Fixed Income Valuation PDF

S2.Book Chapter 3: Introduction to fixed Income Valuation 1. Introduction to Fixed-Income Valuation The fixed-income market plays a central role in providing financing for businesses and governments. It is larger than the equity market in terms of out...

S2.Book Chapter 3: Introduction to fixed Income Valuation 1. Introduction to Fixed-Income Valuation The fixed-income market plays a central role in providing financing for businesses and governments. It is larger than the equity market in terms of outstanding market value, making it an essential investing avenue for institutions such as pension funds, mutual funds, insurance companies, and sovereign wealth funds. Retirees also rely on fixed-income securities for a stable income stream. Understanding how to value fixed-income securities, particularly option-free, fixed-rate bonds, is critical for investors, issuers, and analysts. This chapter focuses primarily on such bonds but also touches on floating-rate notes and money market instruments. The chapter covers: Basic bond valuation, including pricing bonds using market discount rates and spot rates. Bond price and yield calculations. Yield curves and forward rates. Yield spread measures, including the relationship between risk and higher yields. 2. Bond Prices and the Time Value of Money Bond pricing is fundamentally about discounted cash flow analysis. Bonds pay periodic interest (coupon) and return the principal at maturity. 💬 Bond's price = the present value of all future cash flows, discounted at the market discount rate, which reflects the return investors demand given the bond's risk. 2.1 Bond Pricing with a Market Discount Rate A bond’s price is calculated as the present value of its expected cash flows. For a traditional bond, these cash flows consist of regular coupon payments and the repayment of the principal (face value) at maturity. The market discount rate is the rate of return required by investors given the risk of the investment in the bond. S2.Book Chapter 3: Introduction to fixed Income Valuation 1  Bond Price Formula: For a bond with N periods to maturity: Example: For a bond with a 4% coupon rate, 5 years to maturity, and a market discount rate of 6%, the price is 91.575 (per $100 of face value). This bond is trading at a discount because its coupon rate (4%) is less than the market rate (6%). You are essentially receiving smaller cash flows than the market expects, so the bond's price is lower. Key Relationships: The price of a fixed-rate bond, relative to par value, depends on the relationship of the coupon rate to the market discount rate. Coupon rate < Market discount rate: The bond trades at a discount. Coupon rate > Market discount rate: The bond trades at a premium. Coupon rate = Market discount rate: The bond trades at par value. 2.2 Yield-to-Maturity (YTM) YTM = the internal rate of return (IRR) on a bond, assuming the bond is held to maturity, the issuer makes all payments on time, and coupon payments are reinvested at the same rate. It’s essentially the return an investor can expect from holding the bond, and it takes into account both coupon payments and any capital gains or losses due to differences between the purchase price and par value. YTM formula: Solve for r in the bond pricing equation using the market price. Example If a bond is priced at $105 but has a 5% coupon rate, the YTM will be less than 5% because the investor is paying a premium. The YTM is calculated by solving for the rate rrr in the bond pricing formula. This can be done using a financial calculator or trial-and-error. YTM Assumptions: 1. The bond is held to maturity. 2. The issuer makes all payments in full and on time. 3. Coupon payments are reinvested at the same YTM. S2.Book Chapter 3: Introduction to fixed Income Valuation 2  2.3 Relationships Between Bond Price and Bond Characteristics Four key relationships govern bond prices: 1. Inverse relationship: Bond prices decrease as market discount rates increase. 2. Convexity effect: A decrease in discount rates leads to a larger percentage increase in price than an equivalent increase in rates would lead to a price decrease. 3. Coupon effect: Lower-coupon bonds are more price-sensitive (volatile) than higher-coupon bonds when discount rates change. 4. Maturity effect: Longer-term bonds experience larger price changes than shorter-term bonds for the same rate change, although there are rare exceptions for low-coupon bonds. Demonstrate how bond prices respond to changes in yield Show the convexity of bon price changes when yield vary The constant-yield price trajectory illustrates the change in the price of a fixed-income bond overy time. This trajectory shows the “pull to par” effect on the price of a bond trading at a premium or a discount to par value. If the issuer does not default, the price of a bond approaches par value as its time-to-maturity approaches zero. S2.Book Chapter 3: Introduction to fixed Income Valuation 3  3. Prices and Yields: Conventions for Quotes and Calculations In practice, there’s a difference between the quoted price (clean or flat price) and the price paid (dirty or full price) due to accrued interest. 3.1 Flat Price, Accrued Interest, and Full Price When a bond trades between coupon payment dates, the buyer pays the flat price (clean price) plus accrued interest. 💬 Flat Price = The quoted or "clean" price of a bond, excluding accrued interest. 