Fixed Income Valuation PDF

Introduction to Fixed- Income Valuation Fixed-income valuation is a critical skill for investors, issuers, and financial analysts. This chapter explores the fundamentals of valuing fixed-rate bonds, floating-rate notes, and money market instruments. We'll examine how to calculate bond prices using m...

Introduction to Fixed- Income Valuation Fixed-income valuation is a critical skill for investors, issuers, and financial analysts. This chapter explores the fundamentals of valuing fixed-rate bonds, floating-rate notes, and money market instruments. We'll examine how to calculate bond prices using market discount rates and spot rates, understand the relationships between price, coupon rate, maturity, and yield, and analyze various yield measures. We’ ll also cover matrix pricing, yield conversions, and the maturity structure of interest rates, providing a comprehensive overview of fixed-income valuation techniques. Read: Chapter 3, Pettit, B. (2019) Fixed Income Analysis  CONTENTS Introduction Bond Prices and the Time Value of Money Prices and Yields: Conventions for Quotes and Calculations The Maturity Structure of Interest Rates Yield Spreads  INTRODUCTION The fixed-income market is a key source of financing for business and governments. Similarly, the fixed-income market represents a significant investing opportunity for institutions and individuals. Understanding how to value fixed-income securities is important to investors, issuers, and financial analysts.  BOND PRICES AND THE TIME VALUE OF MONEY Bond pricing is an application of discounted cash flow analysis. Bond price should be equal to the value of all discounted future cash flows. On an option-free fixed-rate bond, the promised future cash flows are a series of coupon interest payments and repayment of the full principal at maturity. The market discount rate is used to obtain the present value. The market discount rate is the rate of return required by investors given the risk of the investment in the bond. Bond Pricing Basics 1 Identify Cash Flows Determine the bond's coupon payments and principal repayment at maturity. 2 Determine Discount Rate Establish the market discount rate based on the bond's risk and market conditions. 3 Calculate Present Value Use time value of money calculations to find the present value of all future cash flows. 4 Sum Present Values Add up the present values to determine the bond's price. Formula for Calculating the Bond Price given the discount rate 𝑃𝑀𝑇 1 𝐹𝑉 𝑃𝑉 = 1− + 𝑟 (1 + 𝑟)𝑁 1+𝑟 𝑁 PV is the present value (price) of the bond PMT is the coupon payment per period is the future value paid at maturity, or the bond’s par FV value r Market discount rate, or required rate of return per period N Number of evenly spaced periods to maturity  Examples Calculate the value of: a) five-year, 4% annual payment coupon bond with a market discount rate of 6% b) three-year, 8% semiannual payment coupon bond with a market discount rate of 6%. The bond price is 91.575 per 100 of par value. The bond price is 105.417 per 100 of par value.  Price of a Fixed-Rate Bond The price of a fixed-rate bond, relative to par value, depends on the relationship of the coupon rate to the market discount rate. If the bond price is higher than par value, This happens when the coupon rate is the bond is said to be greater than the market discount rate. traded at a premium. If the bond price is lower than par value, This happens when the coupon rate is the bond is said to be less than the market discount rate. traded at a discount. If the bond price is equal to par value, the This happens when the coupon rate is bond is said to be equal to the market discount rate. traded at par. The market discount rate, also called the required yield or required rate of return, is used to calculate the present value of a bond's cash flows. The relationship between the coupon rate and market discount rate determines whether a bond trades at a discount, premium, or par value.  