43. A 000 par value 5-year bod with 8.0% semiaual coupos was bought to yield 7.5% covertible semiaually. Determie the amout of premium amortized i the 6 th coupo paymet. (A).00 (B).08 (C).5 (D).5 (E).34 Course 44 May 000

40. Amog a compay s assets ad accoutig records, a actuary fids a 5-year bod that was purchased at a premium. From the records, the actuary has determied the followig: (i) (ii) (iii) The bod pays semi-aual iterest. The amout for amortizatio of the premium i the d coupo paymet was 977.9. The amout for amortizatio of the premium i the 4 th coupo paymet was 046.79. What is the value of the premium? (A) 7,365 (B) 4,784 (C) 6,549 (D) 48,739 (E) 50,445 Course 40 Form 00B

44 CHAPTER 4 which is the latest possible redemptio date, time k. It may be the case that the redemptio amout may vary accordig to whe redemptio takes place. If so, the a more detailed aalysis is ecessary to determie the miimum price of the bod, as illustrated i the Example 4.6. EXAMPLE 4.5 (Callable bod) (a) A 0% bod with face amout,000,000 is issued with the coditio that redemptio ca take place o ay coupo date betwee ad 5 years from the issue date. Fid the price paid by a ivestor wishig a miimum yield of (i) i () =., ad (ii) i () =.08. (b) Suppose the ivestor pays the maximum of all prices for the rage of redemptio dates. Fid the yield rate if the issuer chooses a redemptio date correspodig to the miimum price i each of cases (i) ad (ii) of part (a). (c) Suppose the ivestor pays the miimum of all prices for the rage of redemptio dates. Fid the yield rate if the issuer chooses a redemptio date correspodig to the maximum price i each of cases (i) ad (ii) of part (a). SOLUTION P =,000,000 + (.05.06) a.06, where is the umber of coupos util redemptio, = 4,5,,30. The rage of the price for this rage of redemptio dates is from 874,496 for redemptio at = 4 to 86,35 for redemptio at = 30. It is most prudet for the ivestor to offer a price of 86,35. As oted above, for a bod bought at a discout, the miimum price will occur at the latest possible redemptio date. (ii) The rage of prices is from,5,470 if redemptio occurs at years, to,7,90 if redemptio is at 5 years The prudet ivestor would pay,5,470. (a) (i) From Equatio (4.) [ ] (b) If the ivestor i (i) pays the maximum price of 874,496 (based o redemptio at = 4), ad the bod is redeemed at the ed of 5 years, the actual omial yield is.80%. If the ivestor i (ii) pays,7,90 (based o redemptio at = 30), ad the bod is redeemed at the ed of years, the actual omial yield is 7.76%.

BOND VALUATION 45 (c) If the ivestor i (i) pays 86,35 (based o 5 year redemptio) ad the bod is redeemed after years, the actual omial yield is.%, ad if the ivestor i (ii) pays,5,470 (based o year redemptio) ad the bod is redeemed after 5 years, the actual omial yield is 8.%. Equatio (4.) shows that for a bod bought at a discout, the loger the time to redemptio, the lower will be the price, with the reverse beig true for a bod bought at a premium. Thus for a callable bod for which the ivestor desires a miimum yield rate that is larger tha the coupo rate (a bod bought at a discout), the price should be based o the latest optioal redemptio date, ad for a callable bod for which the ivestor desires a yield rate that is smaller tha the coupo rate (bought at a premium), the price should be based o the earliest optioal redemptio date. If somethig more tha the miimum price is paid, the ivestor rus the risk of havig redemptio occur at a time which is to the ivestor s disadvatage (as i part (b) of Example 4.5), so that the actual yield to maturity is less tha the desired miimum. O the other had, if the ivestor pays the miimum price ad the actual redemptio date is other tha the oe o which that miimum price is based, the the ivestor will ear a yield rate greater tha the miimum desired (as i part (c) of Example 4.5). Suppose a bod is bought at a discout, so that P< F. The sooer the bod is redeemed, the sooer the ivestor will realize the gai of F P, so it is to the ivestor s disadvatage to have a later redemptio date. Sice the ivestor prices the bod assumig the redemptio will occur to his greatest disadvatage, the ivestor assumes the latest possible redemptio date. Similar reasoig i the case of a bod bought at a premium results i a ivestor choosig the earliest possible redemptio date for calculatig the price, sice if P> F the ivestor takes a loss of F P whe the bod is redeemed. It is to the ivestor s disadvatage to have this loss come early. Whe the first optioal call date arrives, the bod issuer, based o market coditios ad its ow fiacial situatio, will make a decisio o whether or ot to call (redeem) the bod prior to the maximum term. If the issuer is ot i a positio to redeem at a early date, uder appropriate market coditios, it still might be to the issuer s advatage to redeem the bod ad issue a ew bod for the remaiig term. As a simple illustratio of this poit, suppose i Example 4.5(a) that years after the issue date, the yield rate o a 3-year bod is 9%. If the issuer redeems the bod ad immediately

