CHEAT SHEET MIDTERM 2

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Duquesne University *

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FINC-324

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Apr 3, 2024

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Duration is shorter than maturity for all bonds except zero coupon bonds – Duration is equal to maturity for zero coupon bonds An investor buys a 20-year semiannual bond with a coupon rate of 5% for $950. He plans to hold the bond for 8 years and then sell it. The investor expects to reinvest the first 6 coupon payments at 4.5% and the next 10 payments at 5.5%. He also expects the bond’s YTM at the end of the holding period to be 6%. Find annualized total return. 20-8=12 N=24 If the price and/or yield are not given, just assume the bond is a new bond that will trade at par value initially. Question #1 (6 points) Calculate the duration (in years) of a 5-year bond that offers a 6% coupon, paid semiannually. Question #2 (6 points) Calculate the duration of a bond identical to that described in question #1 except with a coupon rate of 10%. Comment on the difference in your answer. The duration of a 5-year, 10% coupon bond is 4.14 years, a little less than in (1) because more cash flow is paid sooner. Duration Changes: The duration of the 10-year, 6% bond is 7.66 years, less than twice the duration of the 5-year 6% bond. The reason that the duration does not increase 1 for 1 with maturity has to do with discounting. Recall that duration is just weighted-average maturity, where the weights are the fraction of the present value of the bond paid in each period. We know that discounting is nonlinear...longer-term cash flows are discounted more heavily than shorter term cash flows, and discounting progresses at an increasing rate. This is why the duration of perpetuity is not infinite. It only stands to reason that if the duration of a 10- year bond was twice that of a 5-year bond, then the duration of perpetuity would have to be infinite. With perpetuities, the effect of discounting at very long maturities becomes so great, that the weights in the duration calculation get very, very close to zero. Expectations vs Liquidity : Whether I believe the expectations theory or the liquidity premium theory has no bearing whatsoever on how I calculate forward rates. Forward rates are just mathematical definitions .... the rate that equates the returns on a long-term asset to a series of short-term assets. The thing that DOES change depending on which theory you believe is the process you go through to convert forward rates into forecasts of expected spot rates in the future. If you believe the expectations hypothesis, the expected spot rate equals the forward rate. If you believe the liquidity premium hypothesis, the forward rate equals the expected spot rate plus or minus a liquidity premium (plus if investors are short-term oriented, minus if investors are long-term oriented. forward rate = expected spot rate expectations view forward rate = exp. spot rate + liq. premium if people like ST forward rate = exp. spot rate - liq. premium if people like LT In all cases, the calculation of the forward rate is the same. A 6% coupon bond paying interest annually has a modified duration of 10 years, sells for $800, and is priced at a yield to maturity of 8%. If the YTM increases to 9%, the predicted change in price, using the duration concept, decreases by: Delta P/P=-D*(delta Y) Delta P/800=-10*0.01 Delta p=80$ Suppose your company has 3 mm in short-term liabilities with an average modified duration of 1.5, and 7 mm in intermediate-term liabilities with an average modified duration