Yield-To-Maturity If the market price of a bond is known, the PV equation can be used to calculate its yield-to-maturity. The yield-to-maturity is the internal rate of return on a bond’s cash flows. It is the implied market discount rate. The yield-to- maturity (YTM) is The investor holds the bond to maturity. the rate of return The issuer does not default on coupon on the bond to an or principal payments. investor provided The investor is able to reinvest coupon three conditions payments at that same yield. are met: Therefore, the yield-to-maturity is the promised yield. Yield-to-Maturity Calculation 1 Definition 2 Calculation 3 Assumptions Yield-to-maturity is the It is calculated by solving for The yield-to-maturity internal rate of return on a the discount rate that calculation assumes the bond's cash flows, assuming equates the present value of investor can reinvest coupon the investor holds the bond future cash flows to the payments at the same yield. to maturity and all payments bond's current market price. are made as scheduled. Yield-to-maturity is a crucial concept in bond valuation. It represents the total return an investor can expect if the bond is held until it matures, assuming all coupon and principal payments are made on schedule.  Example Suppose that a four-year, 5% annual coupon paying bond is priced at 105 per 100 of par value. The yield-to-maturity is the solution for the rate, r, in this equation: 5 5 5 105 105 = + + + (1 + 𝑟)1 (1 + 𝑟)2 (1 + 𝑟)3 (1 + 𝑟)4 where r = 0.03634, or 3.634%. The bond is traded at a premium because its coupon rate is greater than the yield required by investors. Bond Price and Characteristic Relationships The price of a fixed-rate bond will change whenever the market discount rate changes. Inverse Effect Bond price is inversely related to the market discount rate. When the market discount rate increases, the bond price decreases (the inverse effect). Convexity Effect For the same coupon rate and time-to-maturity, the percentage price change is greater when the market discount rate decreases than when they increases. Coupon Effect For the same time-to-maturity, lower coupon bonds have greater percentage price changes than higher coupon bonds when their market discount rates change by the same amount. Maturity Effect Longer-term bonds generally have greater percentage price changes than shorter-term bonds when their market discount rates change by the same amount. Understanding the relationships between bond prices and various bond characteristics is essential for fixed- income investors: the inverse effect, convexity effect, coupon effect, and maturity effect. These relationships help explain how bond prices change in response to changes in market discount rates and other factors. Relationships between Bond Prices and Bond Characteristics Discount Rates Go Discount Rates Go Coupon Price at Down Up Bond Maturity Rate 20% Price at % Price at % 19% Change 21% Change A 10% 10 58.075 60.950 4.95% 55.405 –4.60% B 20% 10 100.000 104.339 4.34% 95.946 –4.05% C 30% 10 141.925 147.728 4.09% 136.487 –3.83% D 10% 20 51.304 54.092 5.43% 48.776 –4.93% E 20% 20 100.000 105.101 5.10% 95.343 –4.66% F 30% 20 148.696 156.109 4.99% 141.910 –4.56% Constant-yield price trajectory  Market Discount Rates (Spot Rates) 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 correspond to the cash flow dates. Spot rates are yields-to- These market discount maturity on zero-coupon rates are called “spot bonds maturing at the rates.” date of each cash flow. General formula for calculating a bond price given the sequence of spot rates: where Z1, Z2, and ZN are spot rates for period 1, 2, and N, respectively. Spot Rates and Bond Valuation 1 2 3 4 Identify Cash Obtain Spot Discount Cash Sum Present Flows Rates Flows Values Determine the Find the appropriate Use each spot rate to Add up the present bond's coupon spot rate for each discount its values to g et the payments and cash flow date. corresponding cash bond's price. principal repayment. flow. Spot rates, or zero-coupon yields, provide a more precise method for valuing bonds. This approach allows each future cash flow to be discounted at a rate associated with its timing. Spot rates are used to calculate a bond's "no-arbitrag e value“.  