Chapter 11 - Duration, Convexity and Immunization

1 Chapter 11 - Duration, Convexity and ImmunizationSection - DurationConsider two opportunities for an investment of $1, : Pays $610 at the end of year 1 and $1,000 at the end of year 3B: Pays $450 at the end of year 1, $600 at the end of year 2 and$500 at the end of year have a yield rate ofi=.25 because( ) 1=.8,1000= (.8)(610) + (.8)3(1000)and1000= (.8)(450) + (.8)2(600) + (.8)3(500).11-1 The repayment patterns of these two investments are quite differentand we seek to compare them on the basis of the timing of therepayments. Getting the repayments sooner would be advantageousif reinvestment yield rates are above the current yield of thisinvestment (in the above settingi=.25), whereas delaying therepayments is advantageous if the reinvestment interest rates arelower than the current yield of Equated Time (See section ) provides a simple answerto measure the timing of the repayments:HereRtdenotes a return (Rt>0 is a payment back to the investormade at timet).

2 11-2 Example: (from page 11-1)A:t=1(610)+3(1000)610+1000= :t=1(450)+2(600)+3(500)450+600+500= money is returned faster under investment - - - - - - - - - -A better index would also take into account the current value of thefuture repayments:Macaulay Duration:Here the investment yield rate is used in . The quantitydis adecreasing function : (from page 11-1)A:d=1(.8)(610)+3(.8)3(1000)(.8)(610 )+(.8)3(1000)= :d=1(.8)(450)+2(.8)2(600)+3(.8)3(500)(.8 )(450)+(.8)2(600)+(.8)3(500)= - - - - - - - - - -Bothtanddare weighted averages of the return times. Inttheweights are the return amounts and indthe weights are the- - - - - - - - - - -The (net) present value of a set of returns is:It represents the value of an investment today. We now focus on it asa function of the current interest interest rates frequently change, the volatility of the presentvalue to changes iniis very important.

3 It is measured withVolatility:The minus sign is included becauseP(i)is a decreasing function ofiand henceP (i)<0. So including the minus sign makes the valuepositive and therefore makes larger values of indicate morevolatility (susceptibility to changes ini), relative to the magnitude ofP(i).We now relate volatility to duration by examining their P (i)P(i)= ddi( nt=11(1+i)tRt)( nt=11(1+i)tRt)Thus our measure of volatility is often called modified Duration, even though its purpose is quite different from that of : (from page 11-1)A: = (.8)( ) = : = (.8)( ) = does make sense that an investment that takes longer to achieveits return will be more susceptible to changes in the interest also that by the definition of ,P (i) = P(i) implieslimh 0P(i+h) P(i)h= P(i) orP(i+h) P(i).= h P(i)which produceswhenhis small. Typically, this approximation produces a value thatis below the actual value ofP(i+h)whenh6= continuous compounding at a constant force of interest ,d= = nt=1te tRt( nt=1e tRt)that is, Macaulay duration and modified duration are the same.

4 Seepages 455-456 in the - - - - - - - - - -Example:Consider a zero coupon bond that makes one payment of C at theend ofnperiods, with a effective interest rate ofifor each nC nC=n =n 11-8 Example:Consider an annuity immediate with payments ofkat the end ofeach ofnperiods and an interest rate nt=1tk nt=1k=kn(n+1)2nk=n+12d= nt=1t tk nt=1 tk =d 11-9 Example:Consider a perpetuity immediate with payments ofkat the end ofeach period and an interest rate ofiand =11+ t=1tk t=1k(This is undefined.)d= t=1t tk t=1 tk= t=1t t 1limn ( (1 n)(1 ))= (1 ) t=1[dd t]= (1 )dd [ t=1 t]= (1 )dd [ 1 ]= (1 )(1 ) + (1 )211-10 Exercise 11-6: The current price of an annual coupon bond is derivative of the price of the bond with respect to the yield tomaturity is -650. The yield to maturity is an effective rate of 7%.(a) Calculate the Macaulay duration of the bond.

5 (b) Estimate the price of the bond using the approximation formulaon page 11-7 when the yield is 8% instead of 7%.11-11 Section - ConvexityTypically the present value of a set of cash flows decreases as afunction of the interest ratei. In fact, this function is most often aconvex decreasing function. A second order Taylor series expansionwill capture the curvature in addition to the trend and will often wellapproximate the changes in the function asichanges, at least forsmall changes ini. In the previous section we let = P (i)P(i)where the minus sign was inserted becauseP (i)is usuallynegative. Similarly, we now letwhich is called the Convexity of the present value of the cash second order Taylor series approximation ofP(i)then producesWe also note thatd di=ddi[ P (i)P(i)]= P(i)P (i) + [P (i)]2[P(i)]2 Recall that describes the sensitivity ofP(i)to changes ,cplays a role in describing the sensitivity of to thatP(i) =n t=1(1+i) tRt,P (i) =n t=1 t(1+i) (t+1)RtandP (i) =n t=1t(t+1)(1+i) (t+2) : Annuity Immediate (See page 11-9)c= nt=1t(t+1) t+2k nt=1 tk=(1 ) (1 n)n t=1t(t+1) t+211-14In continuous compounding settings with a constant force of interestdescribed by ,P( ) =n t=1e tRt,P ( ) =n t=1te tRtandP ( ) =n t=1t2e these settings, the Macaulay Convexity is defined as:P ( )P( )= nt=1t2e tRt nt=1e tRt11-15 Exercise 11-11.

