Chapter 8 Valuation of Known Cash Flows Bonds

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

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 buyer • The remaining life of the mortgage is 60 months, with payment of $400. Your required rate of return is 1. 5% / month 3

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

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

Change in Required Rate • If your required rate of return increased to 1. 6% / month 6

Using Present Value Formulas to Value Known Flows • Observe 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 security • The proof is very easy 7

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

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 down • Similarly, i down -> PV up 9

Bond Prices Rise as the Interest Rates Fall • Basic 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 10

Note • Volatile market rates imply volatile market values 11

Finding the Correct Discount Rate • Bond 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 12

Yield Curve • A typical yield curve: 13

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8. 2 The Basics Building Blocks: Pure Discount Bonds • We 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 date 15

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

Pure Discount Bonds • The 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: 17

Pure Discount Bonds • 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 18

Pure Discount Bonds • Example – 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? 19

Pure Discount Bonds 20

8. 3 Coupon Bonds, Current Yield, and Yield to Maturity • A 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 21

Coupon Rate • The 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 22

Par, premium, and Discount Bonds • A 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 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!) 23

Bonds Trading at Par • 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: 24

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 bond • An excellent way to compute the ytm is given in Chapter 4 25

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 -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 26

First Solution Method 27

Second Solution Method 28

Conclusion • The 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 bond – cash flows are then evaluated using them 29

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 YTM • We use the financial calculator, but a numerical method was provided in chapter 4 for this class of problems 30

The YTM of the Coupon Bond 31

Observation • The yield to maturity on the 3 -year pure discount bond was 7. 28% and the yieldto-maturity on the 3 -year coupon bond was 7. 10% • The yield-curve for default-free bonds is not a unique value 32

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

Proof of Relationship between YTM and Current Yield • For 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 34

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Proof of Relationship between Current and Coupon Yields • For 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 36

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How to Remember Principles • Imagine 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: 38

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High Yield T-Bond Funds • Yield 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 40

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Clarification • The last example used a flat yield curve • Let 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 42

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Investment Implications • Assume a 20 -year bond with a coupon rate of 6% • Purchase for $1016. 54 when the lower curve prevails • When yield curve rises, the bond is worth only $814. 05 – This is a massive capital risk • Additionally, long-term rates are more volatile than short-term rates 44

8. 4 Reading Bond Listings • There 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) 45

$Reading Bond Listings • Take care that the fractional part of a number is$

Reading Bond Listings • Take care that the fractional part of a number is understood – Is it 16 ths, 32 nds, 64 ths, 100 ths or some other convention? • Ask price: dealer’s selling price • Bid price: dealer’s buying price 46

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 47

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 par • Coupon bonds are more complex, and their price may exceed their par value, but at maturity they reach their par value 48

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

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A Different View of the Yield Curve • The dynamics of the short-term and longterm interest rates are shown next • Note 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) 51

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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 sedate • The path is similar to the x-component of the random flight of a moth to a flame 53

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