1 Chapter 8: Valuation of Known Cash Flows: Bonds Copyright © Prentice Hall Inc 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 8.1 Using Present Value Formulas to Value Known Flows You have been offered the opportunity to purchase a mortgage. It was originally part of a creative financing package where the original owner financed the buyerYou have been offered the opportunity to purchase a mortgage. It was originally part of a creative financing package where the original owner financed the buyer The remaining life of the mortgage is 60 months, with payment of $400. Your required rate of return is 1.5% / monthThe remaining life of the mortgage is 60 months, with payment of $400. Your required rate of return is 1.5% / month

4 Calculation Using the present value of an annuity formula discussed in chapter 4, you will pay no more thanUsing the present value of an annuity formula discussed in chapter 4, you will pay no more than

5 Financial Calculator Alternatively, using your financial calculator (remember to set the correct defaults) you obtainAlternatively, using your financial calculator (remember to set the correct defaults) you obtain

6 Change in Required Rate If your required rate of return increased to 1.6% / monthIf your required rate of return increased to 1.6% / month

7 Using Present Value Formulas to Value Known Flows Observe that the maximum you would pay for the bond has decreasedObserve that the maximum you would pay for the bond has decreased An increase in the required rate of return always leads to a decrease in the value of a fixed income securityAn increase in the required rate of return always leads to a decrease in the value of a fixed income security The proof is very easyThe proof is very easy

8 Bond Prices Rise as the Interest Rates Fall Write the PV of the fixed income security as the sum termsWrite the PV of the fixed income security as the sum terms

9 Bond Prices Rise as the Interest Rates Fall If i goes up, 1+i goes up, 1/(1+i) goes down for i > -1, (1/(1+i)) j goes down for i > 0. So if the payments are positive, then the sum must also go downIf i goes up, 1+i goes up, 1/(1+i) goes down for i > -1, (1/(1+i)) j goes down for i > 0. So if the payments are positive, then the sum must also go down Similarly, i down -> PV upSimilarly, i down -> PV up

10 Bond Prices Rise as the Interest Rates Fall Basic principle in evaluating known flowsBasic principle in evaluating known flows –A change in market interest rates causes a change in the opposite direction in the market values of all existing contracts promising fixed payments in the future

11 Note Volatile market rates imply volatile market valuesVolatile market rates imply volatile market values

12 Finding the Correct Discount Rate Bond analysis is not as easy as this analysis appears to implyBond analysis is not as easy as this analysis appears to imply –We need an interest rate to use in the formula –We saw in Chapter 2 that interest rates are a function of time-to-maturity –Two default-free bonds with identical maturities may have different YTMs

13 Yield Curve A typical yield curve:A typical yield curve:

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The Basics Building Blocks: Pure Discount Bonds We can always analyze any fixed income contract into a sum of pure discount bondsWe can always analyze any fixed income contract into a sum of pure discount bonds A pure discount bond is a security that promises to pay 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 promises to pay a specified single cash payment (face value or par value) at a specified date called its maturity date

16 Pure Discount Bonds NoteNote –There is no cash flow associated with interest –Pure discount bonds are purchased at a discount from their face or par value

17 Pure Discount Bonds The pure discount bond is an example of the present value of a lump sum equation we analyzed in Chapter 4The pure discount bond is an example of the present value of a lump sum equation we analyzed in Chapter 4 Solving this, the yield-to-maturity on a pure discount bond is given by the relationship:Solving this, the yield-to-maturity on a pure discount bond is given by the relationship:

18 Pure Discount Bonds In this equation,In this equation, –P is the present value or price of the bond –F is the face or future value –n is the investment period –i is the yield-to-maturity

19 Pure Discount Bonds ExampleExample –You can purchase a pure discount bond for $9,000, and it matures in two years with a face value of $10,000 –What is the ytm?

20 Pure Discount Bonds

Coupon Bonds, Current Yield, and Yield to Maturity A coupon bond obligates the issuer toA coupon bond obligates the issuer to –make periodic payments of interest (called coupon payments) to the bond holder until the bond matures –at which time the face value of the bond is also paid to the bond holder –and the contract is satisfied

22 Coupon Rate The coupon rate is the interest rate applied to the face value to compute the coupon paymentThe coupon rate is the interest rate applied to the face value to compute the coupon payment –A bond with a face value of $1,000 and a coupon rate of 10% pays an annual coupon of $100 –At maturity, the payment is $1,000+$100

23 Par, premium, and Discount Bonds A coupon bond with its current price equal to its par value is a par bondA coupon bond with its current price equal to its par value is a par bond If it is trading below par it is a discount bondIf it is trading below par it is a discount bond If it is trading above par it is a premium bond (not to be confused with the U.K. lottery bond of the same name!)If it is trading above par it is a premium bond (not to be confused with the U.K. lottery bond of the same name!)

