This paper is intended to equip the candidate with knowledge, skills and attitudes that will enable him/her to value, analyse and determine risk associated with fixed income securities.


A candidate who passes this paper should be able to:

  • Identify various types of fixed income instruments
  • Assess various types of risks associated with fixed income instruments
  • Analyse interest rate volatility using the term structure of interest rate approach
  • Value and analyse fixed income instruments
  • Value bonds using interest rate models
  • Determine the value of bonds using yield and spread analysis
  • Apply pricing strategies and evaluate the risk of fixed income securities
  • Apply the models to determine credit default


  • Overview of fixed income securities
  • Basic features of fixed income securities
  • Types of fixed income securities
  • Bond indenture; affirmative and negative covenants; effect of legal, regulatory and tax considerations on the issuance and trading of fixed income securities
  • Structure of cash flows of fixed income securities; contingency provisions affecting the timing and/or nature of cash flows of fixed income securities
  • Risks associated with fixed income securities

14.2       Markets of fixed income securities: Issuance, trading and funding

  • Classifications of global fixed income markets
  • Interbank offered rates as reference rates in floating-rate debt; mechanisms available for issuing bonds in primary markets; secondary markets for bonds; securities issued sovereign governments, non-sovereign governments, government agencies and supranational entities; debt securities issued corporations; short-term funding alternatives available to banks; repurchase agreements (repos)

14.3       Fundamentals of fixed income valuation

  • Determination of price of the bond given a market discount rate
  • Relationships among a bond’s price, coupon rate, maturity and market discount rate (yield-to- maturity)
  • Bonds price quotation: spot rates; flat price(clean price), accrued interest and the full price of a bond(dirty price)
  • Matrix pricing of a bond
  • Yield measures for fixed-rate bonds, floating-rate notes and money market instruments
  • Term structure of interest rate: pure expectation theory, liquidity preference theory, market segmentation theory; implications of the yield curve for the yield-curve theories
  • Spot curves, yield curve on coupon bonds, par curve and forward curve
  • Forward rates; determination of spot rates from forward rates, forward rates from spot rates and the price of a bond using forward rates; yield spread measures
  • Bond refinancing/refunding

14.4       Fixed income risk and return

  • Return from investing in a fixed-rate bond
  • Bond duration measures: Macaulay duration, modified duration and effective durations, portfolio duration ; money duration of a bond and price value of a basis point (PVBP)
  • Effective duration as a measure of interest rate risk for bonds with embedded options
  • Key rate duration as a measure of sensitivity of bonds to changes in the shape of the benchmark yield curve
  • Effect of a bond’s maturity, coupon, embedded options and yield level to its interest rate risk
  • Bond convexity: approximate convexity; effective convexity; determination of percentage price change of a bond for a specified change in yield, given the bond’s approximate duration and convexity
  • Effect of term structure of yield volatility on the interest rate risk of a bond; relationships among a bond’s holding period return, its duration and the investment horizon
  • Effect of changes in credit spread and liquidity on yield-to-maturity of a bond and how duration and convexity can be used to estimate the price effect of the changes

14.5       Credit risk management

  • Credit risk and credit-related risks affecting corporate bonds; seniority rankings of corporate bonds; potential violation of the priority of claims in a bankruptcy proceeding; corporate issuer credit ratings; issue credit ratings ; rating agency practice of “notching”; risks in relying on ratings from credit rating agencies; components of traditional credit analysis
  • Financial ratios used in credit analysis; credit quality of a corporate bond issuer given key financial ratios of the issuer and the industry
  • Factors influencing the level and volatility of yield spreads; determination of return impact of spread changes; special considerations when evaluating the credit of high yield, sovereign and municipal debt issuers and issues

14.6       The term structure and interest rate dynamics

  • Relationships among spot rates, forward rates, yield to maturity, expected and realised returns on bonds and the shape of the yield curve
  • Forward pricing and forward rate models: determination of forward and spot prices and rates using those models
  • Assumptions concerning the evolution of spot rates in relation to forward rates implicit in active bond portfolio management; the strategy of riding the yield curve
  • Swap rate curve: its use in valuation market participants; determination and interpretation of the swap spread for a default-free bond; the Z-spread; treasury and Euro dollar(TED) spread and London interbank offer rate(LIBOR) – OIS spreads
  • Review of traditional theories of the term structure of interest rates; the implications of each theory to forward rates and the shape of the yield curve
  • Modern term structure models and their use; measuring the bond’s exposure to each of the factors driving the yield curve and how these exposures can be used to manage yield curve risks; maturity structure of yield volatilities and their effect on price volatility