46 CHAPTER 4 issues a ew 3-year bod with the same coupo ad face amout, the issuer must pay,000,000 to the bodholder, but the receives,05,789 for the ew 3-year bod, which is bought at a yield rate of 9%. A callable bod might have differet redemptio amouts at the various optioal redemptio dates. It might still be possible to use some of the reasoig described above to fid the miimum price for all possible redemptio dates. I geeral, however, it may be ecessary to calculate the price at several (or all) of the optioal dates to fid the miimum price. Example 4.6 (Varyig redemptio amouts) A 5-year 8% bod with face amout 00 is callable (at the optio of the issuer) o a coupo date i the 0 th to 5 th years. I the 0 th year the bod is callable at par, i the th or th years at redemptio amout th 5, or i the 3, 4 th or 5 th years at redemptio amout 35. (a) What price should a ivestor pay i order to esure a miimum omial aual yield to maturity of (i) %, ad (ii) 6%? (b) Fid the ivestor s miimum yield if the purchase price is (i) 80, ad (ii) 0. Solutio (a) (i) Sice the yield rate is larger tha the coupo rate (or modified coupo rate for ay of the redemptio dates), the bod will be bought at a discout. Usig Equatio (4.E) from Exercise 4..5, we see that durig ay iterval for which the redemptio amout is level, the lowest price will occur at the latest redemptio date. Thus we must compute the price at the ed of 0 years, years ad 5 years. The correspodig prices are 77.06, 78.60 ad 78.56. The lowest price correspods to a redemptio date of 0 years, which is ear the earliest possible redemptio date. This example idicates that the pricipal of pricig a bod bought at a discout by usig the latest redemptio date may fail whe the redemptio amouts are ot level. (ii) For redemptio i the 0 th year ad the th or th years, the yield rate of.03 every six moths is smaller tha the modified coupo rate of.04 (for redemptio i year 0) or 00(.04) =.0348 5 (for redemptio i years or ). The modified coupo rate is

BOND VALUATION 47.096 <.03 for redemptio i the 3 th to 5 th years. Thus the miimum price for redemptio i the 0 th year occurs at the earliest redemptio date, which is at 9 years, ad the miimum price for redemptio i the th or th years also occurs at the earliest date, which is at 0 years. Sice g < j i the 3 th to 5 th years, the miimum price occurs at the latest date, which is at 5 years. Thus we must calculate the price of the bod for redemptio at 9 years, 0 years ad 5 years. The prices are 4.3, 3.48, ad 34.0. The price paid will be 4.3, which correspods to the earliest possible redemptio date. (b) (i) Sice the bod is bought at a discout (to the redemptio value), it is to the ivestor s disadvatage to have the redemptio at the latest date. Thus we fid the yield based o redemptio dates of 0 years, years ad 5 years. These omial yield rates are.40%,.75% ad.77% The miimum yield is.40%. (ii) Sice the bod is bought at a premium to the redemptio value i the 0 th year ad i the th ad th years, the miimum yield to maturity occurs at the earliest redemptio date for those periods, which is 9 years for the 0 th year ad 0 years for the th ad th years The bod is bought at a discout to the redemptio amout i the 3 th to 5 th years, so the miimum yield occurs at the latest redemptio date, which is 5 years. We fid the yield based o redemptio at 9 years, 0 years ad 5 years. These omial yield rates are 5.9%, 6.38% ad 7.5% The miimum is 5.9%. Through the latter part of the 980 s, bods callable at the optio of the issuer became less commo i the marketplace. The icreased competitio for fuds by govermets ad corporatios durig that period produced various icetives that are occasioally added to a bod issue. Oe such icetive is a retractable-extedible feature, which gives the bodholder the optio of havig the bod redeemed (retracted) o a specified date, or havig the redemptio date exteded to a specified later date. This is