Example Suppose that the one-year spot rate is 2%, the two-year spot rate is 3%, and the three-year spot rate is 4%. Calculate the price of a three-year 5% annual coupon paying bond: The bond price is 102.960. Calculate what is the YTM implied by this price and then check what would be the PV if cash flows are discounted by it. The present values of the individual cash flows discounted using spot rates differ from those using yield-to-maturity, but the sum of the present values is the same. Thus, the same price is obtained using either approach. Flat Price, Accrued Interest, and Full Price Flat Price The quoted or "clean" price of a bond, excluding accrued interest. Accrued Interest Interest earned but not yet paid since the last coupon payment date. Full Price The total price paid by the buyer, including flat price and accrued interest. When a bond trades between coupon payment dates, its price consists of two components: the flat price and accrued interest. The flat price is typically the price which is observed in the bond market and at which a trader buys or sells a bond. Once a trader ag rees to buy or sell a bond at its flat price, it is important to be able to calculate what will be the full price, which is the actual amount paid (if buying ) or received (if selling ) on the settlement date.  Prices and yields: conventions for quotes and calculations Bond price consists of two components Flat (clean) price Accrued interest (AI) (PVFlat) The sum of flat price and accrued interest is the full (dirty) price (PVFull). PVFull = PVFlat + AI Buyers pay the full Bond dealers price for the bond on usually quote the settlement date. the flat price. Day Count Conventions Convention Days in Year Days in Month Common Use Actual/Actual 365 or 366 Actual Government Bonds 30/360 360 30 Corporate Bonds Actual/360 360 Actual Money Market Day count conventions are methods used to calculate accrued interest on bonds. Different conventions are used in various markets and for different types of securities.  Accrued interest: counting days Accrued interest is the proportional share of the next coupon payment: 𝑡 AI = × PMT 𝑇 where t is the number of days from the last coupon payment to the settlement date; T is the number of days in the coupon period; t/T is the fraction of the coupon period that has gone by since the last payment; and PMT is the coupon payment per period. 30/360 is Actual/actual common for The two most common is common for corporate conventions to count government bonds. days in bond markets: bonds.  Formula: PVFull The full price of a fixed-rate bond between coupon payments given the market discount rate per period (r) can be calculated as: where PV is the value of the bond on the most recent coupon payment date if its yield were r. PV can be calculated using the standard bond price formula provided previously.  Example Example. A 6% German corporate bond is priced for settlement on 18 June 2019. The bond makes semiannual coupon payments on 19 March and 19 September of each year and matures on 19 September 2030. Using the 30/360 day-count convention, calculate the full price, the accrued interest, and the flat price per EUR100 of par value if the annual yield to maturity is 5.80% (2.90% per six months): The value of the bond at the beginning of the period on 19 March (when 23 semiannual periods remain until maturity) is: The price of the bond would have been EUR101.661589 on 19 March 2019 if its market discount rate at that time had been 5.80%. Example (continued): There are 89 days between 19 March and 18 June using a 30/360 day-count convention. The full price on 18 June 2019 is: The accrued interest is: The flat price is:  Matrix pricing Matrix pricing is an estimation process used for bonds that are not actively traded. In matrix pricing, market discount rates are extracted from comparable bonds (i.e., bonds with similar time-to-maturity, coupon rate, and credit quality). 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. Matrix pricing is also used in underwriting new bonds to get an estimate of the required yield spread over the benchmark rate. The benchmark rate is typically the yield-to-maturity on a government bond having the same, or close to the same, time-to-maturity. The spread is the difference between the yield-to-maturity on the new bond and the benchmark rate. The yield spread is the additional compensation required by investors for the difference in the credit risk, liquidity risk, and tax status of the bond relative to the government bond. This spread is sometimes called the “spread over the benchmark.” Matrix Pricing Identify Comparable Bonds Find actively traded bonds with similar characteristics. Create Yield Matrix Org anize yields by maturity and coupon rate. Interpolate Yields Estimate the yield for the targ et bond. Calculate Price Use the estimated yield to price the targ et bond.  