MFE8812 Bond Portfolio Management

MFE8812 Bond Portfolio Management William C. H. Leon Nanyang Business School January 16, 2018 1 / 63 William C. H. Leon MFE8812 Bond Portfolio Management 1 Overview Value of Cash Flows Value of a Bond Present Value Formula 2 Overview 2 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Overview Bond pricing is typically performed by taking the discounted value of the bond cash flows. We shall review the basics of the mathematics of discounting, including time basis and compounding conventions, and introduce various types of interest rates involved in the fixed-income world, including the notions of coupon rate, current yield, yield to maturity, spot rate, forward rate and bond par yield. 3 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Present Value of a Cash Flow Consider a cash flow CF T that occurs at time T : Discount rate r T 0 Present Cash Flow CF T 1 2 3... T Time t Future PV = CF T 1 + r T ) T CF T 4 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Present Value of a Stream of Cash Flows Consider a stream of cash flow CF 1, CF 2, CF 3,...,CF T : Discount rate r t for maturity t 0 Present CF 1 CF 2 CF 3 1 2 3... CF T... T Future Time t PV = CF 1 1+r 1 + CF 2 1 + r 2 ) 2 + CF 3 1 + r 3 ) 3 + + CF T 1 + r T ) T = T t=1 CF t 1 + r t ) t. 5 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Valuing a Bond Valuing a bond can be viewed as a three-step process: Step 1: obtain the cash flows the bondholder is entitled to. Step 2: obtain the discount rates for the maturities corresponding to the cash flow dates. Step 3: obtain the bond price as the discounted value of the cash flows. 6 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Obtaining Cash Flows of a Bond If we are dealing with a straight default-free, fixed-coupon bond, the value of cash flows paid by the bond are known with certainty ex ante, i.e., on the date when pricing is performed. In general, two parameters that are needed to fully describe the cash flows on a bond. The first is the maturity date of the bond, on which the principal or face amount of the bond is paid and the bond retired. The second parameter needed to describe a bond is the coupon rate. For a non-standard bond, the following are some of the problems associated with estimating cash flows: The due date for the payment of the principal may be altered. The coupon payments may be reset periodically. There may be an option to convert or exchange one security for another security. 7 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Example A Canadian Government bond issued in the domestic market pays one-half of its coupon rate times its principal value every 6 months up to and including the maturity date. Thus, a bond with an 8% coupon and $5,000 face value maturing on December 1, 2xx5, will make future coupon payments of 4% of principal value, that is, $200 on every June 1 and December 1 between the purchase date and the maturity date. 8 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Obtaining Discount Rates Given that the cash flows are known with certainty ex ante, only the time-value needs to be accounted for, using the present value rule, which can be written as the follow PV CF t )= CF t 1+R0, t) ) t = B0, t) CF t, where PV CF t ) is the present value of the cash flow CF t received at date t, R0, t) is the annual spot rate or discount rate) at date 0 for an investment up to date t, andb0, t) isthepriceatdate0today)ofazero-couponbondor pure discount bond) paying $1 on date t. Note that it would be easy to obtain B0, t) if we can find zero-coupon bonds corresponding to all possible maturities. 9 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Obtaining Bond Price The price of a bond P 0 = PV Bond) = T t=1 CF t 1+R0, t) ) t = T B0, t) CF t. t=1 We must address the following important issues to turn the above simple principle into sound practice: Where do we get the discount factors B0, t) from? Do we use the equation to obtain bond prices or implied discount factors? Can we deviate from this simple rule? Why? 10 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Mathematics of Discounting Suppose that all cash flows and all discount rates across various maturities are identical and, respectively, equal to CF and y. Then T ) ) T CF PV CF = ) t = CF 1 1 ) 1+y y T. 1+y t=1 t=1 For a bond, we have P 0 = C y 1 1 1+y ) T ) + N 1+y ) T. where P 0 is the present value of the bond, T is the maturity of the bond, N is the nominal value of the bond, C = c N is the coupon payment, c is the coupon rate and y is the discount rate. 11 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Example Consider the problem of valuing a bond with a 5% coupon rate, annual coupon payments, a 10-year maturity and a $1,000 face value; assuming all discount ratesequalto6%. Step 1: The cash flows of the bond is CF 1 = CF 2 = = CF 9 = $50 and CF 10 =$1, 050. Step 2: The discount rate y =0.06. Step 3: The value of the bond is 9 50 P 0 = 1.06 + 1050 50 = t 1.0610 0.06 t=1 = $926.39913. 