14.7       The arbitrage-free valuation framework

  • Overview of arbitrage-free valuation of a fixed-income instrument
  • Computation of the arbitrage-free value of an option-free, fixed-rate coupon bond
  • Binomial interest rate tree framework: the backward induction valuation methodology and computation of the value of a fixed-income instrument given its cash flow at each node; process of calibrating a binomial interest rate tree to match a specific term structure
  • Pricing using the zero-coupon yield curve and pricing using an arbitrage-free binomial lattice; path wise valuation in a binomial interest rate framework and computation of the value of a fixed-income instrument given its cash flows along each path
  • Monte Carlo forward-rate simulation and its application

14.8       Valuation and analysis of bonds with embedded options

  • Overview of fixed-income securities with embedded options
  • Relationships between the values of a callable or putable bond, the underlying option-free (straight) bond and the embedded option; Use of the arbitrage-free framework to value a bond with embedded options
  • Effect of interest rate volatility on the value of a callable or putable bond
  • Effect of changes in the level and shape of the yield curve on the value of a callable bond
  • Determination of the value of a callable or putable bond from an interest rate tree; option – adjusted spreads (OAS); effect of interest rate volatility on option-adjusted spreads
  • Effective duration of callable, putable and straight bonds; use of one-sided durations and key rate durations to evaluate the interest rate sensitivity of bonds with embedded options
  • Effective convexities of callable, putable and straight bonds
  • Determination of the value of a capped or floored floating-rate bond
  • Defining features of a convertible bond; components of a convertible bond’s value; valuation of convertible bond in an arbitrage-free framework; risk–return characteristics of a convertible bond, straight bond and underlying common

14.9       Credit analysis models

  • Overview of credit analysis models; probability of default, loss given default, expected loss and present value of the expected loss and relative importance of each across the credit spectrum
  • Credit scoring and credit ratings ; ordinal rankings
  • Strengths and weaknesses of credit ratings
  • Structural models of corporate credit risk: reasons for equity being viewed as a call option on the company’s assets; reduced form models of corporate credit risk
  • Reasons for debt being valued as the sum of expected discounted cash flows after adjusting for risk
  • Assumptions, strengths and weaknesses of both structural and reduced form models of corporate credit risk
  • Determinants of the term structure of credit spreads; present value of the expected loss on a bond over a given time horizon
  • Credit analysis required for asset-backed securities
  • Credit analysis of corporate debt

14.10    Emerging issues and trends




TOPIC                                                                                          PAGE

  1. Overview of fixed income securities……………………………8
  2. Markets of fixed income securities, trading and funding……..18
  3. Fundamentals of fixed income valuation……………………….29
  4. Fixed income risk and return………………………………….61
  5. Credit risk management…………………………………….…77
  6. The term structure and interest rate dynamics……………….82
  7. The arbitrage-free valuation framework……………………….99
  8. Valuation and analysis of bonds with embedded options……..109
  9. Credit analysis models………………………………………….117





 Fixed Income Investments are financial assets or securities that promise a fixed rate of return. E.g. Bonds & Preference shares.


This is a fixed income investment issued the Government or a corporate firm.

 Properties of Bonds

Main properties of Bonds

Coupon Rate

Is the interest rate the issuer promises to pay the holder. It will determine the  cash flows that investors receive.

Par Value / Face / Redemption Value

This is the amount that the issuer promises to pay the holder at maturity period. The per value influence the cash flow that investor receives.

Currency denomination

This is the currency in which the bond is issued. There are 2 main types

  1. A bond issued in the USD is known as dollar denominated bond.
  2. The bond issued in any other currency other than the USD is known as non – dollar denominated bond.
  3. Maturity

Is the period it takes for the issuer to pay the final interest + principal amount. Maturity is important because:-

  1. It indicates the period it takes for the holder to receive the final interest + principal amount.
  2. The price volatility of the bond is a function of maturity.
  3. It influences the Yield to Maturity .i.e. the longer t is the higher the YTM & vice versa.

