Harmen Simons B. Course in Investments - Bond Pricing 1 Bond Pricing Principles: General Overview (I) Principles explaining and predicting the movement of Bond Prices. The key variables of bond pricing. The Concept of Duration. The Concept of Convexity. The Realized Compound Yield to Maturity (RCYTM).

Harmen Simons B. Course in Investments - Bond Pricing 2 Bond Pricing Principles: General Overview (II) The basis of our analysis will be the Straight Bond: = An IOU that obligates the issuer to pay the Bondholder a fixed sum of money (F) at maturity, along with fixed periodic interest payments (C) Many Bonds also have special features to enhance a bond´s appeal to investors, for example –Convertible Bonds: confer the right to convert bonds into shares of common stock –“Putable” Bonds: have a “Put” feature that grants bondholders the right to sell their bonds back at a special “Put” price. But All Bonds have the “Straight Component”. Therefore the Straight Bond is important as the Basic Unit of Analysis.

Harmen Simons B. Course in Investments - Bond Pricing 3 Bond Pricing Principles: The Effect of Time on Bond Price Movements Bond prices change over time, due to the mere passage of time, even if interest rates do not change at all. An investor must realize that the price of a Premium Bond, i.e. a bond whose market price exceeds its face value, must fall toward its Face Value (F) even if interest rates remain constant. If a Bond is selling at a Discount, i.e. a bond whose current price is less than its face value, its Price must rise, reaching the Par Value by the Maturity Date.

Harmen Simons B. Course in Investments - Bond Pricing 4 Bond Pricing Principles: The Effect of interest rates on Bond Pricing (I) The effect of a given change in interest rates on the price of a bond depends upon 3 key variables: –The Maturity of the bond (M or T) –The Coupon rate (C) –The level of Interest rates at the time of the change in interest rates The basic valuation equation of finance, applied to bonds, states that the price of a bond (P), equals the present value of all expected cash flows from the bond (C(t)), when those cash flows are discounted to the present at the bond’s Yield To Maturity (YTM; y): P =  C t / (1 + y) t where the sum extends from t = 1 to t = M Some other, frequently used, measures are (see next slide…)

Harmen Simons B. Course in Investments - Bond Pricing 5 Bond Pricing Principles: The Effect of interest rates on Bond Pricing (II) Measures of BOND YIELDS –Coupon Rate: = Annual Coupon divided by Face Value, (C/F). Expresses the annual coupon as a % of Face Value. –Current Yield: = Annual Coupon divided by Current Market Price, (C/P). –YTM: = Annual Rate of Return that investors would earn if the Bond were held to Maturity. If there are Semiannual Coupon Payments at the “true” interest rate y, then: YTM=(1+y) 2 - 1; YTM solves: P =  {C/(1+YTM) t } + F/ (1+YTM) n (annual coupon), or: P =  {C/(1+y) 2t } + F/ (1+y) 2n (semiannual C)

Harmen Simons B. Course in Investments - Bond Pricing 6 Bond Pricing Principles: The Effect of interest rates on Bond Pricing (III) Measures of BOND YIELDS (continues…) PAR Bond: Current Market Price (P) = Face Value (F) Premium Bond: => P > F Discount Bond: => P < F PAR Bond: CR = CY = YTM; P = F Premium Bond: CR > CY > YTM; P > F Discount Bond: CR < CY < YTM; P < F –Note: C, F = Constant. P changes as we move towards maturity => CY = C/P  reliable guide for actual yield!