of 3.7. Both short and intermediate liabilities have the same yield. Suppose also that the company has 8 mm in assets. Find the duration of assets that allows net worth immunization of the company’s position. First find the average duration of the liability portfolio Weight of short term and long term liabilities are 0.3 = 3mm/10mm and 0.7= 7mm/10mm Then 0.3*1.5+0.7*3.7=3.04=D L Now we are using immunization identity 3.04*10=x*8 So the duration of the asset is 3.8 A 6% coupon bond with semiannual coupons has a convexity (in years) of 120, sells for 80% of par, and is priced at a yield to maturity of 8%. If the YTM increases to 9.5%, the predicted contribution to the percentage change in price, due to convexity, would be: % change= ½ convexity*(delta Y)^2= ½*120*0.015^2=0.0135=1.35% A bond with an annual coupon payment has a coupon rate of 8%, YTM of 10%, and a Macaulay duration of 9. The bond’s modified duration is: (9/(1+0.10)= mod duration of 8.18 years Which statement is true for the Macaulay duration of a zero-coupon bond? The Macaulay duration of a zero- coupon bond : Duration of the zero = time to maturity The interest rate risk of a bond normally is: Lower for higher coupons When interest rates decline, the duration of a 30-year bond selling at a premium: Increases Which bond has the longest duration: 15 year 6% coup You are managing a portfolio of $1 mm. Your target duration is 10 years, and you can choose from two bonds: a zero-coupon bond with a maturity of 5 years, and perpetuity, each currently yielding 5%.How much of each bond will you hold in your portfolio? How will these fractions change next year if the target duration is now nine years? The duration of perpetuity is 1.05/0.05=21 years. If w is the weight of the zero then W5+(1-w)21=10 W=11/16=0.6875 =vlent taxable yield: ytax=ytaxexempt/(1-T) Tax exempt ytm: ytaxexempt=ytaxable(1-T) T= marginal rate of marginal investor Term structure of interest rates ( or the Yield Curve) is a static function that relates the term to maturity to the yield to maturity for a sample of bonds at a given point in time. It represents a cross-section of yields for a category of bonds that are comparable in all respects but maturity Downward sloping yield curve preceds falling interst rates The two year forward one year (from now) , defined 1 f 2 , is the rate we expect to see on two-year bonds in one year.  The YTM on zero coupon bonds right now is called the spot rate andto be denoted s The zero will now have a duration of 4 years, and perpetuity still has 21 years duration. To get a portfolio duration of 9 years W4+(1-w)21=9 W=12/17=0.7059 Offering yield when adding call option to bond: The call feature provides a valuable option to the issuer since it can buy back the bond at a given call price even if the present value of the scheduled remaining payments are worth more than the call price. The investor will demand, and the issuer will be willing to pay, a higher yield on the issue as compensation for this feature. Call option on Duration and Convexity: The call feature will reduce both the duration (interest-rate sensitivity) and the convexity of the bond. The bond will not experience as large a price increase if interest rates fall. Moreover, the usual curvature that would characterize a straight bond will be reduced by a call feature, the price-yield curve flattens out as the interest rates fall and the option to call the bond becomes more attractive. In fact, at very low interest rates the bond exhibits negative convexity. Assume quote is given Feb 1st Forward rate: lending for 1 year starting from 2 years from today FR=((1+that yrs spot rate)^yrs)/(1+prev yrs spot rt)^that yr)-1