1 1 ) + 1000 1.06 10 1.06 10 12 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Priced at Par When the coupon rate is equal to the discount rate i.e., c = y or equivalently C = yn), then the bond value is equal to its face value. ) P 0 = cn 1 N 1 ) y T + ) T 1+y 1+y = yn y 1 1 1+y ) T ) + N 1+y ) T = N. 13 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Premium, Par & Discount More generally, depending on the relation between coupon rate, c, and discount rate, y, a bond may be priced: at a premium, P 0 > N; at par, P 0 = N; or at a discount, P 0 < N. P 0 = N c y 1 1 1+y ) T ) + ) 1 ) T, 1+y c > y c = y c < y P 0 > N, i.e., at a premium P 0 = N, i.e., at par P 0 < N, i.e., at a discount 14 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Perpetual Bond A bond paying a given coupon amount every year over an unlimited horizon is known as a perpetual bond. The price of such bond is ) T C C 1 P 0 = lim ) t = lim 1 ) T 1+y T y T 1+y = C y. t=1 15 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Annual vs. Semiannual Coupons Consider a bond with a coupon rate c, amaturityt year, a face value N, and a discount rate y. If the bond delivers coupon annually, its price T cn N P 0 = ) t + ) 1+y T = cn 1 1+y y t=1 1 1+y ) T ) + N 1+y ) T. If the bond delivers coupon semiannually, its price P 0 = 2T t=1 = cn y cn/2 1+y/2 ) t + 1 N 1+y/2 ) 2T 1 1+y/2 ) 2T ) + N 1+y/2 ) 2T. 16 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Time Basis & Compounding Frequency Conventions To apply present value formulas, one must have information about both the time basis usually interest rates are expressed on an annual basis) and the compounding frequency. Careful attention needs to be paid to the question of how an interest rate is defined: An amount $x invested at the T -year interest rate R T,n expressed on an annual basis and compounded n times per year grows to the amount after T years. x 1+ R ) nt T,n n 17 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Example If you invest $100 at a 6% 2-year annual rate with semiannual compounding, you would get: $100 ) 1+ 6% 2 after 6 months, $100 ) 1+ 6% 2 2 after 1 year, $100 ) 1+ 6% 3 2 after 1.5 year, $100 ) 1+ 6% 4 2 after 2 years. If you invest $100 at a 4% 3-year semiannual rate with semiannual compounding, you would get: $100 1 + 4%) after 6 months, $100 1 + 4%) 2 after 1 year, $100 1 + 4%) 3 after 1.5 year,. $100 1 + 4%) 6 after 3 years. 18 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Example If you invest $100 at a 3% 1-year semiannual rate with monthly compounding, you would get: $100 ) 1+ 3% 6 after 1 month, $100 ) 1+ 3% 2 6 after 2 months, $100 ) 1+ 3% 3 6 after 3 months,. $100 ) 1+ 3% 12 6 after 1 year. 19 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Effective Annual Yield The effective equivalent annual i.e., compounded once a year) yield R is defined as the solution to x 1+ R ) nt T,n = x 1 + R) T n or R = 1+ R ) n T,n 1. n Bond yields are often expressed on a yearly basis with semiannual compounding in the United States and in the United Kingdom, they are expressed on a yearly basis with annual compounding in France or Germany. To compare bond yields, one can always turn a bond yield into an effective annual yield EAY), that is, an interest rate expressed on a yearly basis with annual compounding. 20 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Continuous Compounding It seems desirable to have a homogeneous convention in terms of compounding frequency. This is where the concept of continuous compounding is useful. Let the compounding frequency increase without bound, lim x 1+ R ) nt T,n = x e Rc T n n where R c expressed on an annual basis is a continuously compounded rate. Note that R c = lim R T,n = lim ln 1+ R ) n T,n. n n n 21 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Continuous Compounding: Proof Since ln 1+ R ) n T,n = n ln 1+ R ) ln T,n = R T,n n n 1+ R T,n n R T,n n ) ln 1, R and lim T,n ln1+y) ln 1 n = 0 and lim n y 0 y lim ln 1+ R T,n n n = 1, we have ) n = lim n R T,n = R c. 22 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Future & Present Value Using Continuous Compounding The future and present value of any cash flow are, respectively, FV t CF 0 )=CF 0 e Rc t, PV CF t )=CF t e Rc t, where FV t CF 0 ) is the future value at date t of a cash flow CF 0 invested at date 0 at a R c continuously compounded rate, and PV CF t )isthepresent value at date 0 of a cash flow CF t received at date t. 23 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Effective Annual Yield Using Continuous Compounding The effective equivalent annual i.e., compounded once a year) rate R is the solution to or x e Rc T = x 1 + R) T R = e Rc 1. Note that the difference R R c is positive and it is actually small when R is small. Indeed, we know from numerical analysis that e y =1+y + y 2 2! + y 3 3! + + y k k! +, so that R R c as a first-order approximation. 24 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Different Types of Rates & Yields There are a host of types of interest rates involved in the fixed-income jargon. Coupon rates. Current yields. Yields to maturity. Spot zero-coupon rates. Forward rates. 