the entire pattern (both size and timing) of the cash flows over the life of the bond • The weighted average maturity of a bond’s cash flows  Weights determined by present value of cash flows

Calculating Duration

• Need to time-weight present value of cash flows from bond

D



t n

 

1 PV(CF t ) Market Price



t

• Duration depends on three factors    Maturity of the bond Coupon payments Yield to maturity

Duration Relationships

• Duration increases a decreasing rate with time to maturity, but at  For coupon paying bonds, duration is always less than maturity  For zero coupon-bonds, duration equals time to maturity • Duration increases with lower coupons • Duration increases with lower yield to maturity

Why is Duration Important?

• Allows comparison of effective lives of bonds that differ in maturity, coupon • Used in bond management strategies, particularly immunization • Measures bond price sensitivity to interest rate movements, which is very important in any bond analysis

Convexity

• Refers to the degree to which duration changes as the yield to maturity changes  Price-yield relationship is convex • Duration equation assumes a linear relationship between price and yield • Convexity largest for low coupon, long-maturity bonds, and low yield to maturity

Duration Conclusions

• To obtain maximum price volatility, investors should choose bonds with the longest duration • Duration is additive  Portfolio duration is just a weighted average • Duration measures volatility, which is not the only aspect of risk in bonds

Appendix 11-A Treasury Bill Yields and Prices

• T-bill are sold in Canada on a discount basis

r BEY



Face



P P

 365  100

n P



Face

( 1 

r BEY



n

365 )

Appendix 11-C Convertible Bonds

• Convertible Bonds  Bonds that are convertible into a specified number of C/S • Terminology Conversion Price (CP)  Par Value Conversion (M) Ratio (CR)

So

, CR  M CP Iie. How many shares per bond

Appendix 11-C Convertible Bonds

Conversion Value (CV)  CR  Current C/S Price Straight Bond Value (SBV) the value of the convertibl e bond if it were a " straight" bond SBV  C      1 (1  1 r) n r      

M

  ( 1  1 r) n   Where “r” is the required rate of return on identical (similar) non-convertible bonds.

Appendix 11-C Convertible Bonds

Conversion Premium = Market price of convertible – Conversion value Minimum (Floor) Value = Maximum (straight bond value; conversion value)

Why Issue/Buy Convertibles?

• • Investors  income (interest) but at lower rate  participate in share price appreciation Firm  “lower” coupons  delayed equity financing (when share price rises)  usually have call feature attached

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