6 A loan is to be repaid with payments of $1,000 atthe end of year 1, $2,000 at the end of year 2, and $3,000 at the endof year 3. The effective rate of interest isi=.25. Find (a) theamount of the loan, (b) the duration, (c) the modified duration, and(d) the - Interest Sensitive Cash FlowsSome cash flow settings have present values that are quite sensitiveto changes inibecause the returns themselves depend are callable bonds and mortgages without a prepaymentpenalty. To better capture the volatile nature of the present value, thefunctionP (i)is approximated viaP (i).=P(i+h) P(i h)2hand for smallhthe effective volatility is described bywhere the order in the numerator is reversed to make the for smallh, the effective Convexity is described (i h) P(i)h P(i) P(i h)hhP(i)Again the order of these differences is chosen to make this positive,sinceP(i h) +P(i+h)>2P(i)for a decreasing convex :A homebuyer takes out a 30-year $100,000 loan at 6% convertiblemonthly.

7 At the end of 15 years, the homebuyer can pay off the loanif interest rates fall, but will keep the existing loan if they rise or staythe same. Finddeandceusing 7% and 5%, that ish=. - - - - - -100,000=P(.06) =360 t=1( ) t(monthly pmt)produces(monthly pmt)=100,000 360t=1( ) t=(1 )100,000 (1 360)= here = (1+.0612) 1= ( ) then computeP(.07) =360 t=1(1+.0712) t( ) =90, addition we find=111, we have used the outstanding loan balance at the end of 15years to be( )a180|.0512=75, follows , 90, (.02)(100,000)= , +90, 2(100,000)(.01)2(100,000)= - Analysis of PortfoliosCompanies, investment funds, etc. all have multiple securities, eachof which produce a separate cash flow. The present value of theportfolio is the sum of the present values of the securities thatcomprise it, that isP=P1(i1) +P2(i2) + +Pm(im),with each security having its individual yield rate.

8 The modifiedduration of the portfolio is : = P P=P1(i1)P( P 1(i1)P1(i1))+ +Pm(im)P( P m(im)Pm(im))=P1(i1)P( 1)+ +Pm(im)P( m)which is a weighted average of the modified durations with weightsthat are the fraction of the total present value in the the Convexity of the portfolio becomesc=P P=P1(i1)P(c1)+ +Pm(im)P(cm),a weighted average of the individual assessing a portfolio, separate securities have different startdates and conversion periods. Thus it becomes necessary tomeasure duration at any point in time, not just at start dates orconversion periods. When measuring duration of any single security,we note that its duration decreases over time. We also note that, asan average time until future payment, the duration increases slightlyright after a payment is made, creating a zig-zag plot ofdover the securities in a portfolio differ in their yield rates andconversion periods, it is difficulty to measure the effect of increasingiby 100 basis points (100 basis points = 1%).

9 So when assessing aportfolio, it is typical to first standardize the conversion periods andyield rates to annual values before they are 11-22: A $60K portfolio is constructed with $10K used tobuy 2-year zero coupon bonds, $20 used to buy 5-year zero couponbonds and $30K used to buy 10-year zero coupon bonds. The yieldrates of the bonds are unknown. Calculate the Macaulay convexityof the portfolio at - - - -c=1060(22e2 110e2 110)+2060(52e5 220e5 220)+3060(102e10 330e10 330)=16(4) +26(25) +36(100) = 11-20:A 3-year loan at 10% effective is being repaid with level annualpayments at the end of each year.(a) Calculate the jump in duration at the time of the first payment.(b) Rework (a) at the time of the second payment.(c) Compare the answers to (a) and (b) and verbally explain - Matching Assets and LiabilitiesFinancial institutions must have the assets available to coverliabilities when they arise.

10 Many types of liabilities are known inadvance. It is therefore possible to set up investments, like bonds,so that the inflow of cash from the bonds will match the outflowneeded to cover the liabilities due at each point in time. This strategyis calledExample A company has a $10,000 liability due at the end of year 1and a $12,000 liability due at the end of year 2. It can purchase1-year zero coupon bonds at 8% effective and 2-year zero couponbonds at 9% effective. What is the cost of implementing an absolutematching strategy today?$10, +$12,000( )2= $9, + $10, $19, :Suppose the liabilities in the previous example are financed with a1-year zero coupon bond with a yield rate of 6% and a 2-year 5%annual coupon bond with a yield rate of 7%. What is the cost today?- - - - - - - -(a) 2-year bond:At the end of year 2 we needSo F= 11, . Also with = , the price of this 2-year bond is(b) 1-year bond:At the end of year 1 we need= (11, )(.)