Chapter 10 Bond Prices and Yields PDF
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Douglas College
2021
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This PowerPoint presentation details bond prices and yields, covering calculation methods, the importance of yield to maturity, interest rate risk, and other key aspects of fixed-income investments.
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Chapter 10 Bond Prices and Yields Copyright © 2021 McGraw-Hill Education. All rights reserved. Learning Objectives Bonds can be an important part of portfolios. You will learn: 1. How to calculate bond prices and yields. 2. The importance of yield to maturity. 3. I...
Chapter 10 Bond Prices and Yields Copyright © 2021 McGraw-Hill Education. All rights reserved. Learning Objectives Bonds can be an important part of portfolios. You will learn: 1. How to calculate bond prices and yields. 2. The importance of yield to maturity. 3. Interest rate risk and Malkiel’s theorems. 4. How to measure the impact of interest rate changes on bond prices. Bond Prices and Yields, II. Our goal is to understand the relationship between bond prices and yields. In addition, we will examine some fundamental tools that fixed-income portfolio managers use when they assess bond risk. Bond Basics, I. A straight bond is an IOU that obligates the issuer of the bond to pay the holder of the bond: A fixed sum of money (called the principal, par value, or face value) at the bond’s maturity Constant, periodic interest payments (called coupons) during the life of the bond U.S. Treasury bonds are straight bonds. Special features may be attached: Convertible bonds Callable bonds Putable bonds Bond Basics, II. Two basic yield measures for a bond are its coupon rate and its current yield. Annual coupon Coupon rate Par value Annual coupon Current yield Bond price Straight Bond Prices and Yield to Maturity The price of a bond is found by adding together: the present value of the bond’s coupon payments, and the present value of the bond’s face value. The yield to maturity (YTM) of a bond is the discount rate that equates today’s bond price with the present value of all the future cash flows of the bond. The Bond Pricing Formula The price of a bond is found by adding together the present value of the bond’s coupon payments and the present value of the bond’s face value. The formula is: In the formula, C represents the annual coupon payments (in $), FV is the face value of the bond (in $), M is the maturity of the bond, measured in years, and YTM is the yield to maturity of the bond, expressed as an annual percent. Example: Using the Bond Pricing Formula What is the price of a straight bond with: $1,000 face value, coupon rate of 8%, a YTM of 7%, and a maturity of 20 years? (Note that we simplified the bond pricing formula a C 1 FV Bond Price 1 little.) YTM 1 YTM 2 2M 1 YTM 2 2M 80 1 1000 Bond Price 1 0.07 1 0.07 2 220 2 1 0.07 220 (1,142.857 0.747428) 252.5725 $1,106.78. Example: Calculating the Price of this Straight Bond Using Excel Excel has a function that allows you to price straight bonds, and it is called PRICE. =PRICE(“Settlement”,“Maturity”,Coupon Rate,YTM,100,2,3) Enter “Today” and “Maturity” in quotes, using Date() function. Enter the Coupon Rate and the YTM as a decimal. The "100" tells Excel to use $100 as the par value. The "2" tells Excel to use semi-annual coupons. The "3" tells Excel to use an actual day count with 365 days per year. Note: Excel returns a price per $100 face in this example. Spreadsheet Analysis, I. Premium, Par, and Discount Bonds, I. Bonds are given names according to the relationship between the bond’s selling price and its par value. Remember: C Y P F (Can You Pass Finance?) Premium bonds: If Coupon rate > YTM then Price > Face (par value) Discount bonds: If Coupon rate < YTM then Price < Face (par value) Par bonds: If Coupon rate = YTM then Price = Face (par value) Do you see the relationship between C Y and P F? Premium, Par, and Discount Bonds, II. Premium, Par, and Discount Bonds, III. In general, when the coupon rate and YTM are held constant: For premium bonds: the longer the term to maturity, the greater the premium over par value. For discount bonds: the longer the term to maturity, the greater the discount from par value. Relationships among Yield Measures For premium bonds: coupon rate > current yield > YTM For discount bonds: coupon rate < current yield < YTM For par value bonds: coupon Current yield rate =coupon is the annual current yield = YTM payment divided by the bond’s current market price. Calculating Yield to Maturity, I. Suppose we