A Closer Look at Bond Risk: Duration

Transcription

1 W E B E X T E S I O 4C A Closer Look t Bond Risk: Durtion This Extension explins how to mnge the risk of bond portfolio using the concept of durtion. BOD RISK In our discussion of bond vlution in Chpter 4, we discussed interest rte nd reinvestment rte risk. Interest rte (price) risk is the risk tht the price of debt security will fll s result of increses in interest rtes, nd reinvestment rte risk is the risk of erning return less thn expected when debt principl or interest pyments re reinvested t rtes less thn the originl yield to mturity. To illustrte how to reduce interest rte nd reinvestment rte risks, we will consider firm tht is obligted to py worker lump sum retirement benefit of $10,000 t the end of 10 yers. Assume tht the yield curve is horizontl, the current interest rte on ll Tresury securities is 9%, nd the type of security used to fund the retirement benefit is Tresury bonds. The present vlue of $10,000, discounted bck 10 yers t 9%, is $10,000(0.4224) $4,224. Therefore, the firm could invest $4,224 in Tresury bonds nd expect to be ble to meet its obligtion 10 yers hence. 1 Suppose, however, tht interest rtes chnge from the current 9% rte immeditely fter the firm hs bought the Tresury bonds. How will this ffect the sitution? The nswer is, It ll depends. If rtes fll, then the vlue of the bonds in the portfolio will rise, but this benefit will be offset to greter or lesser degree by decline in the rte t which the coupon pyment of 0.09($4,224) $ cn be 1 For the ske of simplicity, we ssume the firm cn buy frction of bond.

2 4C-2 Web Extension 4C A Closer Look t Bond Risk: Durtion reinvested. The reverse would hold if interest rtes rose bove 9%. Here re some exmples (for simplicity, we ssume nnul coupons): 1. The firm buys $4,224 of 9%, 10-yer mturity bonds; rtes fll to 7% immeditely Portfolio vlue t Future vlue of 10 interest pyments Mturity the end of 10 yers of $ ech compounded t 7% vlue $5,252 $4,224 $9,476 Therefore, the firm cnnot meet its $10,000 obligtion, nd it must contribute dditionl funds. 2. The firm buys $4,224 of 9%, 40-yer bonds; rtes fll to 7% immeditely fter the purchse nd remin t tht level: Portfolio vlue t $5,252 Vlue of 30-yer 9% bonds when r d 7% the end of 10 yers $5,252 $5,272 $10,524 In this sitution, the firm hs excess cpitl t the end of the 10-yer period. 3. The firm buys $4,224 of 9%, 10-yer bonds; rtes rise to 12% immeditely Portfolio vlue t Future vlue of 10 interest pyments Mturity the end of 10 yers of $ ech compounded t 12% vlue $6,671 $4,224 $10,895 This sitution lso produces funding surplus. 4. The firm buys $4,224 of 9%, 40-yer bonds; rtes rise to 12% immeditely Portfolio vlue t $6,671 Vlue of 30-yer 9% bonds when r d 12% the end of 10 yers $6,671 $3,3 $9,874 This time, shortfll occurs. Here re some generliztions drwn from the exmples: 1. If interest rtes fll nd the portfolio is invested in reltively short-term bonds, then the reinvestment rte penlty exceeds the cpitl gins, so net shortfll occurs. However, if the portfolio hd been invested in reltively long-term bonds, drop in rtes would produce cpitl gins tht would more thn offset the shortfll cused by low reinvestment rtes. 2. If interest rtes rise nd the portfolio is invested in reltively short-term bonds, then gins from high reinvestment rtes will more thn offset cpitl losses, nd the finl portfolio vlue will exceed the required mount. However, if the portfolio hd been invested in long-term bonds, then cpitl losses would more thn offset reinvestment gins, nd net shortfll would result.

3 Durtion 4C-3 IMMUIZATIO DURATIO If compny hs mny csh obligtions expected in the future, the complexity of estimting the effects of interest rte chnges is obviously excerbted. Still, methods hve been devised to help del with the risks ssocited with chnging interest rtes. Severl methods re discussed in the following sections. Bond portfolios cn be immunized ginst interest rte nd reinvestment rte risk, much s people cn be immunized ginst the flu. In brief, immuniztion involves selecting bonds with coupons nd mturities such tht the benefits or losses from chnges in reinvestment rtes re exctly offset by losses or gins in the prices of the bonds. In other words, if bond s reinvestment rte risk exctly mtches its interest rte price risk, then the bond is immunized ginst the dverse effects of chnges in interest rtes. To see wht is involved, refer bck to our exmple of firm tht buys $4,224 of 9% Tresury bonds to meet $10,000 obligtion 10 yers hence. In the exmple, we see tht if the firm buys bonds with 10-yer mturity nd interest rtes remin constnt, then the obligtion cn be met exctly. However, if interest rtes fll from 9% to 7%, shortfll will occur becuse the coupons received will be reinvested t rte of 7% rther thn the 9% reinvestment rte required to rech the $10,000 trget. But, suppose the firm hd bought 40-yer rther thn 10-yer bonds. A decline in interest rtes would still hve the sme effect on the compounded coupon pyments, but now the firm would hold 9% coupon, 30-yer bonds in 7% mrket 10 yers hence, so the bonds would hve vlue greter thn pr. In this cse, the bonds would hve risen by more thn enough to offset the shortfll in compounded interest. Cn we buy bonds with mturity somewhere between 10 nd 40 yers such tht the net effect of chnges in reinvested csh flows nd chnges in bond vlues t yer 10 will lwys be positive? Yes, s we explin below. The key to immunizing portfolio is to buy bonds tht hve durtion equl to the yers until the funds will be needed. Durtion cnnot be defined in simple terms like mturity, but it cn be thought of s the weighted verge mturity of ll the csh flows (coupon pyments plus mturity vlue) provided by bond, nd it is exceptionlly useful to help mnge the risk inherent in bond portfolio. The durtion formul nd n exmple of the clcultion re provided below, but first we present some dditionl points bout durtion: 1. Durtion is similr to the concept of discounted pybck in cpitl budgeting, in the sense tht the longer the durtion, the longer funds re tied up in the bond. 2. To immunize bond portfolio, buy bonds tht hve durtion equl to the number of yers until the funds will be needed. In our exmple, the firm should buy bonds with durtion of 10 yers. 3. Durtion is mesure of bond voltility. The percentge chnge in the vlue of bond (or bond portfolio) will be pproximtely equl to the bond s durtion times the percentge point chnge in interest rtes. 2 Therefore, 2 percentge 2 Actully, durtion mesures the negtive of the percentge chnge in bond price. Durtion is mesure of the bond price s elsticity with respect to the interest rte. This elsticity mesure is vlid only for smll chnges in the interest rte.