Example An analyst is pricing a three-year, 4% semiannual coupon corporate bond with no active market to derive the appropriate YTM. He finds two bonds with a similar credit quality: A two-year bond is traded at a YTM of 3.8035%, and a five-year bond is traded at a YTM of 4.1885%. Using linear interpolation, the estimated YTM of a three-year bond will be 3.9318%: Using 3.9318% as the estimated three-year annual market discount rate (see next slide), the three-year, 4% semi-annual coupon payment corporate bond has an estimated price of 100.191 per 100 of par value.  Example (cont.) In the matrix we have two two-year bonds: one with a 3% coupon and the other with a 5% coupon. We want to price a bond that has a coupon in the middle of the way between them (4%) and 1 year more in terms of maturity. So the YTM of the bond we want to price should be at least equal to average of these two bonds’ YTM, but also taking into account for the fact that it has 1 year more in terms of maturity (if we don’t, we would be estimating the YTM of a two-year, instead of three-year, 4% coupon bond). And in the matrix we also have two five-year bonds: one with a 2% coupon and the other with a 4% coupon. It is useful to use all the market pricing information we can get in the interpolation. With these longer term bonds, we can account for the fact that the bond we are trying to price has 1 year more maturity than the two two-year bonds. So if we also compute the average YTM for these two bonds, we now have the YTM of equivalent (same risk as we are trying to price) five- and two-year bonds. The five-year YTM has embedded in it three years of “time premium” over the two-year YTM: this is what you see in the denominator of the ratio adjusting this premium. In the numerator, it is indicated that we are trying to capture a one-year “time premium” (since the three-year bond we are trying to price has one year more maturity than the two-year YTM we extracted from the two two-year bonds). So the calculation you see can be read as follows: starting with the estimated YTM for a two-year 4% coupon bond, we add to it (by simple linear interpolation) a one-year “time premium” which is one third ( (3-2) / (5-2) ) of the three-year (five-year over two- year) “time premium” implied by the estimated five- and two-year YTM’s.  Yield Measures for Fixed-Rate Bonds Investors use standardized yield measures to allow for comparison between bonds with varying maturities. For bonds maturing in more than one For money market instruments of less year: than one year to maturity: An annualized and compounded These are annualized but not yield-to-maturity is used. compounded. An annualized and compounded yield on a fixed-rate 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. For example, converting a YTM of 4.96% from a periodicity of 2 (semiannual) to a periodicity of 4 (quarterly) gives a YTM of 4.93%:  Other Yield Measures Street True yield-to- Government Current yield: Simple yield: convention maturity: equivalent yield: The sum of The sum of yield-to- The internal rate Restatement of a coupon coupon maturity: of return on the yield-to-maturity payments payments plus The internal rate cash flows using based on a received over the the straight-line of return on the the actual 30/360 day-count year divided by amortized share cash flows, calendar of to one based on the flat price of the gain or assuming the weekends and actual/actual loss, divided by payments are bank holidays the flat price made on the scheduled dates (no weekends or holidays) Yield measures for floating-rate Notes The interest The principal on the floater is typically non- payments on a amortizing and is redeemed in full at maturity. floating-rate note vary from The reference rate is determined at the period to beginning of the period, and the interest period payment is made at the end of the period. depending on the current The most common day-count conventions for level of a calculating accrued interest on floaters are reference actual/360 and actual/365. interest rate. Floating-Rate Note Basics Reference Rate The base interest rate, often a short-term money market rate like LIBOR. Quoted Margin The fixed spread added to the reference rate to determine the coupon. 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 chang e. 