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## Bond Mathematics

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**Bond Mathematics**Finance 298 Analysis of Fixed Income Securities**A General Valuation Model**• The basic components of valuing any asset are: • An estimate of the future cash flow stream from owning the asset • The required rate of return for each period based upon the riskiness of the asset • The value is then found by discounting each cash flow by its respective discount rate and then summing the PV’s (Basically the PV of an Uneven Cash Flow Stream)**The formal model**• The value of any asset should then be equal to:**A Basic Bond**• A bond is basically a debt contract issued by a corporation or government entity. • The buyer is lending the issuer an amount of money (the par value). The issue agrees to pay interest at specified intervals (coupon payments) to the buyer, and return the par value at the end of the contract (the maturity date).**Components of a bond:**• Par Value: • Initial issue amount • Coupon Payment: • Interest payments on the par value. • Coupon Rate: • The rate that determines the coupon payments. • Maturity Date: • The point in time when the par value and final coupon payment are made. • Embedded Options (Call and Put Provisions): • The issuer may be able to “call” the bond prior to its maturity. • Market Price: • The current price the bond is selling for in the market.**Applying the general valuation formula to a bond**• What component of a bond represents the future cash flows? • Coupon Payment: The amount the holder of the bond receives in interest at the end of each specified period. • The Par Value: The amount that will be repaid to the purchaser at the end of the debt agreement.**Basic Bond Mathematics**• Given • r: The interest rate per period or return paid on assets of similar risk • CP: The coupon payment • MV: The Par Value (or Maturity Value) • n: the number of periods until maturity • The value of the bond is represented as:**The coupon payment is the same every period and can be**factored out, this shows that the first part is just an annuity. Therefore the formula can be rewritten as:**A Simple Example**• A Bond with 20 years left until maturity, with a 9% coupon rate and a par value of $1,000. Assume for now that the discount rate is also 9% and that it makes annual coupon payments. • What are the relevant PVIFA and PVIF? • PVIFA9%,20 = 9.1285 PVIF9%,20=.1748 • How much is the coupon payment? • 1,000(.09) = $90**Applying the formula**• On a Financial Calculator: 20 N 9 I 90 PMT 1,000 FV PV = 1,000**The Discount Rate**• So far we assumed that the interest rate is the same as the coupon rate. When this is true the value of the bond equals the par value. • Are the two usually the same? No, the discount rate should represent the current required return on assets of similar risk. This changes as the level of interest rates in the economy changes**Continuing our example**• Assume that we bought our 9% yearly coupon bond with 20 years left to maturity and one year later the required return decreased to 7%. • What is the value of the bond? 19 N 7 I 90 PMT 1,000 FV PV=1,206.71**Why did the price increase?**• New bonds of similar risk are only paying a 7% return. This implies a coupon rate of 7% and a coupon payment of $70. • The old bond has a coupon payment of $90, everyone will want to buy the old bond, (the increased demand increases the price) • Why does it stop at $1,206.71? • If you bought the bond for $1206.71 and received $90 coupon payments for the next 19 years you receive a 7% return.**Changes in Bond Value Over Time**• Assuming that interest rate stay constant, what happens to the price of the bond as it gets closer to maturity? N= # of Payments 19 10 1 I = k 7% 7% 7% PMT = Coup Pay 90 90 90 FV = Par Value 1000 1000 1000 PV = Bond Value 1206.71 1140.47 1018.69**Calculating Return**• Total Return (yearly) – The combined capital gains yield and interest (current) yield from holding the bond one year. • Yield to Maturity – The yearly return if you purchase the bond today and hold it until maturity • Yield to Call – The yearly return if you purchase the bond today and hold it until it is called. • Yield to Put – the yearly return if you purchase the bond today and hold it until the put option is exercised.**Total Return Example**If you bought the bond in the previous example for 1,000 and then sold it after the rate change for 1206.71 what is your total return from owning the bond?**Yield to Maturity**• Before we were looking for the “value” of the bond given a required rate of return. • Now given the current market price we want to find the interest rate that makes the cash flows from the bond equal to its market price - this rate is known as the Yield to Maturity. • The YTM is the return you earn IF you buy the bond today and hold it until maturity.**Calculating YTM**• To solve for YTM we are solving for the interest rate (r) in the bond valuation formula: • We cannot solve for r algebraically, only by trial and error**Calculating YTM**• Unfortunately calculating YTM is difficult: • You can approximate it by using the PVIFA and PVIF tables • Solve for I on the Financial Calculator (make sure to enter both (-) and (+) CF’s • on excel use the Yield command • =Yield(settlement, maturity, rate, price, redemption value, frequency, basis)**YTM Vs. Total Return**• Yearly total return equals the YTM only if the required return does not change over the year. • In the previous example assume you bought the 7% YTM bond with 19 periods left and held it one year. • The price at the end of the year is 1,201.18**YTM and Risk**• The YTM will change as the level of interest rates in the economy change and as the risk associated with the firm and its projects change. • The YTM is a representation of the probability of default and the current level of interest rates in the economy.**Promised or Expected Return**• You will earn the YTM if the bond does not default and you hold it to maturity. • The expected return should encompass the chance of default, probability the bond is called or a put option is exercised, and the possibility of interest rate fluctuations. • The YTM is only the expected return if the prob. of default is zero, the prob. of call or put is zero, and interest rates remain unchanged.**Yield to Call**• The yield to call is the yield paid on the bond assuming that a call option is exercised, given the current market price. • It represents the yield you would earn if you bought the bond today and held it until the call option was exercised.