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**Session 3 Introduction to the Valuation of Debt Securities**

Fixed Income Analysis Session 3 Introduction to the Valuation of Debt Securities

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**Introduction to the Valuation of Debt Securities by Frank J. Fabozzi**

PowerPoint Slides by David S. Krause, Ph.D., Marquette University Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express permission of the copyright owner is unlawful. Request for futher information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages caused by the use of these programs or from the use of the information contained herein.

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**Chapter 5 Introduction to the Valuation of Debt Securities**

Major learning outcomes: The valuation – which is the best process of determining the fair value of a fixed financial asset: Single discount rate Multiple discount rates This process is also called pricing or valuing. Only option-free bond valuation is presented in this chapter.

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Key Learning Outcomes Describe the fundamental principles of bond valuation. Explain the steps in the valuation process (i.e., estimate expected cash flows, determine an appropriate discount rate or rates, and compute the present value of the cash flows). Define a bond’s cash flows. Describe the difficulties of estimating the expected cash flows for some types of bonds. Compute the value of a bond, given the expected cash flows and the appropriate discount rates. Explain how the value of a bond changes if the discount rate increases or decreases. Explain how the price of a bond changes as the bond approaches its maturity date. Compute the value of a zero-coupon bond.

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Key Learning Outcomes Compute the value of a bond that is between coupon payments. explain the deficiency of the traditional approach to bond valuation. Explain the arbitrage-free bond valuation approach and the role of Treasury spot rates in the valuation process. Explain how the process of stripping and reconstitution forces the price of a bond towards its arbitrage-free value. Demonstrate how a dealer can generate an arbitrage profit if a bond is mispriced. Compute the price of the bond given the term structure of default free spot rates and the term structure of credit spreads. Explain the basic features common to models used to value bonds with embedded options.

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Valuation Valuation is the process of determining the fair value of a financial asset. The process is also referred to as ‘‘valuing’’ or ‘‘pricing’’ a financial asset. The fundamental principle of financial asset valuation is that its value is equal to the present value of its expected cash flows. This principle applies regardless of the financial asset. Thus, the valuation of a financial asset involves the following three steps: Step 1: Estimate the expected cash flows. Step 2: Determine the appropriate interest rate or interest rates that should be used to discount the cash flows. Step 3: Calculate the present value of the expected cash flows found in step 1 using the interest rate or interest rates determined in step 2.

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Estimating Cash Flows Cash flow is the expected future amount(s) to be received from an investment. The cash flows for a simple bond are the coupon or interest payments and the principal value. Cash flows for a bond become more complicated when: The issuer has the option to change the contractual due date for the payment of the principal (callable, putable, mortgage-backed, and asset-backed securities); The coupon rate is reset periodically by a formula based on come value or reference rates, prices, or exchange rates (floating-rate securities); and The investor has the choice to convert or exchange the bond into common stock (convertible bonds).

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Estimating Cash Flows Whether or not callable, putable, mortgage-backed, and asset-backed securities are exercised early is determined by the movement of interest rates; If rates fall far enough, the issuer will refinance If rates rise far enough, the lender has an incentive to refinance Therefore, to properly estimate cash flows it is necessary to incorporate into the analysis how future changes in interest rates and other factors might affect the embedded options.

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Estimating Cash Flows For any fixed income security which neither the issuer nor the investor can alter the payment of the principal before its contractual due date, the cash flows can easily be determined assuming that the issuer does not default. The difficulty in determining cash flows arises for securities where either the issuer or the investor can alter the cash flows, or the coupon rate is reset by a formula dependent on some reference rate, price, or exchange rate.

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Discount Rates On-the-run Treasury yields are viewed as the minimum interest rate an investor requires when investing in a bond. The risk premium or yield spread over the interest rate on a Treasury security investors require reflects the additional risks in a security that is not issued by the U.S. government. For a given discount rate, the present value of a single cash flow received in the future is the amount of money that must be invested today that will generate that future value.

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**The Appropriate Discount Rate**

Interest rate and yield are used interchangeably. The minimum interest rate that a U.S. investor should demand is the yield on a Treasury security. This is why the Treasury market is watched closely. For basic or traditional valuation, a single interest rate is used to discount all cash flows; however, the proper approach to valuation uses multiple interest rates each specific to a particular cash flow and time period.

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**Discounting the Cash Flows**

The present value of the future expected cash flows is computed by discounting by the appropriate interest rate. The present value of all future cash flows is summed to compute the net present value of the bond. (Be able to use PV tables, financial calculators, spreadsheets, or long-hand to compute bond values)

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**Present Value Properties**

For a given discount rate, the present value of a single cash flow received in the future is the amount of money that must be invested today that will generate that future value. The present value of a cash flow will depend on when a cash flow will be received (i.e., the timing of a cash flow) and the discount rate (i.e., interest rate) used to calculate the present value The sum of the present values for a security’s expected cash flows is the value of the security. The lower the present value, the further into the future the cash flow will be received. The higher the discount rate, the lower a cash flow’s present value and since the value of a security is the sum of the present value of the cash flows, the higher the discount rate, the lower a security’s value. The price/yield relationship for an option-free bond is convex.