25 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Coupon Rates The coupon rate is the stated interest rate on a security, referred to as an annual percentage of face value. It is called the coupon rate because bearer bonds carry coupons for interest payments. Each coupon entitles the bearer to a payment when a set date has been reached. Today, most bonds are registered in holders names, and interest payments are sent to the registered holder, but the term coupon rate is still widely used. Coupon or interest payment is commonly made twice a year in the United States, for example) or once a year in France and Germany, for example). The coupon rate is essentially used to obtain the cash flows and shall not be confused with the current yield. 26 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Current Yield The current yield y c is obtained using the following formula y c = cn P, where c is the coupon rate, N is the nominal value and P is the current price. For example, a par $1,000 bond has an annual coupon rate of 7%, so it pays $70 a year. If you buy the bond for $900, your actual current yield is 7.78% = $70 $900. If you buy the bond for $1,100, your actual current yield is 6.36% = $70 $1, 100. 27 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Yield to Maturity The yield to maturity YTM) is the single rate that sets the present value of the cash flows equal to the bond price, i.e., the YTM y solves the equation T t=1 CF t 1+y ) t = P, where CF t is the cash flow at time t, T is the number of cash flows, and P is the current price. 28 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Yield to Maturity Continue) More precisely, the bond price P is found by discounting future cash flows back to their present value as indicated in the two following formulas depending on the coupon frequency: When coupons are paid annually, P = T t=1 CF t 1+y ) t, where the yield denoted by y is expressed on a yearly basis with annual compounding, and T is the number of annual periods. When coupons are paid semiannually, P = 2T t=1 CF t 1+ y 22 ) t, where the yield denoted by y 2 is expressed on a yearly basis with semiannual compounding, and 2T is the number of semiannual periods. 29 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Exercise Consider a $1,000 face value 3-year bond with 10% annual coupon, which sells for 101. What is the yield to maturity of this bond? Consider a $1,000 nominal value 2-year bond with 8% coupon paid semiannually, which sells for 103 23 i.e., 103 23 ). What is the yield to 32 maturity of this bond? 30 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Answer 31 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Yield to Maturity & Internal Rate of Return The YTM is the internal rate of return IRR) of the series of cash flows. Hence, each cash flow is discounted using the same rate. We implicitly assume that the yield curve is flat at a point in time. An IRR is an average discount rate assumed to be constant over the different maturities. It is equivalently the unique rate that would prevail if the yield curve happened to be flat at date t which of course is not generally the case). It is computed by trial and error, but may be easily determined by using built-in functions in financial calculators or spreadsheet softwares e.g., the IRR function in Microsofts Excel). 32 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Yield to Maturity as a Total Return Rate A YTM may also be seen as a total return rate. Consider a bond with a 3-year maturity and a YTM of 10% selling for $875.66. The bond pays an annual coupon of $50 and a principal of $1,000 at maturity. If we buy the bond today, we will receive $50 at the end of the first year, $50 at the end of the second year and $1,050 after 3 years. Assuming that we reinvest the intermediate cash flows, i.e., the coupons paid after 1 year and 2 years, at an annual rate of 10%. The total cash flows we receive at maturity are $50 1 + 10%) 2 + $50 1 + 10%) + $1, 050 = $1, 165.50. Our investment therefore generates an annual total return rate y over the period, such that 1 + y) 3 = 1, 165.50 875.66 = y = 10%. 33 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Yield to Maturity & Price Under certain technical conditions, there exits a one-to-one correspondence between the YTM and the price of a bond. Thus, giving a YTM for a bond is equivalent to giving a price for the bond. It should be noted that this is precisely what is actually done in the bond market, where bonds are often quoted in YTM, i.e., YTM is just a convenient way of reexpressing bond price. A bond YTM is not a very meaningful number. This is because there is no reason one should discount cash flows occurring on different dates with a unique discount rate. Unless the term structure of interest rates is flat, there is no reason one would consider the YTM on a T -year bond as the relevant discount rate for a T -year horizon. The relevant discount rate is the T -year pure discount rate. In other words, YTM is a complex average of pure discount rates that makes the present value of the bonds payments equal to its price. 