**YTC Example**• Assume you bought the bond in the previous example and that it had a call option that could be exercised in 9 years. If exercised, the firm is required to pay a $90 premium.**Yield to Worst**• After calculating all the possible Yields (yield to call, yield to put) the one with the lowest return is the termed the yield to worst.**Quick Facts**• If the level of interest rates in the economy increases the bond price decreases and vice versa. • If r>Coupon rate the price of the bond is below the par value - it is selling at a discount. • If r<Coupon rate the price of the bond is above the par value - it is selling at a premium. • Keeping everything constant the value of the bond will move toward par value as it gets closer to maturity.**Complications**• Most bonds make payments every six months instead of each year. • We have assumed that the next coupon payment is exactly 6 months away, often that is not the case. When the time frame is less than 6 months you need to account for interest over the shortened period. • We have assumed that the interest rate is constant, some bonds pay a floating rate of interest.**Semiannual Compounding**• Most bonds make coupon payments twice a year, to account for this: • Divide the annual coupon interest payment by 2. • Multiply the number of periods by 2. • Divide the annual interest rate by 2**Example:Semiannual Compounding**• What is the most you would be willing to pay for a 10% coupon bond that makes semiannual coupon payments, 30 years left to maturity and an annual required return of 12% • PMT = 1,000(.10) / 2 = 50 (each 6 months) • I = 12%/2 = 6% each six months • N = 30 (2) = 60 FV = 1,000 • PV = ? = -838.3857**Floating Interest Rates**• A useful measure is the effective margin (Or spread compared to a base rate). • Use the spread to calculate a value and compare that to the current price. If the two are different then there should be a change to the margin. You then develop an estimate of the margin the market is currently using to price the bond.**Bond Price Volatility**• Assuming an option free bond, we have shown that the price and yield move in an opposite direction, however there are some important details: • Given similar bonds that differ only in maturity or coupon rate, The % price change associated with the same size change in yield will differ. • For a given bond the % price change associated with a small change in yield is the same regardless of whether the yield increases or decreases.**Bond Price Volatility continued**• For a given bond • the % price change associated with a large increase in yield will not be the same as the % price change associated with the same size decrease in yield • For a large change in yield the % price increase is greater than the % change decrease associated with the same size yield change.**Different Coupons**Compare two 30 year semiannual coupon bonds, both with a current yield of 12%. Let Bond A have a coupon rate of 10% and Bond B have a coupon rate of 8%. What is the associated price change if the yield changes to 11%? 13%? Bond A 10% couponBond B 8% coupon Yield Price ChangePrice Change • 11% 912.7507 738.2522 • 74.3650 (8.87%)61.4808(9.08%) • 12%838.3857676.7714 • 63.8802 (7.62%)52.5955(7.77%) • 13%774.5055 624.1759**Impact of Maturity**Compare two 10% coupon semiannual bonds, both with a current yield of 12%. Let Bond A have 30 years to maturity and Bond B have 15 years. What is the associated price change if the yield changes to 11%? 13%? Bond A 30 YearsBond B 15 Years Yield Price ChangePrice Change • 11% 912.7507 927.3312 • 74.3650 (8.87%)64.9795 (7.54%) • 12%838.3857862.3517 • 63.8802 (7.62%)58.2318 (6.75%) • 13%774.5055 804.1199**Measuring Bond Price Volatility**• Price value of a basis point • Measures the price change for a one basis point (.0001 or.01%) change in the yield of the bond. • Yield value of a basis point • Measures the change in the yield of the bond for a given price change. • Duration • Measures the price elasticity of the bond.**Price Value of a Basis Point (PVBP)**• For small changes in yield the price change will be very close regardless of the whether the yield change is an increase or a decrease. • Use the semi annual bond above (30 years, 10% coupon, 12% YTM). YTM Price Change % change 12.01% 837.6986 -0.6871 -.081955% 12.00% 838.3857 11.99% 839.0739 0.6882 .082086%**PVBP**• In the previous example whether the yield increased or decreased the price changed by approximately 82 cents. • To find a larger price change you can scale the price change. For example if you had a 1% (100Bp) change you cold estimate the price change to be 100(.82) or $82.**Yield Value of a Price Change**• The change in yield for a given change in the price of a bond. • Example using the semi annual bond above, what is the yield change if the price changes by 1/32 of 1% of par value? Price Yield Change 838.0732 12.004546% .004546% 838.3857 12.0000% 838.6982 11.995458% .004542%**Duration: The Big Picture**• Duration: Measures the sensitivity of the PV of a cash flow stream to a change in the discount rate. • Keeping everything else constant the change in PV is greater: • The longer the time prior to receiving the cash flow • The larger the cash flow • (we just showed both of these)**Duration: The Big Picture**• Calculation: Given the PV relationships, we need to weight the Cash Flows based on the time until they are received. In other words we are looking for a weighted maturity of the cash flows where the weight is a combination of timing and magnitude of the cash flows**Calculating Duration**• One way to measure the sensitivity of the price to a change in discount rate would be finding the price elasticity of the bond (the % change in price for a % change in the discount rate)**Duration Mathematics**• Macaulay Duration is the price elasticity of the bond (the % change in price for a percentage change in yield). • Formally this would be:**Estimating Duration**• There are multiple methods for estimating the duration of a bond we will look at three different approaches. • Weighted Discounted Cash Flows (Macaulay) • Modified Duration • Averaging the price change**Duration Mathematics**• Taking the first derivative of the bond value equation with respect to the yield will produce the approximate price change for a small change in yield.**Duration Mathematics**The approximate price change for a small change in r**Duration MathematicsMacaulay Duration**substitute