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.  Quoted and Required Margin The required margin (i.e., The specified yield spread discount margin) is the yield over the reference rate is spread over, or under, the called the “quoted margin” reference rate such that the on the FRN. FRN is priced at par value on a rate reset date. Valuation of Floating-Rate Notes Determine Current Coupon Calculate the current interest payment based on the reference rate and quoted marg in. Estimate Future Coupons Project future interest payments using forward rates or assuming constant rates. Calculate Discount Margin Determine the yield spread required by investors g iven the FRN ’s risk. Discount Cash Flows Use the discount marg in to calculate the present value of future cash flows. Valuing floating -rate notes requires a different approach than fixed-rate bonds due to their variable cash flows. The process of valuing FRN s involves calculating the current coupon, estimating future coupons, and determining the discount marg in. FRN ’s are subject to the "pull to par“ effect and is it is important to understand how chang es in credit risk affect their valuation.  Simplified FRN pricing Model where: PV is the present value/price of the FRN Index is the annual reference rate QM is the quoted margin (annualized) FV is the value at maturity m is the periodicity of the FRN DM is the annualized discount margin N is the number of evenly spaced periods to maturity  Example Suppose that a five-year FRN pays three-month Libor plus 0.75% on a quarterly basis. Currently, three-month Libor is 1.10%. The price of the floater is 95.50 per 100 of par value. Calculate the discount margin: 0.0110+0.0075 ×100 0.0110+0.0075 ×100 0.0110+0.0075 ×100 +100 4 4 4 95.50 = 0.0110+𝐷𝑀 1 + 0.0110+𝐷𝑀 2 +.. + 0.0110+𝐷𝑀 20 1+ 1+ 1+ 4 4 4 This has the same format as the general PV equation , which can be used to solve for the market discount rate per period, r = 0.7045%. Solving for DM, DM = 1.718%, or 171.8 bps Money Market Instruments O verview Short-Term Issuers Yield Measures Low Risk Money market Include governments, Use simple interest rather Generally considered to instruments typically banks, and corporations than compound interest be among the safest have maturities of one seeking short-term for annualized rates. fixed-income year or less. funding. investments. 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.  Yield Measures There are several important differences in yield measures between the money market and the bond market: Money market Money market The rate of return instruments often are instruments having on a money quoted using different times-to- market nonstandard interest maturity have instrument is rates and require different stated on a simple different pricing periodicities for the interest basis. equations than those annual rate. used for bonds. Discount Rate vs. A dd-O n Rate Discount Rate Add-On Rate Comparison Used for instruments like T-bills Used for instruments like CD s and Discount rates understate the true and commercial paper. Interest is repos. Interest is added to the yield compared to add-on rates for deducted from face value at principal at maturity. the same instrument. issuance.“ Discount rate” has a unique meaning in the money market. It is a specific type of quoted rate. 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 Pricing formula for money Days market instruments quoted PV = FV × 1 − × DR Year on a discount rate basis: Pricing formula for money FV market instruments quoted PV = Days 1+ × AOR on an add-on rate basis: Year where Days is the number of days between settlement and maturity Year is the number of days in a year (365 or 360) DR is the discount rate, stated as an annual percentage rate AOR is the add-on rate, stated as an annual percentage rate.  