34 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Spot Zero-Coupon Rate The spot zero-coupon rate R0, t) is implicitly defined by B0, t) = 1 1+R0, t) ) t where B0, t) isthemarketpriceatdate0ofabondpayingoff$1atdatet. Note that such a bond may not exist in the market. The yield to maturity and the zero-coupon rate of a strip bond i.e., a zero-coupon bond) are identical. In practice, when we know the spot zero-coupon yield curve t R0, t), we are able to obtain spot prices for all fixed-income securities with known future cash flows. Zero-coupon rates make it possible to find other very useful forward rates and par yields. 35 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Forward Rate If R0, t) istherateatwhichyoucaninvesttodayinat period bond, we can define an implied forward rate or forward zero-coupon rate) between years t 1 and t 2 as 1+R0, t 2 ) F 0, t 1, t 2 t 1 )= 1+R0, t 1 ) ) t2 ) t1 1 t 2 t 1 1. It is the forward rate as seen from date t =0,startingatdatet = t 1,and with residual maturity t 2 t 1. Basically, it is the rate at which you could sign a contract today to borrow or lend between periods t 1 and t 2. In practice, it is very common to draw the forward curve τ F 0, t,τ) with rates starting at date t. Denote its continuously compounded equivalent as F c 0, t 1, t 2 t 1 ). 36 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Forward Rate: A Rate That Can Be Guaranteed The forward rate is the rate that can be guaranteed now on a transaction occurring in the future. Suppose we simultaneously borrow and lend $1 repayable at the end of 2 years and 1 year, respectively. The cash flows generated by this transaction are as follows: Today In 1 Year In 2 Year Borrow 2 Year) 1 1+R0, 2) ) 2 Lend 1 Year) 1 1+R0, 1) Net Cash Flow 0 1 + R0, 1) 1+R0, 2) ) 2 This is equivalent to borrowing 1 + R0, 1) in 1 year, repayable in 2 years at the amount 1+R0, 2) )2. 37 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Forward Rate: A Rate That Can Be Guaranteed Continue) Borrowing 1 + R0, 1) in 1 year and repaying 1+R0, 2) )2 in2yearshas the implied rate on the loan given by 1+R0, 2) ) 2 1+R0, 1) 1=F 0, 1, 1). Thus, F 0, 1, 1) is the rate that can be guaranteed now for a loan starting in 1 year and repayable after 2 years. 38 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Spot Zero-Coupon Rate & Forward Rate The spot zero-coupon rate R0, ) andforwardratesf 0,, ) are related as follow: R0, t) = 1+R0, 1) ) 1+F 0, 1, 1) ) 1+F 0, 2, 1) )...... 1+F 0, t 1, 1) )) 1 t 1 and F 0, t 1, t 2 t 1 )= 1+F 0, t1, 1) ) 1+F 0, t 1 +1, 1) ) 1+F 0, t 1 +2, 1) )...... 1+F 0, t 2 1, 1) )) 1 t 2 t 1 1. 39 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Forward Rate as a Break-Even Point The forward rate may be considered as the break-even point that equalizes the rates of return on bonds which are homogeneous in terms of default risk) acrosstheentirematurityspectrum. Suppose that a zero-coupon yield curve today from which we derive the forward yield curve starting in 1 year is as follow: Zero-coupon Rate Forward Rate Starting in 1 Year R0,1) 4.00% F0,1,1) 5.002% R0,2) 4.50% F0,1,2) 5.504% R0,3) 5.00% F0,1,3) 5.670% R0,4) 5.25% F0,1,4) 5.878% R0,5) 5.50% 40 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Forward Rate as a Break-Even Point Continue) The rate of return for the coming year of the 1-year zero-coupon bond is of course 4%, while the return on the 2-year bond depends on the selling price of the bond in 1 year. What is the level of the 1-year zero-coupon ratein1yearthatwouldensurethatthe2-yearbondalsohasa4%rate of return? The answer is 5.002%. With this rate, the price of the 2-year bond will rise from the initial 91.573 = 100/1.045 2 ) to 95.236 = 100/1.05002) in 1 year, generating a return of 4% over the period. The forward rate F 0, 1, 1) at 5.002% is the future level of the 1-year zero-coupon rate that makes the investor indifferent between the 1-year and the 2-year bonds during the year ahead. If the forward rate F 0, 1, 2) is 5.504%, a zero-coupon bond with a 3-year maturity also returns 4% for the coming year. Consequently, all the bonds have the same 4% return rate for the year ahead. Breakeven is therefore the future scenario that balances all bond investments. 41 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Instantaneous Forward Rate The instantaneous forward rate f t, s) is the forward rate seen at date t, starting at date s and maturing an infinitely small instant later on. It is a continuously compounded rate. The instantaneous forward rate is defined mathematically as follow: f t, s) = lim τ 0 F c t, s,τ). Note that f t, t) =rt) is the short-term interest rate at date t. Typically, this is the rate with a 1-day maturity in the market. By varying s between1dayand30years,itispossibletoplotthelevelof instantaneous forward rates at dates that are staggered over time. This is what is called the instantaneous forward yield curve. In practice, the market treats the instantaneous forward rate as a forward rate with a maturity of between 1 day and 3 months. It is especially useful for modeling purposes. 