know the current price of a bond ($110), its coupon rate (4%), and its time to maturity (8 years). How do we calculate the YTM? We can use the straight bond formula, trying different yields until we find the one that produces the current bond price. $40 1 $100 $110 1 YTM 1 YTM 2 28 1 YTM 2 28 This process is tedious. So, to speed up the calculation, we use financial calculators and spreadsheets. Calculating Yield to Maturity, II. We can use the YIELD function in Excel: =YIELD(“Settlement”,“Maturity”,Coupon Rate,Price,100,2,3) Enter “Settlement” and “Maturity” in quotes, use Date() Function. Enter the Coupon Rate as a decimal. Enter the Price as a percent of face value. Note: As before, The "100" tells Excel to use $100 as the par value. The "2" tells Excel to use semi-annual coupons. The "3" tells Excel to use an actual day count with 365 days per year. Using dates 8 years apart, a coupon rate of 4%, and a price (per hundred) of $110 results in a YTM of 2.607%. Spreadsheet Analysis, II. A Quick Note on Bond Quotations, I. We have seen how to calculate bond prices. We have calculated the price as if the next coupon payment is six months off. If you buy a bond between coupon dates, you will receive the next coupon payment (and might have to pay taxes on it). When you buy the bond between coupon payments, you must compensate the seller for any accrued interest. A Quick Note on Bond Quotations, II. The convention in bond price quotes is to ignore accrued interest. This results in what is commonly called a clean price (i.e., a quoted price net of accrued interest). Sometimes, this price is also known as a flat price. The price the buyer actually pays is called the dirty price. The buyer must pay accrued interest and the clean price. Note: The price the buyer actually pays is sometimes known as the full price, or invoice price. A Quick Note on Bond Quotations, III. Suppose you buy a 10-year bond with a 12 percent annual coupon, payable semiannually. The next coupon payment is in four months. The bond’s quoted price (i.e., its clean price) is $1,060. (What is the YTM? About 11%, actually 10.996%). Or, if you are given 10.996% YTM, 12% coupon, 10 years, semiannual coupons, the flat, or clean, price of the bond is $1,060. But, you actually pay $1,080 for this bond. That is, the dirty, or invoice, price of the bond is $1,080. Notice that the next coupon will be $60. The accrued interest on a bond is calculated by taking the fraction of the coupon period that has passed, in this case two months out of six, and multiplying this fraction by the next coupon, $60. So, the accrued interest in this example is 2/6 × $60 = $20. Invoice Price ($1,080) = Clean price ($1,060) plus accrued interest ($20). Callable Bonds Thus far, we have calculated bond prices assuming that the actual bond maturity is the original stated maturity. Most bonds, however, are callable bonds. A callable bond gives the issuer the option to buy back the bond at a specified call price anytime after an initial call protection period. For callable bonds, YTM might not be useful. Yield to Call Yield to call (YTC) is a yield measure that assumes a bond will be called at its earliest possible call date. The formula to price a callable bond is: C 1 CP Callable Bond Price 1 YTC 1 YTC 2T 2 1 YTC 2 2T In the formula, C is the annual coupon (in $), CP is the call price of the bond, T is the time (in years) to the earliest possible call date, and YTC is the yield to call, with semi-annual coupons. As with straight bonds, we can solve for the YTC, if we know the price of a callable bond. Spreadsheet Analysis, III. Interest Rate Risk Holders of bonds face interest rate risk. Interest rate risk is the possibility that changes in interest rates will result in losses in the bond’s value. The yield actually earned or “realized” on a bond is called the realized yield. Realized yield is almost never exactly equal to the yield to maturity, or promised yield. Interest Rate Risk and Maturity Malkiel’s Theorems, I. 1. Bond prices and bond yields move in opposite directions. As a bond’s yield increases, its price decreases. Conversely, as a bond’s yield decreases, its price increases. 2. For a given change in a bond’s YTM, the longer the term to maturity of the bond, the greater the magnitude of the change in the bond’s price. Malkiel’s Theorems, II. 3. For a given change in a bond’s YTM, the size of the change in the bond’s price increases at a diminishing rate as the bond’s term to maturity lengthens. 