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**Present Value Properties**

For any given discount rate, the further into the future a cash flow is received, the lower is its present value. The value of a bond is equal to the present value of the coupon payments plus the present value of the maturity value. Exhibit 1 shows the relationship between bond price and discount rate is convex (bowed in from the origin). We’ll later see how this convexity has implication for the price volatility of a bond.

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**Price/Discount Relationship**

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**Relationship Between Coupon Rate, Discount Rate, and Price Relative to Par Value**

The following relationships hold: If the coupon rate = market yield, the price will = par value. If coupon rate < market yield, the price < par value (discount). If coupon rate > market yield, the price > par value (premium).

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**Bond Pricing Between Coupon Periods**

When a bond is purchased between coupon periods, the buyer pays a price that includes accrued interest, called the full price or dirty price. The clean price or simply price of a bond is the full price minus accrued interest. In computing accrued interest, day count conventions are used to determine the number of days in the coupon payment period and the number of days since the last coupon payment date. The traditional valuation methodology is to discount every cash flow of a security by the same interest rate (or discount rate), thereby incorrectly viewing each security as the same package of cash flows.

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**Valuing a Bond Between Coupon Payments**

For coupon bonds, it is likely that a purchase or sale is going to occur on a non-interest payment date. The amount that the buyer pays the seller in such cases is the present value of the cash flow with the next payment encompassing two components: Interest earned by seller Interest earned by the buyer The interest earned by the seller is the interest that has accrued – called accrued interest. At the time of purchase, the buyer must compensate the seller for the accrued interest.

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**Valuing a Bond Between Coupon Payments**

When the price of a bond is computed using the traditional present value approach, the accrued interest is embodied in the price – this is referred to as the full or ‘dirty’ price. From the full price, the accrued interest must be deducted to determine the price of the bond, referred to as the clean price.

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**Valuing a Bond Between Coupon Payments – Full Price**

To compute the full price of a bond between coupon payment dates it is necessary to determine the fractional periods between the settlement date and the next coupon payment date. w periods = (days between settlement date and next coupon payment date)/days in coupon period Then the present value of the expected cash flow to be received t periods from now using discount rate I assuming the first coupon payment is w periods from now: Present value t = expected cash flow / (1+i)t-1+w

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**Valuing a Bond Between Coupon Payments – Full Price**

This is called the “Street method” for calculating the present value of a bond purchased between payment dates. The example in the book computes the full price (which includes the accrued interest the buyer is paying the seller).

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**Change in Value as Bond Moves Toward Maturity**

As a bond gets closer to maturity, its value changes: Value decreases over time for bonds selling at a premium. Value increases over time for bonds selling at a discount. Value is unchanged if a bond is selling at par. At maturity at bond is worth par value so there is a “pull to par value” over time. Exhibit 2 shows the time effect on a bond’s price based on the years remaining until maturity.

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**Change in Value as Bond Moves Toward Maturity**

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The Price Path of a Bond What would happen to the value of a bond if its required rate of return remained at 10%, or at 13%, or at 7% until maturity? ”Pull to par” Years to Maturity 1,372 1,211 1,000 837 775 kd = 7%. kd = 13%. kd = 10%. VB

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**Traditional and Arbitrage-Free Approaches to Bond Valuation**

The traditional valuation methodology is to discount every cash flow of a security by the same interest rate (or discount rate), thereby incorrectly viewing each security as the same package of cash flows. The arbitrage-free approach values a bond as a package of cash flows, with each cash flow viewed as a zero-coupon bond and each cash flow discounted at its own unique discount rate.

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**Valuation: Traditional versus Arbitrage-Free Approaches**

Traditional approach – This is also called the relative price approach. A benchmark or similar investment’s discount rate is used to value the bond’s cash flows (i.e. 10-year Treasury bond). The flaw is that it views each security as the same package of cash flows and discounts all of them by the same interest rate. It will provide a ‘close’ approximation, but not necessarily the most accurate.

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**Valuation: Traditional versus Arbitrage-Free Approaches**

Arbitrage-free pricing approach – Assumes that no arbitrage profits are possible in the pricing of the bond. Each of the bond’s cash flow (coupons and principal) is priced separately and is discounted at the same rate as the corresponding zero-coupon government bond. Since each bond’s cash flow is known with certainty, the bond price today must be equal to the sum of each of its cash flows discounted at the corresponding – or arbitrage is possible.

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Arbitrage Arbitrage is the simultaneous buying and selling of an asset at two different prices in two different markets. The arbitrageur buys low in one market and sells for a higher price in another. The fundamental principle of finance is the “law of one price.” If arbitrage is possible, it will be immediately exploited by arbitrageurs. If a synthetic asset can be created to replicate anther asset, the two assets must be priced identically or else arbitrage is possible.

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**Arbitrage-Free Valuation**

The Treasury zero-coupon rates are called Treasury spot rates. The Treasury spot rates are used to discount the cash flows in the arbitrage-free valuation approach. To value a security with credit risk, it is necessary to determine a term structure of credit rates. Adding a credit spread for an issuer to the Treasury spot rate curve gives the benchmark spot rate curve used to value that issuer’s security. Valuation models seek to provide the fair value of a bond and accommodate securities with embedded options.

