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%
100 100 1100
Price = + 2 + 3 = 913.1
(1.10) (1.12) (1.14)
– Yield-to-Maturity (IRR of cash flows)
100 100 1100
913.1 = + +
(1 + y ) (1 + y ) (1 + y ) 3
2
100 100 1100
= + +
(1.137 ) (1.137 ) (1.137 ) 3
2
y = 13.7%

21. Zero Coupon Bond Example
• Price of 3-year zero coupon bond with face value =
1000
– Spot rates: r1=10%, r2=12%, r3=14%
1000
Price = = 675
(1.14) 3
– Yield-to-Maturity
1000
675 =
(1 + y ) 3
y = 14%

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 $114.8775 versus a present value of
$115.2621 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 $114.8775 and “strip” it to credit 20 zero-coupon bonds worth a
combined $115.2621
– 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.