Harmen Simons B. Course in Investments - Bond Pricing 7 Bond Pricing Principles: The Effect of interest rates on Bond Pricing (IV) Measures of BOND YIELDS (continues…) If a Bond is “Callable”, its YTM is no longer a useful measure of Bond Yield. Instead we use YTC: = A measure of Return that assumes a bond will be redeemed at the earliest “Call Date” ( T) for the specified Call Price (CP). YTC solves: P =  {C/(1+YTC) t } + CP/ (1+YTC) T (annual coupon), or: P =  {C/(1+y) 2t } + CP/ (1+y) 2T (semiannual C), where: –T= time in years until earliest possible call date –CP= Contractually specified Call Price of the Bond

Harmen Simons B. Course in Investments - Bond Pricing 8 Bond Pricing Principles: The Effect of interest rates on Bond Pricing (V) The BOND Pricing Principles: (1) Bond prices move inversely with interest rates, i.e. a decrease in interest rates creates an increase in the price of any bond, and any increase in interest rates will cause the price of any bond to fall. However, the amount of the price change depends on the particular features of the bond. (2) The longer the Maturity (M) of a bond, the more sensitive is its price to a change in interest rates, ceteris paribus. (3) The price sensitivity of bonds increases with maturity, but at a decreasing rate. (4) The lower the Coupon Rate (C) on a bond, the more sensitive is its price to a change in interest rates, cet.par. (5) For a given bond, the capital gain caused by a yield decrease exceeds the capital loss caused by a yield increase of the same magnitude.

Harmen Simons B. Course in Investments - Bond Pricing 9 Bond Pricing Principles: The Effect of interest rates on Bond Pricing (VI) Example 1: Prices are given for 8% Coupon Bonds with Yields and Maturity as stated Yields5 years10 years20 years 7%$ $ $ %$ $ $ Price Change:$ 81.14$ $

Harmen Simons B. Course in Investments - Bond Pricing 10 Bond Pricing Principles: The Effect of interest rates on Bond Pricing (VII) Example 2: 20-Year Bond Prices and Yields (first row states Coupon Rates) Yields6%8%10% 6%$1,000$1,231.15$1, %$ $1,000$1, %$ $ $1,000

Harmen Simons B. Course in Investments - Bond Pricing 11 Bond Pricing Principles: The Need for a Summary Measure The 5 Pricing Principles are important for understanding bond investing. However, they are Ceteris Paribus relations: Each of the principles assumes that all other factors are held constant, except for the one under consideration, so, it is still difficult to compare bonds that differ in many ways. Example: Consider the following bonds: –Bond A: 30-year, 14% coupon, yielding 10%, current price $ –Bond I: 20-year, 7% coupon, yielding 10%, current price $ (verify!) Looking at M you would say that Bond A is more sensitive to a given change in yield, while, if you look at the Coupon, you would think that Bond I is more sensitive => we need a measure of a bond’s price sensitivity that allows direct comparisons between bonds.

Harmen Simons B. Course in Investments - Bond Pricing 12 Bond Pricing Principles: The Need for a Summary Measure Example: Relative Price Changes for Bonds A and I BondP at 9%P at 10%P at 11% A$151.60$137.86$ (+9.97%)(-8.48%) I$ 81.60$ 74.26$ (+9.88%)(-8.55%) Note the similarity in price sensitivity despite the fact that Maturity and Coupon are very different.

Harmen Simons B. Course in Investments - Bond Pricing 13 Bond Pricing Principles: Duration As all factors can change simultaneously, it would be convenient to have a summary measure that reflects all factors affecting a bond’s price sensitivity. Such a measure exists, Duration. It is given by the following equation:

Harmen Simons B. Course in Investments - Bond Pricing 14 Bond Pricing Principles: Duration Duration can also be defined as the Holding Period that balances the Price Effect against the Reinvestment Effect. Price Effect: The increase/ decrease in Current Price as a result of a decrease / increase in Interest Rates. Can be viewed as the capital gain or loss, respectively, as a result of a change in interest rates. Reinvestment Effect: Rate at which the periodic Coupon payments can be reinvested. Note that both effects work in opposite directions The Holding Period determines which of the effects dominates There exists a point t  M, at which the Price Effect and the Reinvestment Effect just offset each other: Duration.