💬 Accrued interest = the portion of the next coupon payment that the seller has earned but has not yet received, since the last coupon payment date. Accrued Interest Formula: Where: Accrued interest: counting days The two most common conventions to count days in bond markets: 30/360 is common for corporate bonds Actual/actual is common for government bonds. 💬 The full price (also called the dirty price)= the bond's quoted price (or flat price, also known as the clean price) plus this accrued interest. S2.Book Chapter 3: Introduction to fixed Income Valuation 4  This means the buyer pays the seller the flat price of the bond plus the accrued interest to cover the coupon the seller earned but has not yet received Example For a bond with a 4.375% coupon rate, paying semi-annually, if 43 days have passed in a 184-day coupon period, the accrued interest is: If the flat price is 101, the full price will be: 3.2 Matrix Pricing 💬 Matrix pricing = used to estimate the price of bonds that are not actively traded by comparing them with similar bonds that do trade. These comparable bonds should have similar maturities, coupon rates, and credit quality. Example Suppose you need to price a 4% coupon, 3-year bond. You identify two similar bonds: 2-year bond with a 3% coupon, priced at 98.50. 5-year bond with a 4% coupon, priced at 99.125. By calculating the average yield of these bonds and using linear interpolation, you can estimate the yield and price of the illiquid bond. S2.Book Chapter 3: Introduction to fixed Income Valuation 5  Matrix pricing is a method used to estimate the price of illiquid or newly issued bonds. This technique involves using the quoted prices of more frequently traded comparable bonds to estimate the market discount rate and price Yield Measures for Fixed-Rate Bonds Investors use standardized yield measures to allow for comparison between bonds with varying maturities. maturity >1 year =annualized and compounded yield-to- maturity is used. maturity < 1 year = annualized but not compounded. An annualized and compounded yield on a fixed6rate bond depends on the periodicity of the annual rate. The periodicity of the annual market discount rate for a zero-coupon bond is arbitrary because there are no coupon payments. The effective annual rate helps to overcome the problem of varying periodicity. It assumes there is just one compounding period per year. Semiannual bond equivalent yield Another way to overcome a problem of varying periodicities is to calculate a semiannual bond equivalent yield (i.e., a YTM based on a periodicity of two). General formula to convert yields based on different periodicities: where APR is the annual percentage rate and m and n are the number of payments/compounding periods per year, respectively Other Yield Measures S2.Book Chapter 3: Introduction to fixed Income Valuation 6  Yield measures for floating-rate Notes Floating -rate notes (FRN s) differ from fixed-rate bonds in that their interest payments vary based on a reference rate. FRN s aim to offer investors securities with less market price risk in volatile interest rate environments. Reference Rate The base interest rate, often a short-term money market rate like LIBOR (determined at the beginning of the period, and the interest payment is made at the end of the period) Quoted Margin The fixed spread added to the reference rate to determine the coupon =/ The required margin (i.e., discount margin) is the yield spread over, or under, the reference rate such that the FRN is priced at par value on a rate reset date Reset Frequency How often the coupon rate is adjusted based on chang es in the reference rate Price Stability FRN s typically have more stable prices than fixed-rate bonds when interest rates change S2.Book Chapter 3: Introduction to fixed Income Valuation 7  Valuation of Floating-Rate Notes Linear interpolation You can use the yields from these bonds to estimate the yield for the illiquid bond and calculate its price. This process uses linear interpolation to estimate yields for periods between those of the available bonds. Steps: 1. Identify comparable bonds with similar maturity, coupon, and risk. 2. Calculate yields for the comparable bonds. 3. Interpolate the yield for the illiquid bond. 4. Yield Measures for Varying Compounding Periods Different yield measures are often used for comparison between bonds with different coupon payment frequencies (semi-annual, quarterly, etc.). Yield-to-maturity is typically annualized, and different compounding periods can result in different yields. 4.1 Effective Annual Rate (EAR) The Effective Annual Rate (EAR) is the yield adjusted for compounding frequency. For example, a bond with semi-annual compounding will have an EAR lower than a bond with annual compounding, even if the nominal rate is the same. Conversion Formula: S2.Book Chapter 3: Introduction to fixed Income Valuation 8  4.2 Money Market Instruments Money Market Overview: Money market instruments are short-term debt securities with maturities typically ranging from overnight to one year. Differently from Bonds (long-term debt securities), simple interest yield measures are used in the money market. Discount Rate vs. Add-On Rate Discount Rate Add-On Rate Used for instruments like T-bills and commercial Used for instruments like CD s and repos. paper. Interest is deducted from face value at Interest is added to the principal at maturity. issuance.