42 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Bond Par Yield A par bond is a bond with a coupon identical to its YTM, and its price is equal to its principal. We define the par yield ct )sothatat-year maturity bond paying annually a coupon rate of ct ) with a $100 face value quotes par, i.e., ct ) $100 1+R0, 1) + ct ) $100 1+R0, 2) ) 2 + + 1+cT ) ) $100 1+R0, T ) ) T = $100. Hence, ct )= 1 T t=1 1 1+R0, T ) ) T = 1 ) t 1+R0, t) 1 B0, T ) T. t=1 B0, t) 43 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Bond Par Yield Continue) In practice, the YTM curve suffers from the coupon effect. Two bonds having the same maturity but different coupon rates do not necessarily have the same YTM. In the case of an upward sloping curve, the bond that pays the highest coupon has the lowest YTM. To overcome this coupon effect, it is customary to plot the par yield curve. We can extract the par yield curve t ct), 0 < t T,whenwe know the zero-coupon rates R0, 1), R0, 2),...,R0, T ). Typically, the par yield curve is used to determine the coupon level of a bond issued at par. 44 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Overview The value of a bond moves in the opposite direction to a change in interest rates. If interest rates increase decrease), the price of a bond will decrease increase). A key to measuring the interest rate risk is the accuracy in estimating the value of the position after an adverse interest rate change. A valuation model determines the value of a position after an adverse interest rate move. Consequently, if a reliable valuation model is not used, there is no way to properly measure interest rate risk exposure. There are two approaches to measuring interest rate risk: 1 the full valuation approach, and 2 the duration & convexity approach. 45 / 63 William C. H. Leon MFE8812 Bond Portfolio Management The full valuation approach approach requires the re-valuation of a bond or bond portfolio for a given interest rate change scenario. The full valuation approach is a straightforward approach but can be very time consuming. If one has a good valuation model, assessing how the value of a bond will change for different interest rate scenarios measures the interest rate risk of the bond. This approach is sometimes referred to as scenario analysis because it involves assessing the exposure to interest rate change scenarios. 46 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Example Consider a $10 million par value position in a 20-year bond with 9% semiannual coupon. The current price of the bond is 134.67216 with a yield to maturity of 6%. The market value of the position is $13,467,216 = 134.67216% $10 million). Suppose that yields change instantaneously for the following three scenarios: 1 50 basis point increase; 2 100 basis point increase; and 3 200 basis point increase. The following table shows what will happen to the bond position if the yield on the bond increases from 6% to 1) 6.5%, 2) 7%, and 3) 8%: Yield New New New Percentage Scenario Change bp) Yield Price Value $) Change 1 50 6.5% 127.76054 12,776,054 5.13% 2 100 7.0% 121.35507 12,135,507 9.89% 3 200 8.0% 109.89639 10,989,639 18.40% 47 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Continue) A common question that often arises when using the full valuation approach is which scenarios should be evaluated to assess interest rate risk exposure. There may be specified scenarios established by regulators for regulated entities. It is common for regulators of depository institutions to require entities to determine the impact on the value of their bond portfolio for a 100, 200, and 300 basis point instantaneous change in interest rates. Risk managers tend to look at extreme scenarios to assess exposure to interest rate changes. This practice is referred to as stress testing. The state-of-the-art technology involves using a complex statistical procedure such as principal component analysis) to determine a likely set of interest rate scenarios from historical data. The full valuation approach can also handle scenarios where the yield curve does not change in a parallel fashion. 48 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Duration Duration is a measure of the approximate price sensitivity of a bond to interest rate changes. More specifically, duration of a bond, D, is the approximate percentage change in price for a change in rates, i.e. D = ΔP/P Δy = P P + 2 P Δy, 1) where P is the current price, Δy is the change in the yield, P P + is the price when yield decreases by Δy, and is the price when yield increases by Δy. 49 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Price-Yield Relationship for an Option-Free Bond 50 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