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**Arbitrage-Free Valuation**

In this approach, the bond price will reflect its arbitrage-free price. Each cash flow is priced separately and is discounted at the same rate as the corresponding government issue zero-coupon bond. Since each bond cash flow is known with certainty, the bond price today must be equal to the sum of each of its cash flows discounted at the corresponding risk free rate – (i.e. the corresponding government security). If this were not the case, arbitrage profits would be possible.

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**Arbitrage-free Bond Valuation**

By viewing a bond as a package of zero-coupon bonds (Exhibit 4), it is possible to value the bond and the package of zero-coupon bonds. If they are priced differently, arbitrage profits would be possible. To implement the arbitrage-free approach, it is necessary to determine the interest rate that each zero-coupon for each maturity. The Treasury spot rate is used to discount a default-free cash flow with the same maturity. The value of a bond based on spot rates if called the arbitrage-free value.

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**Arbitrage-free Bond Valuation**

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Coupon Bond Example Take a 3-year 10% coupon bond with face value = 1000, assuming annual coupon payments: Spot rates: r1=10%, r2=12%, r3=14% Yield-to-Maturity (IRR of cash flows)

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**Zero Coupon Bond Example**

Price of 3-year zero coupon bond with face value = 1000 Spot rates: r1=10%, r2=12%, r3=14% Yield-to-Maturity

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**Bond Valuation Using Treasury Spot Rates**

Exhibit 5 takes a 10-year, 8% semi-annual coupon bond and creates 20 zero-coupon bonds with different maturities. If given the spot rate (annual discount rate) for each maturity, it is possible to compute the individual present values of the 20 bonds. The summation of the present values is equal to the arbitrage-fee bond value. Exhibit 6 uses a 10-year, 4.8% coupon bond.

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Bond Valuation

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Bond Valuation

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**Why Use Treasury Spot Rates Rather Than the Yield on an 8% 10-year Bond?**

Exhibit 7 takes the 10-year, 8% semi-annual coupon bond in the example and discounts all of the cash flows at 6% - the current yield for a 10-year bond. The present value is $ versus a present value of $ for the sum of the 20 zero-coupon bonds (discounted at the spot rates). The result of these different approaches would result in an arbitrage opportunity because it would be possible to buy the bond for $ and “strip” it to credit 20 zero-coupon bonds worth a combined $ The sum of present value of the arbitrage profits would be $0.384, which could amount to enormous profits for the arbitrageur. On tens of millions of dollars, this would be very profitable!

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Bond Valuation

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**Use of Treasury Spot Rates**

Exhibit 8 and 9 show the opportunities for arbitrage profit. Note: in order to create profits for the 4.8% bond, it would be necessary to “reconstitute” stripped bonds. The process of stripping and reconstituting assures that the price of a Treasury will not depart materially from its arbitrage-free value. The Treasury spot rates can be used to value any default-free security.

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Bond Valuation

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Bond Valuation

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**Credit Spreads and the Valuation of Non-Treasury Securities**

For a non-Treasury bond, the theoretical value is slightly more difficult to determine. The value of a non-Treasury bond is found by discounting the cash flows by the Treasury spot rates plus a yield spread to reflect the additional risks.

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**Credit Spreads and the Valuation of Non-Treasury Securities**

One approach is to discount the non-Treasury bond by the appropriate maturity Treasury spot rate plus a constant credit spread. The problem with this approach is that the credit spread might be different depending upon when the cash flow is received. Credit spreads typically increase with maturity there is a term structure of credit spreads.

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**Valuation of Non-Treasury Securities (Embedded Options)**

To value a security with credit risk, it is necessary to determine a term structure of credit rates. Adding a credit spread for an issuer to the Treasury spot rate curve gives the benchmark spot rate curve used to value that issuer’s security. Valuation models seek to provide the fair value of a bond and accommodate securities with embedded options. The common valuation models used to value bonds with embedded options are the binomial model and the Monte Carlo simulation model. The binomial model is used to value callable bonds, putable bonds, floating-rate notes, and structured notes in which the coupon formula is based on an interest rate.

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Valuation Models The two methods presented in this chapter (traditional and arbitrage-fee) assumed no embedded options. Treasury and non-Treasury bonds without embedded options should be valued using the arbitrage-free method. Binomial and Monte Carlo simulation models are used to value bonds with embedded options.

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**Binomial and Monte Carlo Bond Valuation Features**

They generate Treasury spot rates and they make assumptions about the expected volatility of short-term interest rates – critical to both models. Based on volatility assumptions, different “branches” and “paths” are generated. The models are calibrated to the U.S. Treasury market. Rules are developed to determine when an issuer/borrower will exercise embedded options. Using models like these expose the valuation to modeling risk – the risk that the output of the model is incorrect because the underlying assumptions are incorrect.

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**Binomial and Monte Carlo Bond Valuation Features**

The Monte Carlo simulation model is used to value mortgage-backed and certain asset-backed securities. The user of a valuation model is exposed to modeling risk and should test the sensitivity of the model to alternative assumptions.

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