1. Introduction to the Valuation of Debt Securities by Frank J. Fabozzi
PowerPoint Slides by
David S. Krause, Ph.D., Marquette University
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2. 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.

3. 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.

4. Estimating Cash Flows
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).

5. 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 borrower 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.

6. 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.

7. 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.

8. 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.

9. 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

10. 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).

11. 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.

12. Change in Value as Bond Moves Toward Maturity

13. 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.

14. 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.

15. 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.

16. 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.

17. 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.

18. 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.

19. Arbitrage-free Bond Valuation

20. 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)

21. 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

22. Bond Valuation

23. Bond Valuation

24. 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!

25. Bond Valuation

26. 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.

27. Bond Valuation

28. Bond Valuation

29. 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.

30. 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.

31. 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.

32. 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.

33. 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.

34. 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.