Once you've estimated the spot rates using regression analysis, calculate the 1-year forward rate 1 year from now, and calculate the 3-year forward rate 2 years from now. 1f1 = 5.78% (using my effective spot rates above) 1.0578 = 1.0536^2/1.0494 2f3 = 6.75% 0.0675 = (1.0619^5/1.0536^2)^(1/3)-1 Consider the impact on the 3-year 7% coupon bond if the yield increases by one basis point to 6.01% Term Structure Quiz: Under the expectation hypothesis, if the yield curve is upward-sloping, the marker must expect an increase in short term interest rates: True The liquidity premium theory indicates that all else being equal, longer maturities will have lower yields: False The Market segmentation theory contends that borrowers and lenders prefer particular segments of the yield curve: True The six month treasury bill spot rate is 4% and the one-year treasury bill spot rate is 5%. The implied six-month forward rate for 6 months from now is: 6%  [(1+0.05/1+0.04/2)-1]*2 [(1 + The 1-year Treasury bill spot rate)/(1 + The 6-month Treasury bill spot rate/N) - 1] * N Preferred habitat suggests that, for a sufficiently large premium, short-term investors can be induced to invest in long-term bonds and long-term investors can be induced to invest in short-term bonds 40= semi coupon 11=n*2- periods that already happened First, need to find at which tax rate investors will be indifferent between a corporate bond and municipal security . T=1- 0.09/0.12=25% Investors prefer a corp bond is they are in less than 25% tax bracket A $1000 corporate bond with a coupon rate of 8% maturing March 10, 2028, is purchased with a settlement date of July 26, 2022. The proper discount rate is 6%. Find the full and clean price of the bond. Corp Bond, munis, agency 30/360, Coupon treasury:act/365, MM/Swaps a/360 At which tax rate will investors prefer a corporate bond with a pre- tax yield-to-maturity of 12% and municipal security with a pre-tax yield-to-maturity of 9%? A bond has a current yield of 9% and a yield to maturity of 10%. Is the bond selling above or below par value? Explain. If YTM>CY, the bond must offer the price appreciation as it approaches the maturity date; hence it is selling below par. A 30-year maturity, 8% coupon bond paying coupons semiannually is callable in five years at a call price of $1,100. The bond currently sells at a yield to maturity of 7% (3.5% per half year). a. (3 points) What is the yield to call? Find PV First: N=60, I=3.5, FV=1000, Pmt=40, PV=1124.72 Use PV to find YTC: N=10, FV=1100 PMT=40 PV=-1124.72 YTC=3.368 semiannually b. (3 points) What is the yield to call if the call price is only $1,050? 2.976 c. (3 points) What is the yield to call if the call price is $1,100, but the bond can be called in two years instead of five years? 3.031 Which one of the following statements about the term structure of interest rates is true? The market segmentation theory contends that borrowers and lenders prefer particular segments of the yield curve. The differences between short and forward rates are most closely associated with which one of the following explanations of the term structure of interest rates? Liquidity Premium Theory 1. The yield to maturity on one-year zero coupon bonds is currently 7%; The YTM on two-year zeros is 8%. The Treasury plans to issue a two-year maturity coupon bond, paying coupons once a year with a coupon rate of 9%. The face value of the bond is $100. At what price will the bond sell?: ann coupons: P=9/1.07+109/(1.08)^2=101.86 b. What is YTM : n=2 FV=100 PMT=9 PV=-101.86 I=7.958 If the expectation theory of the yield curve is correct, what is the market expectation of the price that the bond will sell for the next year? 1f1=(1.08)^2/1.07-1=9.01% P=109/1.0901=99.99$ Recalculate your answer to [c] if you believe in the liquidity theory and you believe that the liquidity premium is 1%. If liquidity premium is 1%, then expected future short-term rate is E(r 2 )= 1 f 1 -liq.prem=9.01%-1%=8.01%  109/1.0801=100.92 Cite one major shortcoming for each of the following fixed-income yield measures: I Current yield does not account for capital gains or losses on bonds bought at prices other than par, reinvestment income on coupons. II YTM assumes that the bond is held till maturity and coupons are reinvested at the YTM rate III Realized return is affected by the forecast of reinvestment rates, holding period on the bond at the end of the holding period first calculate FV of reinvested coupons PV=0 PMT=35 N=6 I=3 FV=226.39$ The bond will be selling at par in 3 years 960(1+y) 6 6 =1226.39 y=4.1666% semiannual note: PV=FV/(1+r) n FV= PV*(1+r) n Bonds of Zello Corporation with a par value of $1,000 sell for $960, mature in five years, and have a 7% annual coupon rate paid semiannually. (1 point) Current yield, (3 points) Yield to maturity, (6 points) Realized compound yield for an investor with a three-year holding period and a reinvestment rate of 6% over the period. At the end of three years, the 7% coupon bonds with two years remaining will sell to yield 7%. I Current Yield =Coupon/Price=70/960=7.3% II N=10 PV=-960 FV=1000 PMT=35 YTM=4%*2=8% annually Holding other bond characteristics constant, duration is lower when (check all that applies: b. The coupon rate is higher, Time to maturity shorter , Yield to maturity higher For zero-coupon bonds duration is: Equal to their maturity For large changes in yields actual bond price change is best approximated by: Duration with convexity adjustment All else equal, the call-adjusted duration of a premium callable bond is: Zero Bond and Yield Quiz: Dirty price is always greater or equal to the clean price of a coupon bond: True When the yield is below the coupon rate the bond is priced: At a premium to its par value Bonds with ______ and ______ have higher interest rate sensitivity Longer Maturity / Lower Coupon More liquid bonds have __________ than less liquid bonds: Lower Yields

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