# 1 Chapter 8: Valuation of Known Cash Flows: Bonds Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective Valuation of fixed income securities.

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1 Chapter 8: Valuation of Known Cash Flows: Bonds Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective Valuation of fixed income securities Explain why bond prices change

2 Chapter 8 Contents 1 Using Present Value Formulas to Value Known Flows1 Using Present Value Formulas to Value Known Flows 2 The Basic Building Blocks: Pure Discount Bonds2 The Basic Building Blocks: Pure Discount Bonds 3 Coupon Bonds, Current Yield, and Yield-to- Maturity3 Coupon Bonds, Current Yield, and Yield-to- Maturity 4 Reading Bond Listings4 Reading Bond Listings 5 Why Yields for the same Maturity Differ5 Why Yields for the same Maturity Differ 6 The Behavior of Bond Prices Over Time6 The Behavior of Bond Prices Over Time

3 Yield Curve A typical yield curve:A typical yield curve:

4

5 8.2 and 8.3 - Pure Discount Bonds and Coupon Bonds A pure discount bond is a security that pays a specified single cash payment (face value or par value) at a specified date called its maturity dateA pure discount bond is a security that pays a specified single cash payment (face value or par value) at a specified date called its maturity date We can always analyze any fixed income contract with known cash flows into a sum of \$1-face value pure discount bonds of different maturitiesWe can always analyze any fixed income contract with known cash flows into a sum of \$1-face value pure discount bonds of different maturities

6 Example (based on Prob. 3) Let P1 and P2 be the prices of \$1 face value pure discount bonds maturing at time 1 and 2 respectively.Let P1 and P2 be the prices of \$1 face value pure discount bonds maturing at time 1 and 2 respectively. Hence, P1 = \$0.97 = \$1/(1+i 1 ) 1 and P2 = \$0.90 =\$1/(1+i 2 ) 2Hence, P1 = \$0.97 = \$1/(1+i 1 ) 1 and P2 = \$0.90 =\$1/(1+i 2 ) 2 where i 1 and i 2 are 1-year and 2-year yields on pure discount bondswhere i 1 and i 2 are 1-year and 2-year yields on pure discount bonds

7 Example Continued - The price of a 2-year, 6% coupon bond with a face value of \$100 will equal:The price of a 2-year, 6% coupon bond with a face value of \$100 will equal: \$6/(1+i 1 ) 1 + \$106/(1+i 2 ) 2, or = 6 * P1 + 106 *P2 = 6 * \$0.97 + 106 * \$0.90 = \$101.22\$6/(1+i 1 ) 1 + \$106/(1+i 2 ) 2, or = 6 * P1 + 106 *P2 = 6 * \$0.97 + 106 * \$0.90 = \$101.22

8 Example Continued - Hence, a 2-year, 6% coupon bond with a face value of \$100 is equivalent to the sum of six \$1-face value pure discount bonds maturing in one year and one hundred and six \$1-face value pure discount bonds maturing in two yearsHence, a 2-year, 6% coupon bond with a face value of \$100 is equivalent to the sum of six \$1-face value pure discount bonds maturing in one year and one hundred and six \$1-face value pure discount bonds maturing in two years

9 Example Continued What is the relationship between i 1, i 2, and the two-year yield to maturity, i c on the 6% coupon bond?What is the relationship between i 1, i 2, and the two-year yield to maturity, i c on the 6% coupon bond? \$101.22 = \$6/(1+i 1 ) 1 + \$106/(1+i 2 ) 2, = \$6/(1+i c ) 1 + \$106/(1+i c ) 2,\$101.22 = \$6/(1+i 1 ) 1 + \$106/(1+i 2 ) 2, = \$6/(1+i c ) 1 + \$106/(1+i c ) 2,

10 Example Continued Recall that P1 = \$0.97 = \$1/(1+i 1 ) 1, which implies i 1 = 3.093%Recall that P1 = \$0.97 = \$1/(1+i 1 ) 1, which implies i 1 = 3.093% Also, P2 = \$0.90 =\$1 /(1+i 2 ) 2 which implies i 2 = 5.41%Also, P2 = \$0.90 =\$1 /(1+i 2 ) 2 which implies i 2 = 5.41% Should the yield to maturity i c on a 2-year 6% coupon bond be some complicated average of 3.093% and 5.41%?Should the yield to maturity i c on a 2-year 6% coupon bond be some complicated average of 3.093% and 5.41%?