Examples Suppose that a 91-day US Treasury bill (T-bill) with a face value of USD10 million is quoted at a discount rate of 2.25% for an assumed 360-day year. Enter FV = 10,000,000, Days = 91, Year = 360, and DR = 0.0225. Find the price of the T-bill: 91 𝑃𝑉 = 10,000,000 × 1 − × 0.0225 = 𝐔𝐒𝐃𝟗, 𝟗𝟒𝟑, 𝟏𝟐𝟓 360 Suppose that a Canadian pension fund buys an 180-day banker’s acceptance (BA) with a quoted add-on rate of 4.38% for a 365-day year. If the initial principal amount is CAD10 million, calculate the redemption amount due at maturity: 180 𝐹𝑉 = 10,000,000 × 1 + × 0.0438 = 𝐂𝐀𝐃𝟏𝟎, 𝟐𝟏𝟔, 𝟎𝟎𝟎 365  Discount Rate and Add-on Rate The discount rate is Year FV−PV calculated using the DR = × formula: Days FV The add-on rate is calculated Year FV−𝑃𝑉 using the formula: AOR = × Days PV The first term for both formulas, Year/Days, is the periodicity of the annual rate. The second term for the add-on rate is the interest earned, FV – PV, divided by PV, the amount invested. However, for the discount rate, the denominator in the second term is FV, not PV. Therefore, by design, a money market discount rate understates the rate of return to the investor. Bond Equivalent Yield The bond equivalent yield is a standardized way to express money market rates, facilitating comparisons between different instruments and maturities. 1 Definition A money market rate stated on a 365-day add-on rate basis. 2 Purpose Allows for comparison of yields across different money market instruments. 3 Calculation Converts discount rates or 360-day add-on rates to a common 365-day basis. 4 Limitations Does not account for compounding, unlike bond yields for longer-term securities.  Examples Suppose that an investor is considering an investment in 90-day commercial paper quoted at a discount rate of 5.76% for a 360-day year. Its is FV = 100 and we can obtain a PV = 98.56. Find the paper’s AOR based on a 365-day year: 365 100−98.56 AOR = × = 0.05925, or 5.925% 90 98.56 This converted rate is called a “bond equivalent yield.” Now suppose that an analyst prefers to convert money market rates to a semiannual bond basis. The quoted rate for a 90-day money market instrument is 10%, quoted as a bond equivalent yield (its periodicity is 365/90): 0.10 365/90 APR2 2 1+ = 1+ , APR 2 = 0.10127, or 10.127% 365/90 2 Maturity Structure of Interest Rates Normal Yield Curve Inverted Yield Curve Flat Yield Curve Longer-term yields are higher than Shorter-term yields are higher than Similar yields across different shorter-term yields, reflecting longer-term yields, often seen as a maturities, indicating market expectations of future interest rate predictor of economic recession. uncertainty about future interest increases. rate movements. 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. The maturity structure of interest rates The difference between yields on two bonds might be due to various reasons, such as: currency denomination credit risk liquidity tax status periodicity of the yield varying time-to-maturity The term structure of interest rates is the factor that explains the differences between yields. It involves the analysis of yield curves, which are relationships between yields-to-maturity and times- to-maturity. Examples of yield curves The (government bond) spot curve is a sequence The yield curve on coupon bonds is a sequence of yields-to-maturity on zero-coupon of yields-to-maturity on coupon paying (government) bonds. (government) bonds. Types of Yield Curves Spot Curve Par Curve Based on yields of zero-coupon bonds across Yields on hypothetical coupon-bearing bonds priced different maturities. at par value. Forward Curve Yield Curve on Coupon Bonds Implied future interest rates derived from current Based on yields of actual coupon-bearing bonds in spot rates. the market. There are several types of yield curves used in fixed-income analysis, each serving a specific purpose: spot curves, par curves, forward curves, yield curves on coupon bonds are some of the most used. They are constructed in different ways and are applicable in the valuation of bonds which are more exposed to each of these types of rates and are also an important tool in interest rate forecasting. Constructing the Government Bond Spot Curve 1 Collect Data Gather prices and cash flows of government bonds across various maturities. 2 Bootstrap Short-Term Rates Calculate spot rates for short maturities using T-bills or short-term notes. 3 Iterative Calculation Use longer-term bond prices to solve for spot rates at each maturity. 4 Interpolation Estimate spot rates for maturities between observed data points. Par Curve Calculation Start with Spot Curve Use the previously constructed g overnment bond spot curve. Calculate Par Yields Determine yields for hypothetical bonds priced at par for each maturity. Iterative Process Solve for coupon rates that result in a bond price of 100. Create Par Curve Plot the calculated par yields ag ainst their respective maturities. The par curve represents yields on hypothetical bonds priced at par value for various maturities. The par curve is derived from the spot curve.  