4. For a given change in a bond’s YTM, the resulting percentage change change in the bond’s price is inversely related to the bond’s coupon rate. 5. For a given absolute change in a bond’s YTM, the magnitude of the price increase caused by a decrease in yield is greater than the price decrease caused by an increase in yield. Bond Prices and Yields Duration Bondholders know that the price of their bonds change when interest rates change. But, How big is this change? How is this change in price estimated? Macaulay Duration, or Duration, is a way for bondholders to measure the sensitivity of a bond price to changes in bond yields. That is, given a starting YTM and a change in YTM: Change in YTM Pct. Change in Bond Price Duration 1 YTM 2 Two bonds with the same duration, but not necessarily the same maturity, will have approximately the same price sensitivity to a (small) change in bond yields. Example: Using Duration Example: Suppose a bond has a Macaulay Duration of 11 years and a current yield to maturity of 8%. If the yield to maturity increases to 8.50%, what is the resulting percentage change Pct. Change in Bond Price - 11 0.085in the 0.08 price of the bond? 1 0.08 2 -5.29%. Note that the starting YTM appears in the denominator. Modified Duration Some analysts prefer to use a variation of Macaulay’s Duration, known as Modified Duration. Macaulay Duration Modified Duration YTM 1 2 The relationship between percentage changes in bond prices and changes in bond yields is approximately: Pct. Change in Bond Price - Modified Duration Change in YTM Calculating Macaulay’s Duration, I. Macaulay’s Duration values are stated in years and are often described as a bond’s effective maturity. For a zero-coupon bond, duration = maturity. For a coupon bond, duration = a weighted average of individual maturities of all the bond’s separate cash flows, where the weights are proportionate to the present values of each cash flow. Calculating Macaulay’s Duration for Par Bonds If a bond is selling for par value, the duration formula is: Example: par bond with a 6 percent coupon (and 6 percent YTM, why?) and 10 years to maturity. Verify that the duration is 7.66 years. Calculating Macaulay’s Duration, II. In general, for a bond paying constant semiannual coupons, the formula for Macaulay’s Duration is: In the formula, CPR is the annual coupon rate, M is the bond maturity (in years), and YTM is the yield to maturity, assuming semiannual coupons. Calculating Macaulay’s Duration: Example A bond has a yield to maturity of 7 percent and it matures in 12 years. The coupon rate is 6 percent. Tossing these numbers into the formula: Crank away and verify that the Duration is 8.56 years, and the Modified Duration is 8.27 years. Calculating Macaulay Duration and Modified Duration, using Excel Calculating Duration Using Excel, Explanation We can use the DURATION and MDURATION functions in Excel to calculate Macaulay Duration and Modified Duration. The Excel functions use arguments like we saw before: =DURATION(“Settlement”,“Maturity”,Coupon Rate,YTM,2,3) Example: Verify that a 5-year bond, with a 9% coupon and a 7% YTM, has a Duration of 4.17 and a Modified Duration of 4.03. Duration Properties 1. All else the same, the longer a bond’s maturity, the longer its duration. 2. All else the same, a bond’s duration increases at a decreasing rate as maturity lengthens. 3. All else the same, the higher a bond’s coupon, the shorter is its duration. 4. All else the same, a higher yield to maturity implies a shorter duration. Properties of Duration Bond Risk Measures Based on Duration, I. Dollar Value of an 01: Measures the change in bond price from a one basis point change in yield. Dollar Value of an 01 - Modified Duration Bond Price 0.0001 Yield Value of a 32nd: Measures the change in yield that would lead to a 1/32nd change in the bond price. 1 Yield Value of a 32nd 32 Dollar Value of an 01 In both cases, the bond price is per $100 face value. Bond Risk Measures Based on Duration, II. Suppose a bond has a modified duration of 8.27 years. What is the dollar value of an 01 for this bond (per $100 face value)? What is the yield value of a 32 nd (per $100 face value)? First, we need the price of the bond, which is $91.97. Verify: YTM = 7% Coupon = 6% Maturity = 12 Years. The Dollar Value of an 01 is $0.07606, which says that if the YTM changes one basis point, the bond price changes by 7.6 cents. The Yield Value of a 32nd is.41086, which says that a yield change of.41 basis points changes the bond price by 1/32nd (3.125 cents). Dedicated Portfolios A dedicated portfolio is a bond portfolio created to prepare for a future cash payment, e.g. pension funds. The date the payment is due is commonly called the portfolio’s target date. Dedicated