Issuer / Quality of Issuer

He determines the risk that the holder or investors are exposed to. The risk is normally determined the credit rating of the bond. The higher the rating the lower the risk & vice versa.






 The primary market is a market where securities are issued for the first time.

The primary market for debt functions in manners similar to the primary market for equities.

Typically an investment banker is involved in advising the debt issue & in distributing the debt securities to investors.

When the investment banker actually purchases the entire issue & resells it, they are said to have underrated the issue.

This arrangement is stamped as a firm commitment while the deal is stamped a bought deal.

In underwritten offering of debt securities, the underwriter will typically put together a syndicate of other investment bankers to aid in distributing securities.

The underwriters can reduce their risk pre-selling as much of the offering as possible to their institutional clients & hedging the interest rate risk exposure of the issue for the period they anticipate owning the securities.

An alternative is for the investment banker to agree to sell all the issue they can and this is stamped  “doing the offering on a best efforts basis”

In the above described process since the price paid for the issue & the anticipated sale price is determined between the lead investment bank and the issuing company the offering is stamped a negotiated offering.

Auction Process is another Approach:-  An issuer of debt securities determines the size and terms of auction issue and several investment  banks or underwriting syndicates  of multiple investment banks bid on what  interest rate they require to sell it.

The syndicate with the lowest I.R will be awarded the deal.





 Debt securities that make explicit interest payment have 3 sources of return

  • Periodic coupon interest payments made the issuer.
  • The recovery of principal along with any capital gain or loss that occurs when the bond matures.
  • Re – investment income or the income carried from re – investing periodic coupon payment

NB: A good yield measure should consider all the 3 (three) above consideration

  • Current Yield

 This is an annualized rate of return. It’s the simplest of all return measures but offers limited information. This measure looks at just one source of return i.e. the bond annual interest gain or loss or investment income formulae.

Current yield =


Consider a 10 year 1000 per value, 6% semiannual pay bond that is currently trading at Shs. 802.07. Calculate the current yield.


Annual Cash Flow = 6/100 x 100 = 60

Current yield = 60/802.07 x100 = 7.5%





Bond Valuation

 In order to estimate the value of the bond, the following variables should be established

(i) Coupon Rate:- Is the interest rate that the issuer of the bond promises to pay the bond holder. It will determine the cash flow of an investor.

Cash flow (c) =  X par value

(ii) Interest rate / YTM: Is the prevailing interest rate in the market. This variable is used as a discounting rate.

(iii) Maturity (n)- Is the period it takes for the holder of the bond to receive the final cash flow & the principal amount.

(iv) The par value / face value- is the amount at which the bond was initially issued. Also known as face rate / redemption value etc.

General formula

V0 = C (PV1FArn) + PV (PV1Frn)

Where r = YTM, n = Maturity        Pv = Par Value              Vo = Price of the bond

Other factors that might influence the bond value include

(v) Credit Rating- Is the opinion given rating agencies. It indicates the risk that the investors are exposed to when they invest in a fixed income instrument.

The higher the credit rating the lower the risk and vice versa.





Credit Risk is the risk associated with losses from the failure of the borrower to make timely & full payment of the interest & principal

Credit risk has 2 components;

  • Default risk
  • Loss severity

Default risk: probability that a borrower fails to pay interest or repayment when due.

Loss Serenity or Loss given default-Refers to the value of the bond investor will lose if the issuer defaults

There are four main approaches of gauging the credit risk.

  • Credit rating
  • Traditional credit analysis
  • Credit scoring models
  • Credit risk model
  1. Credit rating

Is an opinion given rating agencies about the risk of default in a particular issue of debt security.

Rating process, surveillance & preview

The rating process begins when a rating agency receives a formal request from an entity planning to issue a bond in which it seeks a rating for the bond issue.

Once credit rating has been assigned to a corporate debt, the rating agency monitors the credit quality of the issue & reassigns a different rating

Typically before an issue rating is changed, the rating agency will announce in advance that it is reviewing the issue with the potential for upgrade or downgrade.

In this case, the issue is said to be on a credit watch or rating watch. In addition, rating agencies will issue a rating outlook.