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): 𝑃𝑀𝑇 PMT PMT + 100 100 = 1 + 2 + ⋯+ 1 + 𝑧1 1 + 𝑧2 1 + 𝑧𝑁 𝑁 Forward Rate and Implied Forward Rate is the interest rate on a bond or money A forward rate market instrument traded in a forward market (future delivery). is calculated from spot rates and is a An implied forward break-even reinvestment rate links the return on an investment in a rate (also known as shorter-term zero-coupon bond to the a forward yield) return on an investment in a longer-term zero-coupon bond. Forward Rates and the Forward Curve Definition Forward rates are interest rates for future periods implied by current spot rates. A forward curve is a series of forward rates, each having the same time frame. These forward rates might be observed on transactions in the derivatives market. Calculation Derived from the relationship between spot rates of different maturities. Interpretation 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. Applications Used in pricing derivatives, making investment decisions, and forecasting. Forward rates play a crucial role in understanding market expectations and valuing fixed-income securities. Forward rates are extracted from spot rates, with which it is possible to construct a forward curve. Implied forward rates can be also be interpreted as breakeven reinvestment rates.  Using Forward Rates in Bond Valuation Forward rates can be used as an alternative to spot rates in bond valuation. 1 Identify Cash Flows Determine the bond's coupon payments and principal repayment. 2 O btain Forward Rates Calculate or observe forward rates for each future period. 3 Discount Cash Flows Use the appropriate forward rates to discount each cash flow. 4 Sum Present Values Add up the discounted cash flows to g et the bond's value. Forward Rate Notation Although finance textbook authors use varying notation, the most common market practice is to name forward rates as in this example: “2y5y” — pronounced “the two-year into five-year rate.” The first number (two) refers to the length of the forward period in years from today. The second number (five) refers to the tenor (time-to-maturity) of the underlying bond. Formula: Spot Rates and Implied Rate 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. Example. Suppose that an investor observes these prices and yields-to- maturity on zero-coupon government bonds: The prices are per 100 of par value. The yields-to-maturity are stated on a semi-annual bond basis. Compute the “1y1y” and “2y1y” implied forward rates, stated on a semi-annual bond basis. (cont.) Formula: Spot Rates and Implied Rate Example (cont.). The “1y1y” implied forward rate is 3.419%. A = 2 (periods), B = 4 (periods), B − A = 2 (periods), z2 = 0.02548/2 (per period), and z4 = 0.02983/2 (per period). The “2y1y” implied forward rate is 2.707%. A = 4 (periods),B = 6 (periods), B − A = 2 (periods), z4 = 0.02983/2 (per period), and z6 = 0.02891/2 (per period). The investor has a three-year investment horizon and is choosing between (1) buying the two-year zero and reinvesting in another one-year zero in two years and (2) buying and holding to maturity the three-year zero. The investor decides to buy the two-year bond. Based on this decision, what is the minimum yield-to-maturity the investor expects on one-year zeros two years from now? The investor’s view is that the one-year yield in two years will be greater than or equal to 2.707%. The “2y1y” implied forward rate of 2.707% is the breakeven reinvestment rate. If the investor expects the one-year rate in two years to be less than that, the investor would prefer to buy the three-year zero. If the investor expects the one-year rate in two years to be greater than 2.707%, the investor might prefer to buy the two- year zero and reinvest the cash flow.  Formula: Forward Curve Because spot rates can be derived using forward rates, bonds can be valued using the forward curve: 𝐏𝐌𝐓 𝐏𝐌𝐓 𝐏𝐌𝐓 + 𝐅𝐕 𝐏𝐕 = + + ⋯+ 𝟏 + 𝒁𝟏 𝟏 + 𝒁𝟏 × (𝟏 + 𝐈𝐅𝐑 𝟏,𝟏 ) 𝟏 + 𝒁𝟏 × 𝟏 + 𝐈𝐅𝐑 𝟏,𝟏 × ⋯ × 𝟏 + 𝐈𝐅𝐑 𝑵−𝟏,𝟏 Example. Suppose that an analyst needs to value a four-year, 3.75% annual coupon payment bond that has the same risks as the bonds used to obtain the forward curve. Using the implied spot rates, the value of the bond is 102.637 per 100 of par value. The bond also can be valued using the forward curve.  