Portfolios: Scenario Suppose the Safety First pension fund estimates that it must pay benefits of about $100 million in five years. Safety First decides to buy coupon bonds yielding 8 percent. These coupon bonds pay semiannual coupons, mature in five years, and are currently selling at par. If interest rates do not change over the next five years, how much money does Safety First need to invest today in these coupon bonds to have $100 million in five years? Dedicated Portfolios: Example To start, we convert the straight bond pricing formula (which is a present value) into a future value by multiplying by : For Safety First, we know the future value is $100,000,000. So: Rearranging, and solving: Safety First needs to invest about $67.5 million today. Dedicated Portfolios: How Does it Work? These bonds sell for par, so $67,556,417 is their total face value. With this face value, the coupon payment every six months is (Recall Coupon Rate = YTM for par bonds): $67,556,417 × 0.08/2 = $2,702,257. If Safety First invests each of these coupons at 8 percent (i.e., the YTM), the total future value of the coupons is $32,443,583. Safety First also receives the face value of the coupon bonds in five years, or $67,556,417. In five years, Safety First has $32,443,583 + $67,556,417 = $100,000,000. Safety First needs about $67.5 million to construct a dedicated bond portfolio to fund a future liability of $100 million. Huge assumption: interest rates unchanged over the next five years. Reinvestment Risk Reinvestment rate risk is the uncertainty about the value of the portfolio on the target date. Reinvestment rate risk stems from the need to reinvest bond coupons at yields not known in advance. Simple solution: purchase zero coupon bonds. Problem with simple solution: U.S. Treasury STRIPS are the only zero coupon bonds issued in sufficiently large quantities. STRIPS have lower yields than even the highest quality corporate bonds. The Cost of Removing Reinvestment Risk Using STRIPS Suppose that Treasury STRIPS have a yield of 7.75 percent. Using semiannual compounding, the present value of these zero coupon bonds providing a principal payment of $100 million in five years is: $68,373,787 − $67,556,417 = $817,370. (Hmm…) Other methods are available at a lower cost. Price Risk Price risk is the risk that bond prices will decrease. Price risk arises in dedicated portfolios when the target date value of a bond is not known with certainty. Price Risk versus Reinvestment Rate Risk For a dedicated portfolio, interest rate increases have two effects: Increases in interest rates decrease bond prices, but Increases in interest rates increase the future value of reinvested coupons For a dedicated portfolio, interest rate decreases have two effects: Decreases in interest rates increase bond prices, but Decreases in interest rates decrease the future value of reinvested coupons Immunization Immunization is the term for constructing a dedicated portfolio such that the uncertainty surrounding the target date value is minimized. It is possible to engineer a portfolio such that price risk and reinvestment rate risk offset each other (just about entirely). Immunization by Duration Matching, I. A dedicated portfolio can be immunized by duration matching—matching the duration of the portfolio to its target date. Then, the impacts of price and reinvestment rate risk will almost exactly offset. This means that interest rate changes will have a minimal impact on the target date value of the portfolio. Immunization by Duration Matching, II. Dynamic Immunization Dynamic immunization is a periodic rebalancing of a dedicated bond portfolio for the purpose of maintaining a duration that matches the target maturity date. The advantage is that the reinvestment risk caused by continually changing bond yields is greatly reduced. The drawback is that each rebalancing incurs management and transaction costs. Chapter Review, I. Bond Basics Straight Bonds Coupon Rate and Current Yield Straight Bond Prices and Yield to Maturity Straight Bond Prices Premium and Discount Bonds Relationships among Yield Measures Chapter Review, II. More on Yields Calculating Yields Yield to Call Interest Rate Risk and Malkiel’s Theorems Promised Yield and Realized Yield Interest Rate Risk and Maturity Malkiel’s Theorems Chapter Review, III. Duration Macaulay Duration Modified Duration Calculating Macaulay’s Duration Properties of Duration Dedicated Portfolios and Reinvestment Risk Dedicated Portfolios Reinvestment Risk Immunization Price Risk versus Reinvestment Rate Risk Immunization by Duration Matching Dynamic Immunization