4 4C-4 Web Extension 4C A Closer Look t Bond Risk: Durtion point increse in interest rtes will lower the vlue of bond with 10-yer durtion by bout %, but the vlue of 5-yer durtion bond will fll by only 10%. As this exmple shows, if Bond A hs twice the durtion of Bond B, then Bond A lso hs twice the voltility of Bond B. A corporte tresurer (or ny other investor) who is terribly concerned bout declines in the mrket vlue of his or her portfolio should buy bonds with low durtions. (This is importnt even if the investor buys bond mutul fund.) 4. The durtion of zero coupon bond is equl to its mturity, but the durtion of ny coupon bond is less thn its mturity. (Remember, the durtion is weighted verge mturity of the csh flows, the only csh flow from zero occurs t mturity, nd coupon bonds hve csh flows prior to mturity.) Further, the higher the coupon rte, the shorter the durtion, other things held constnt, becuse high-coupon bond provides significnt erly csh flows even if it hs long mturity. Durtion is clculted using this formul: t( ) = t( ) V B (4C-1) Here r d is the required return on the bond, is the bond s yers to mturity, t is the yer ech csh flow occurs, nd is the csh flow in yer t ( IT for t nd IT M for t, where IT is the interest pyment nd M is the principl pyment). otice tht the denomintor of Eqution 4C-1 is simply the vlue of the bond, V B. To simplify clcultions in Excel, we define the present vlue of csh flow t (PV of ) s PV of = We cn rewrite Eqution 4C-1 s: t(pv of ) V B (4C-2) To illustrte the durtion clcultion, consider -yer, 9% nnul coupon bond bought t its pr vlue of $1,000. It provides csh flows of per yer for 19 yers, nd $1,090 in the th yer. To clculte durtion, we used n Excel model s shown in Figure 4C-1; see the worksheet Web 4C in the file IFM10 Ch04 Tool Kit.xls for detils. Column 1 of Figure 4C-1 gives the yer ech csh flow occurs, Column 2 gives the csh flows, Column 3 shows the PV of ech csh flow, nd Column 4 shows the product of t nd the PV of ech csh flow. The sum Column 3 is the vlue of the bond, V B. The durtion is equl to the vlue of the bond divided by the sum of Column 4. Since this -yer bond s 9.95 durtion is close to tht of our illustrtive firm s 10-yer libility, if the firm bought portfolio of these -yer bonds nd then

5 Durtion 4C-5 FIGURE 4C-1 Durtion See the worksheet Web 4C in IFM10 Ch04 Tool Kit.xls on the textbook s Web site. A 7 Inputs 8 Yers to mturity = 9 Coupon rte = 10 Annul pyment = 11 Pr vlue = FV = 12 Going rte, r = t (1) B C D E 9.00%.0 $1, % (2) $1,090 V B = PV of (3) $82.57 $75.75 $69.50 $63.76 $58.49 $53.66 $49.23 $45.17 $41.44 $38.02 $34.88 $32.00 $29.36 $26.93 $24.71 $22.67 $.80 $19.08 $17.50 $ $1, Sum of t(pv of ) = V B / Sum of t(pv of ) = t(pv of ) (4) , $9, reinvested the coupons s they cme in, the ccumulted interest pyments, plus the vlue of the bond fter 10 yers, would be close to or exceed $10,000 whether interest rtes rose, fell, or remined constnt t 9%. See IFM10 Ch04 Tool Kit.xls for n exmple. There is lso n Excel function for durtion; the Tool Kit lso demonstrtes its use. Unfortuntely, other complictions rise. Our simple exmple looked t single interest rte chnge tht occurred immeditely fter funding. In relity, interest rtes chnge every dy, which cuses bonds durtions to chnge, nd this, in turn, requires tht bond portfolios be reblnced periodiclly to remin immunized.