Yield Spreads The spread is the difference between the yield-to- maturity and the benchmark. The benchmark is often called the “risk-free rate of return.” Fixed-rate bonds often use a government benchmark (on-the-run) security with the same time-to- maturity as, or the closest time-to-maturity to, the specified bond. A frequently used benchmark for floating-rate notes has been Libor, and recently replaced by market based rates. As a composite interbank rate, it is not a risk-free rate. Credit Spreads and the Credit Curve Credit Rating Impact Term Structure of Credit Spreads Lower-rated bonds typically have hig her credit spreads to C redit spreads often vary by maturity, reflecting compensate for increased default risk. chang ing credit risk perceptions over time. C redit spreads reflect the additional yield required to compensate investors for credit risk. C redit spreads vary across different credit rating s and maturities. A credit curve shows how an issuer's credit spreads chang e across different maturities. Yield-to-maturity Building Blocks Taxation Spread Risk Premium Liquidity Credit Risk Expected Inflation Rate “Risk-Free” Benchmark Rate of Return Expected Real Rate Yield Spreads O ver 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 Spread Risk Premium The base rate, often a government The difference between the yield- Compensation for credit and bond yield, reflecting to-maturity and the benchmark, liquidity risks, and possibly tax macroeconomic factors such as capturing microeconomic factors impacts, provided to investors for inflation, economic growth, and specific to the bond issuer and the holding a specific bond. monetary policy. bond itself. Benchmark Rates and G-Spreads 1 On-the-Run Securities The most recently issued government bonds for each maturity, actively traded and priced close to par value (closest to the current market discount rate for that maturity). 2 Off-the-Run Securities Seasoned government bonds, typically trading at slightly higher yields-to- maturity than on-the-run bonds because of differences in demand for the securities. 3 G-Spread The yield spread in basis points over an actual or interpolated government on-the-run bonds, representing the return for bearing greater risks relative to the sovereign bond. Yield Spread M easures Nominal Spread Zero-Volatility Spread Difference between a bond's yield- Constant spread added to the spot to-maturity and the yield on a curve to match a bond's price. benchmark go vernment bond. Yield spreads measure the additional yield investors require for bearing risks beyond those of go vernment bonds. Two main examples of yield spread measures include the nominal (G-) spread and zero-volatility spread. Yield Spreads Over Benchmark Yield Curves 1 Yield Curve Definition A yield curve shows the relationship between yields-to-maturity and times-to-maturity for securities with the same risk profile. 2 Types of Yield Curves Government bond yield curves and swap yield curves represent different term structures of benchmark interest rates. 3 Term Structure of Credit Spreads Isolating credit risk over varying times-to-maturity gives rise to a distinct term structure of credit spreads for each borrower. 4 Z-Spread The zero volatility spread (Z-spread) is a constant yield spread over a government or interest rate swap spot curve.  Z-Spread Calculated as a constant yield spread over a government (or interest rate swap) spot curve A zero volatility — as opposed to the G-spread and I-spread, spread (Z-spread) which use the same discount rate for each of a bond cash flow 𝐏𝐌𝐓 𝐏𝐌𝐓 𝐏𝐌𝐓+𝐅𝐕 𝐏𝐕 = + + ⋯+ (𝟏+𝒛𝟏 +𝒁)𝟏 (𝟏+𝒛𝟐 +𝒁)𝟐 (𝟏+𝒛𝑵 +𝒁)𝑵 Calculating G- and Z-Spreads Bond Type Coupon Maturity Price YTM Corporate 6% annual 2 years 100.125 5.932% Government 4% annual 2 years 100.750 3.605% The yield-to-maturity for the corporate bond is 5.932%. The yield-to-maturity for the government benchmark bond is 3.605%. The G-spread is calculated as the difference between the yields-to-maturity of the corporate and government bonds: 5.932% - 3.605%= 232.7 bps. The Z-spread of 234.22 bps is demonstrated using one-year and two-year government spot rates of 2.10% and 3.635%, respectively.