A shift in the yield curve refers to the relative change in the yield for each treasury maturity.

A parallel shift in the yield curve refers to a shift in which the change in yield for all maturities is the same.

A non – parallel shift in the yield curve means that the yield for different maturities does not change the same number of

Historically two types of non – parallel shifts have been observed

(a) A twist in the shape of the yield curve

(b) A change in the humpedness / curvature of the yield curve

A twist in the shape of the yield curve refers to a flattening or steepening of the yield curve

A flattening curve – shape of yield curve has decreased whereas steepening of yield curve – slope of yield curve has increased.

The other type of non-parallel shift is a change in the curvature of humpedness of the Y.C

This type of shift involves the movement of the yield at short maturity & long maturity of the Y.C relative to movement of the yield in the intermediate maturity sector of the Y.C

Such non – parallel shift in the Y.C that change its curvature / humpedness are referred to as butterfly shift

A positive butterfly means that the Y.C. becomes less humped

A negative butterfly – the yield curve becomes more humped




This approach argues that the value of the bond using the traditional rate should be equivalent to the value of the bond using the traditional valuation method basically YTM:

V0 = C (PV1FAr=n=) + PV (PV1Fra =n=)

With the arbitrage free valuation approach, we discount each cash flow using a discount rate that is specific to the maturity of each C.F

The discount rates are called spot rates & can be thought of as the required rates of returns on zero coupon bonds, maturing at various times in future.

The arbitrage free valuation approach simply says that the value of a T-bond based on spot rates must be equal to the value of the parts .i.e. the sum of the pv of all expected CFs. If this isn’t the case, there must be an arbitrage opportunity.

If the bond is selling for more than the sum of present value of its expected CFs an arbitrage will buy the bond and sell the pieces.

If the bond is selling for more than the sum of the PV of the pieces i.e. individual CFs one would buy the pieces packages them to make a bond & sell the bond percentage to earn an arbitrage profit.

The 1st Step: In checking for arbitrage free valuation is to value a coupon bond using the appropriate spot rates.

2nd Step: Compare this value to the market price of the bond.

If the market value isn’t given, determine the theoretical value of the bond using the YTM. If the computed value isn’t equal to the market price, there is an arbitrage profit to be earned buying the lower price alternative & selling the higher price alternative.




Valuing a bond with embedded option using Binomial model

A binomial model is a relatively simple single factor interest rate model that given an assumed level of volatility suggest than interest rates have an equal probability of taking on one of the two possible values in the next period.

An interest rate models assumptions about interest rate volatility along with a set of path with interest rate may follow. This set of possible interest rates path is referred to as interest rate tree.

Binomial Interest rate trees

The set of possible interest rate paths that are used to value bonds with a binomial model is called a binomial interest rate tree.


The following information relates to a 6.5% 100 par bond with maturity of 4 years


Year                     Spot rates

1                           3.5%

2                           4.318%

3                           4.17352%

4                           5.2706%



(a) Determine the value of the bond




Cash flow measures


  • The profitability ratios, turnover ratios & liquidity ratios show some downward trend but provide no definite clue to the Company’s impeding bankruptcy.
  • A study of cash flows from operations reveals that company operations were causing an increasing drain rather than providing cash flows.
  • This led to an increased use of external financing which requires a firm to meet interest payment obligation which further drained the cash flows
  • The firm’s statement of cash flows lists separately cash flows from operating activities, investing & financing activities.
  • Cash flow from operating activities is also known as cash flow from operation. By analyzing individual statement of cash flows creditors can examine specific aspects of business such as:
  1. Source of financing for business operations whether through internal or external sources of funds.
  2. Ability of the firm to meet debt obligation i.e. interest & principal payment.
  3. Ability of the company to finance expansion through cash flow from operating activities.
  4. Ability of the company to pay dividends to shareholders.
  5. Flexibility a business has in financing its operations.


A firm that generates cash flows only selling of its assets, obtaining cash flow from investing or issuing more secure securities cannot keep that for very long.

The future prospect and the ability to meet such obligations, the firm must be able to generate cash flows from operations.

Analysts have reformatted the information from the firm’s income statement and statement of cash flows to obtain what is believed as the best description of the company’s activities.


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