51 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Duration Continue) Duration is the linear approximation of the percentage price change. To improve the approximation provided by duration, an adjustment for convexity can be made. Combining duration with convexity to estimate the percentage price change of a bond caused by changes in interest rates is called the duration & convexity approach. Duration is interpreted as the approximate percentage change in price for a 100 basis point change in rates. A common question asked about this interpretation of duration is the consistency between the yield change that is used to compute duration using equation 1) and the interpretation of duration. Note that regardless of the yield change Δy used to estimate duration in equation 1), the interpretation is the same. i.e., if we used a 25 basis point change in yield to compute the prices used in the numerator of equation 1), the resulting duration is still interpreted as the approximate percentage price change for a 100 basis point change in yield. 52 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Modified Duration & Effective Duration Modified duration is the approximate percentage change in a bonds price for a 100 basis point change in yield assuming that the bonds expected cash flows do not change when the yield changes. This means that in calculating the values of P and P + for the value of duration, the same cash flows used to calculate P are used. Therefore, the change in the bonds price when the yield is changed is due solely to discounting cash flows at the new yield level. Effective duration is the approximate percentage change in a bonds price for a 100 basis point change in yield taking into account how the expected cash flows may change when the yield changes. Some valuation models for bonds with embedded options take into account how changes in yield will affect the expected cash flows. Thus, when P and P + are the values produced from these valuation models, the resulting duration takes into account both the discounting at different interest rates and how the expected cash flows may change. 53 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Exercise Consider a 20-years standard par bond with a 6% annual coupon. What is the duration, assuming the yield increases or decreases 10 basis points? 54 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Answer 55 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Application of Duration To approximate the percentage price change, ΔP/P, for a given change in yield, Δy in decimal), and a given duration, D, we may use the following formula: ΔP P = D Δy. The negative sign on the right-hand side of the equation is due to the inverse relationship between price change and yield change e.g., as yields increase, bond prices decrease). 56 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Exercise Consider a 20-years standard par bond with a 6% annual coupon. The duration of the bond is 11.47050 as computed earlier). 1 Suppose the yield increases by 20 basis points. What is the approximate percentage price change using duration? 2 How accurate is the approximation? 3 Repeat the analysis when the yield increases by 200 basis points. 57 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Answer 58 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Error Using Duration The duration measure indicates that the approximate percentage price change is the same regardless of whether interest rates increase or decrease. While for small changes in yield the percentage price change will be the same for an increase or decrease in yield, for large changes in yield this is not true. This suggests that duration is only a good approximation of the percentage price change for small changes in yield. 59 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Convexity Adjustment Duration is a first-order linear approximation for a small change in yield. The approximation can be improved by using a second approximation called the convexity adjustment. It is used to approximate the change in price that is not explained by duration. The formula for the convexity adjustment to the percentage price change is Convexity Adjustment to ΔP P = 1 2 C Δy )2, where Δy is the change in yield for which the percentage price change is sought and C = P + P + 2 P P Δy ) 2. To calculate the convexity adjustment, we may assume that, when the yield changes, the expected cash flows either do not change or they do change. In the former case, the resulting convexity is referred to as modified convexity adjustment; and in the later case, as effective convexity adjustment. 60 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Duration & Convexity Adjustment The following formula provides a better approximation of the percentage price change, ΔP/P, for a given change in yield, Δy in decimal), and a given duration, D, and convexity, C: ΔP P = D Δy + 1 2 C Δy )2. 61 / 63 William C. H. Leon MFE8812 Bond Portfolio Management Exercise Consider a 20-years standard par bond with a 6% annual coupon. The duration of the bond is 11.47050 as computed earlier). 1 Using a 10 basis points change in the yield, compute the convexity of the bond. 2 Suppose the yield increases by 200 basis points. What is the approximate percentage price change using duration and convexity? 3 How accurate is the approximation? 62 / 63 William C. H. Leon MFE8812 Bond Portfolio Management

Answer 63 / 63 William C. H. Leon MFE8812 Bond Portfolio Management