24 Bonds Trading at Par Bond Pricing Principle #1: (Par Bonds)Bond Pricing Principle #1: (Par Bonds) –If a bond’s price equals its face value, then its yield-to-maturity = current yield = coupon rate. Proof:

25 Coupon Bonds, 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 An excellent way to compute the ytm is given in Chapter 4An excellent way to compute the ytm is given in Chapter 4

26 Using Pure Discount Bonds to Value other Bonds Value a bond that pays its $100 coupon at the end of each year for 3-years, and its par value of $1,000 in 3-yearsValue a bond that pays its $100 coupon at the end of each year for 3-years, and its par value of $1,000 in 3-years –You have discovered three pure discount bonds (each with a $1,000 par value) that mature in 1, 2, and 3 years, and that are trading at $960, $890, and $810 respectively

27 First Solution Method

28 Second Solution Method

29 Conclusion The first method uses the fact that a coupon bond is the sum of pure discount bondsThe first method uses the fact that a coupon bond is the sum of pure discount bonds –it is fast and direct The second method first determines the yields-to-maturity of each discount bondThe second method first determines the yields-to-maturity of each discount bond –cash flows are then evaluated using them

30 The YTM of the Coupon Bond We have the price of the coupon bond, and the timing and magnitude of its future cash flows, so we can determine its YTMWe have the price of the coupon bond, and the timing and magnitude of its future cash flows, so we can determine its YTM We use the financial calculator, but a numerical method was provided in chapter 4 for this class of problemsWe use the financial calculator, but a numerical method was provided in chapter 4 for this class of problems

31 The YTM of the Coupon Bond

32 Observation The yield to maturity on the 3-year pure discount bond was 7.28% and the yield- to-maturity on the 3-year coupon bond was 7.10%The yield to maturity on the 3-year pure discount bond was 7.28% and the yield- to-maturity on the 3-year coupon bond was 7.10% The yield-curve for default-free bonds is not a unique valueThe yield-curve for default-free bonds is not a unique value

33 Bond Pricing Principle #2 & 3 Bond Principle # 2: Premium BondsBond Principle # 2: Premium Bonds bond price > face value  ytm face value  ytm < current yield < coupon rate Bond Principle # 3: Discount BondsBond Principle # 3: Discount Bonds bond price current yield > coupon rate

34 Proof of Relationship between YTM and Current Yield For coupon bonds, we have the following relationshipsFor coupon bonds, we have the following relationships –Note the (sensible) restrictions on the variable ranges –Note that 1/((1+i)^n - 1) is always positive

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36 Proof of Relationship between Current and Coupon Yields For coupon bonds, we have the following relationship derived from the bond formulaFor coupon bonds, we have the following relationship derived from the bond formula –Note that the differences between the reciprocals have the same sign, so the actual differences also have the same sign –Note that size relationship is determined by the discount factor which is always < 1

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

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40 High Yield T-Bond Funds Yield curves with large positive slopes make longer-term T-bonds tempting because, like T-bills, they are default-freeYield curves with large positive slopes make longer-term T-bonds tempting because, like T-bills, they are default-free –The above diagram was based on: par = $1000, coupon = $100, n = 10-years, flat –Observe the large effect of modest changes in interest on capital –A close up is given below

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42 Clarification The last example used a flat yield curveThe last example used a flat yield curve Let us look at an example withLet us look at an example with –short-term rates remaining fixed –longer-term rates rising on increased expectation of a general rise in interest rates

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44 Investment Implications Assume a 20-year bond with a coupon rate of 6%Assume a 20-year bond with a coupon rate of 6% Purchase for $ when the lower curve prevailsPurchase for $ when the lower curve prevails When yield curve rises, the bond is worth only $814.05When yield curve rises, the bond is worth only $ –This is a massive capital risk Additionally, long-term rates are more volatile than short-term ratesAdditionally, long-term rates are more volatile than short-term rates

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 364 days outstanding, but a year has only 360 days…(it gets nasty)e.g., a 1 year T-bill has 364 days outstanding, but a year has only 360 days…(it gets nasty)

46 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

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

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

49 Bond Price Trajectory The following diagram shows the dynamic nature of the yield curve as it passes through timeThe following diagram shows the dynamic nature of the yield curve as it passes through time –Think of a yield curve as constant time cross-section of a yield surface in time maturity rate space

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51 A Different View of the Yield Curve The dynamics of the short-term and long-term interest rates are shown nextThe dynamics of the short-term and long-term interest rates are shown next Note that the two rates track each other somewhat, but the long-term rates have a little more volatilityNote that the two rates track each other somewhat, but the long-term rates have a little more volatility –the processes are synthetic, so don’t read too much into them (granular, smoothed)

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53 The Non-Stationary Price Dynamics of a Maturing Bond The following diagram demonstrates that the price history of a long-term coupon bond is quite dynamic in its early days, but as the bond matures price movements becomes much more sedateThe following diagram demonstrates that the price history of a long-term coupon bond is quite dynamic in its early days, but as the bond matures price movements becomes much more sedate The path is similar to the x-component of the random flight of a moth to a flameThe path is similar to the x-component of the random flight of a moth to a flame

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