Harmen Simons B. Course in Investments - Bond Pricing 15 Bond Pricing Principles: Duration Duration is a single number for each bond that summarizes the 3 key factors (M, C, YTM) that affect the sensitivity of a bond’s price to changes in the interest rate In words, the duration equation computes the Present Value of each of the Cash Flows and weights each by the time until it is received. The weighted Cash Flows are then summed and divided by the current price of the bond. The Concept of Modified Duration (MD) equals Macaulay’s duration divided by (1 + r): MD = D m / (1+ r) Using this concept, the Price Change (  P) can be written as:  P  -(MD).(  r).P

Harmen Simons B. Course in Investments - Bond Pricing 16 Bond Pricing Principles: Duration Duration Tracking Errors: –The duration price change formula takes into account the curvature of the Price/Yield line at a given point –It does NOT take into account how the shape of the curve changes, and how duration changes, away from the initial point at which duration is measured –Duration can change dramatically for the same bond with the same maturity and coupon due solely to a change in yields Relation of Duration with yield, coupon rate, and maturity variations: –Duration is inversely related to yield changes, cet. Paribus –Duration is inversely related to the coupon rate, cet. Par. –Duration is proportional to the bond’s maturity, cet. Par.

Harmen Simons B. Course in Investments - Bond Pricing 17 Bond Pricing Principles: Duration and Convexity Convexity: –Convexity (CVX) reflects the way in which duration changes for different yields on the same bond and tries to account for the duration tracking error discussed on previous slide –If we let R = 1 + r, CVX is defined as: M CVX = (1/P*R 2 ) *  { t *(t +1)*C t }/ (R) t t=1 We can use convexity to better approximate the price change for a bond as follows:  P  -(MD).(  R).P + 0.5*CVX.(  R )2.P

Harmen Simons B. Course in Investments - Bond Pricing 18 Bond Pricing Principles: Convexity Convexity and Bond selection: –Other factors being equal, investors should prefer greater convexity. –In a Price/Yield curve, we have seen that duration essentially measures the slope of a curve at a given point. –Convexity, by contrast, reflects the degree of curvature –The greater the degree of this curvature, the better, because it means the price will be higher for any given yield –Therefore, the more convexity the better. The conclusions concerning D m and CVX we have reached so far, are based on 4 assumptions: –(1) Bonds are Not Callable –(2) Restriction to Flat Yield Curves –(3) Restriction to Parallel Shifts in Yield Curves –(4) Restriction to a Single Change in Yields

Harmen Simons B. Course in Investments - Bond Pricing 19 Bond Pricing Principles: The Realized Compound Yield to Maturity (I) Expectations of future changes in levels of interest rates are very important to bond investors for at least two reasons: (1) As we already know, changing rates imply changing bond prices (2) The rate earned on reinvested coupons also depends on how interest rates change The Realized Compound Yield To Maturity (RCYTM) takes into account the rates earned on reinvested coupons Because bond investment is often directed toward some specific date in the future, the rate earned on reinvested coupon payments or on proceeds from maturing bonds is very important in determining the amount of funds available on the target date. A good measure of progress toward wealth maximization on a target date is the RCYTM because it is a geometric mean growth rate over a given period.

Harmen Simons B. Course in Investments - Bond Pricing 20 Bond Pricing Principles: The Realized Compound Yield to Maturity (II) RCYTM = [Terminal Wealth / Initial Wealth] 1/n - 1 Where: –Initial wealth = the value of the funds at the outset of the investment period –Terminal wealth = the value of the funds at the conclusion of the investment period –n = the number of periods between the outset and the conclusion of the investment period There are 3 general rules about reinvestment rates: –If the reinvestment rate for the interim CFs exceeds the YTM, the RCYTM is greater than the YTM –If the reinvestment rate for the interim CFs equals the YTM, the RCYTM equals the YTM –If the reinvestment rate for the interim CFs is less than the YTM, the RCYTM is less than the YTM