“ Discount rate” has a unique meaning Comparison in the money market. It is a specific type of Discount rates understate the trueyield quoted rate. compared to add-on rates for the same instrument. Money market instruments are quoted using either discount rates or add-on rates. These two quoting conventions are calculated differently, so it is important to know how to convert between discount rates and add-on rates for accurate yield comparisons between different money market securities. Pricing Formulas Rate calculation 5. Spot Rates, Yield Curves, and Forward Rates 5.1 Spot Rates and Bond Pricing Because the market discount rates for the cash flows with different maturities are rarely the same, it is fundamentally better to calculate the price of a bond by using a sequence of market discount rates that S2.Book Chapter 3: Introduction to fixed Income Valuation 9  correspond to the cash flow dates. 💬 Spot rates = the yields on zero-coupon bonds maturing at specific dates. Bonds are sometimes valued by discounting each of their future cash flows at the corresponding spot rate. This is known as no-arbitrage pricing because it prevents arbitrage opportunities in perfect markets. Formula using spot rates: Example If a bond has a 5% coupon and three years to maturity, and spot rates are 2%, 3%, and 4% for one, two, and three years, the bond’s price is calculated by discounting each cash flow using the appropriate spot rate. 5.2 Yield Curve The yield curve represents the relationship between yield and time to maturity for bonds of the same quality. It helps investors gauge market expectations of future interest rates. Normal Yield Curve Inverted Yield Curve Flat Yield Curve Longer-term yields are higher Shorter-term yields are higher Similar yields across different than shorter-term yields, than longer-term yields, often maturities, indicating market reflecting expectations of future seen as a predictor of uncertainty about future interest rate increases. economic recession. interest rate movements. S2.Book Chapter 3: Introduction to fixed Income Valuation 10  The maturity structure of interest rates, also known as the term structure, describes the relationship between yields and time-to-maturity for bonds with similar characteristics. There are factors which influence the shape of yield curves, including expectations of future interest rates and economic conditions. Types of Yield Curves Par Curve: A par curve is a sequence of yields-to-maturity such that each bond (for each maturity) is priced at par value. The par curve is obtained from a spot curve using the following formula for each maturity (N) and solving for PMT (zN is the spot rate for the period): 5.3 Forward Rates Forward rates are implied by the current spot rates and represent future expected interest rates. They are essential for understanding how today’s market anticipates future interest rate movements. 💬 Forward rates = the interest rate on a bond or money market instrument traded in a forward market (future delivery). A forward rate is the incremental, or marginal, return for extending the time-to-maturity for an additional time period. It represents market expectations of future short-term interest rates. S2.Book Chapter 3: Introduction to fixed Income Valuation 11  💬 Implied forward rate (also known as a forward yield) = links the return on an investment in a shorter-term zero-coupon bond to the return on an investment in a longer-term zero- coupon bond → is calculated from spot rates and is a break-even reinvestment rate Forward rates can be used as an alternative to spot rates in bond valuation. Forward rates id derived from the relationship between spot rates of different maturities. A general formula for the relationship between the two spot rates and the implied forward rate is: where A is the years from today when the security starts and B – A is the tenor Because spot rates can be derived using forward rates, bonds can be valued using the forward curve: 5.4 Yield Spreads Credit spreads reflect the additional yield required to compensate investors for credit risk. Credit spreads vary across different credit ratings and maturities. A credit curve shows how an issuer's credit spreads change across different maturities. Yield Spreads Over Benchmark Rates Yield spreads play a crucial role in fixed-income security analysis, helping investors understand why bond prices and yields-to-maturity change. Yield spreads over benchmark rates and yield curves provide valuable insights into both macroeconomic and microeconomic factors affecting bond performance. Benchmark Yield The base rate, often a government bond yield, reflecting macroeconomic factors such as inflation, economic growth, and monetary policy. S2.Book Chapter 3: Introduction to fixed Income Valuation 12  Spread The difference between the yield- to-maturity and the benchmark, capturing microeconomic factorsspecific to the bond issuer and the bond itself. Risk Premium Compensation for credit and liquidity risks, and possibly tax impacts, provided to investors for holding a specific bond. Yield Spread Measures Nominal Spread Zero-Volatility Spread Difference between a bond's yield-to-maturity Constant spread added to the spot curve to and the yield on a benchmark go vernment bond. match a bond's price. Two main examples of yield spread measures include the nominal (G-) spread and zero-volatility spread A zero volatility spread (Z-spread) of a bond: The G-spread is calculated as the difference between the yields-to-maturity of the corporate and government bonds S2.Book Chapter 3: Introduction to fixed Income Valuation 13