11 Example Completed The yield to maturity on the 2-year 6% coupon bond with face value of \$100 can be calculated as:The yield to maturity on the 2-year 6% coupon bond with face value of \$100 can be calculated as: \$101.22 = \$6/(1+i c ) 1 + \$106/(1+i c ) 2,\$101.22 = \$6/(1+i c ) 1 + \$106/(1+i c ) 2, i c = 5.34%i c = 5.34%

12 Coupon Stripping Example (based on Prob. 4) Let P1 and P2 be the prices of \$1 face value pure discount bonds maturing at time 1 and 2 respectively.Let P1 and P2 be the prices of \$1 face value pure discount bonds maturing at time 1 and 2 respectively. Hence, P1 = \$0.93 = \$1/(1+i 1 ) 1 and P2 = ? =\$1/(1+i 2 ) 2Hence, P1 = \$0.93 = \$1/(1+i 1 ) 1 and P2 = ? =\$1/(1+i 2 ) 2 where i 1 and i 2 are 1-year and 2-year yields on pure discount bondswhere i 1 and i 2 are 1-year and 2-year yields on pure discount bonds

13 Example Continued - The two cash flows of the 2-year, 7% coupon bond (with a face value of \$1000) are:The two cash flows of the 2-year, 7% coupon bond (with a face value of \$1000) are: \$70 at the end of the first year, and \$1070 at the end of the second year\$70 at the end of the first year, and \$1070 at the end of the second year

14 Example Continued - The price of the 2-year, 7% coupon bond with a face value of \$1000 is given as \$985.30, orThe price of the 2-year, 7% coupon bond with a face value of \$1000 is given as \$985.30, or \$985.30 = \$70/(1+i 1 ) 1 + \$1070/(1+i 2 ) 2, = 70* P1 + 1070*P2 = 70* \$0.93 + 1070*P2 = \$65.10 + 1070*P2\$985.30 = \$70/(1+i 1 ) 1 + \$1070/(1+i 2 ) 2, = 70* P1 + 1070*P2 = 70* \$0.93 + 1070*P2 = \$65.10 + 1070*P2

15 Example Completed - If you unbundle the \$70 cash flow and \$1070 cash flow from the coupon bond and sell them separately, then:If you unbundle the \$70 cash flow and \$1070 cash flow from the coupon bond and sell them separately, then: Sale of first payment = 70* P1 = \$65.10Sale of first payment = 70* P1 = \$65.10 Sale of second payment = \$1070*P2 = \$985.30 - \$65.10 = \$920.20Sale of second payment = \$1070*P2 = \$985.30 - \$65.10 = \$920.20

16 Par, premium, and Discount Bonds A coupon bond with its current price equal to its face value (face value is the same as par value) is a par bondA coupon bond with its current price equal to its face value (face value is the same as par value) is a par bond If it is trading below face value, it is a discount bondIf it is trading below face value, it is a discount bond If it is trading above face value, it is a premium bondIf it is trading above face value, it is a premium bond

17 The Relationship between Coupon Rate, Current Yield, and Yield-to-Maturity The yield-to-maturity is the discount rate that makes the present value of the cash flows from the bond equal to the current price of the bondThe yield-to-maturity is the discount rate that makes the present value of the cash flows from the bond equal to the current price of the bond Coupon Rate = Coupon/Face ValueCoupon Rate = Coupon/Face Value Current Yield = Coupon/PriceCurrent Yield = Coupon/Price

18 Bond Pricing Principles For Par Bonds, bond price = face value  ytm = current yield = coupon rateFor Par Bonds, bond price = face value  ytm = current yield = coupon rate For Prem. Bonds, bond price > face value  ytm face value  ytm < current yield < coupon rate For Disc. Bonds, bond price current yield > coupon rateFor Disc. Bonds, bond price current yield > coupon rate

19 How to Remember Principles Imagine that the bond was issued at parImagine that the bond was issued at par –the yield-to-maturity and current yield move in the opposite direction to price –the coupon rate is unchanging This diagram may help:This diagram may help:

20 Yield Relationships 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 600.00800.001000.001200.001400.001600.001800.00 Price Yield coupon_y current_y y_t_m

21 8.4 Reading Bond Listings There are traditions for reporting yields and computing earned interest that need to be understood before tradingThere are traditions for reporting yields and computing earned interest that need to be understood before trading –Coupon bonds are often quoted in terms of the annual rate compounded semi-annually –T-bills are often quoted on a discount basis e.g., a 1 year T-bill has 365 days outstanding, but a year has only 360 days…(it gets nasty)e.g., a 1 year T-bill has 365 days outstanding, but a year has only 360 days…(it gets nasty)

22 Reading Bond Listings Take care that the fractional part of a number is understoodTake care that the fractional part of a number is understood –Is it 16ths, 32nds, 64ths, 100ths or some other convention? Ask price: dealer’s selling priceAsk price: dealer’s selling price Bid price: dealer’s buying priceBid price: dealer’s buying price

23 8.5 Why Yields for the same Maturity Differ –The fundamental building block of bonds is the pure discount bond: Coupon bonds may be viewed as a portfolio of discount bonds –The rule of one price applies to bonds through pure discount bonds –It is a mistake to assume that coupon bonds with the same life have the same yield--their coupon rates differ, leading to a different % mix of discount bonds

24 8.6 The Behavior of Bond Prices Over Time The expected price of pure discount bonds rises exponentially to the face value with time, and the actual price never exceeds parThe expected price of pure discount bonds rises exponentially to the face value with time, and the actual price never exceeds par Coupon bonds are more complex, and their price may exceed their par value, but at maturity they reach their par valueCoupon bonds are more complex, and their price may exceed their par value, but at maturity they reach their par value

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