CASHFLOW DURATION AND IMMUNIZATION 35 ad the Macaulay duratio is t t= t t= tk ( + i) tkt( + i) D = ( + i) DM = =. P t K ( + i) t= t t (7.3) Kt ( + i) Suppose that i Equatio 7.3 we defie the factor w t as wt =. P The Macaulay duratio ca the be writte i the form D = wt t. t= t Note that sice P Kt ( i) = +, it follows that wt =. The w t t= t= factors ca be thought of as weights, ad the Macaulay duratio is a weighted average of the paymet times from to. I this iterpretatio, the Macaulay duratio is a weighted average of the times at which the paymets are made. The weight applied to the K ( ) t t + i paymet at time t is wt =, which is the fractio of the overall P preset value of the series that is represeted by that particular paymet at time t. For a -year zero coupo bod, there is oly oe paymet, ad it occurs at time, so the weight for that paymet is sice it accouts for the etire preset value, ad hece the Macaulay duratio is =. I geeral, duratio is measured i uits of years. The duratio of a -year zero-coupo bod would be years. A coupo bod has relatively small coupo paymets ad the a large paymet o the maturity date. Therefore, the weights applied to the coupos would be relatively small ad the weight applied to the redemptio paymet would be relatively large. We would expect the Macaulay duratio of a coupo bod to be close to if the coupos are small. As the coupos get larger (relative to the redemptio amout) the duratio should get smaller. This is illustrated i the followig example. t EXAMPLE 7.3 (Duratio of a coupo bod) A bod with aual coupos has face amout F, coupo rate r per year, aual coupos util maturity, ad is valued at yield rate j per year.

35 CHAPTER 7 Calculate the duratio of the bod for all possible combiatios of parameters r =.05,.0,.5; =, 0, 30, 60; ad j =.05,.0,.5. SOLUTION The bod paymets at times,,, are Kt K = F+ Fr. Thus the duratio is = Fr for t =,,, ad D = t= t= t j t Fr v + F v t j Fr v + F v j j. At a yield rate of 5% per year, the duratio values are as show i Table 7.a. Note that the first term i the umerator of D is a icreasig auity with arithmetically icreasig paymets. TABLE 7.a Coupo Rate Coupos Util Maturity 0 30 60.05.95 8.08 6.4 9.876.0.93 7.70 4.38 8.77.5.880 6.797 3.63 8.39 At a yield rate of 0% per year, the duratios are give i Table 7.b. TABLE 7.b Coupo Rate Coupos Util Maturity 0 30 60.05.950 7.66.434.4.0.909 6.759 0.370 0.964.5.875 6.8 9.987 0.90 At a yield rate of 5% per year, the duratios are give i Table 7.c.

CASHFLOW DURATION AND IMMUNIZATION 353 TABLE 7.c Coupo Rate Coupos Util Maturity 0 30 60.05.948 7.70 8.09 7.689.0.905 6.37 7.79 7.67.5.870 5.77 7.55 7.665 For a -year bod with aual coupos at rate r per year ad valued at a effective aual yield rate of i per year, the Macaulay duratio of the + i + i+ ( r i) bod ca be show to be D = i. r[( + i) (See Exercise 7..3.) ] + i 7.. DURATION OF A PORTFOLIO OF SERIES OF CASHFLOWS Suppose that m separate series of aual cashflows are uder cosideratio. Suppose that each cashflow series is a -year series, with the paymets for cashflow series k deoted ( k ) ( k ) ( k c ), c,, c. At effective aual iterest rate i the preset value of cashflow series k is ( k) ( k) ( k) X = c ( + i) + c ( + i) + + c ( + i), for k =,,, m. The k d th k Macaulay duratio of the k cashflow series is Dk = ( + i) di X, so X k that Dk Xk = ( + i) d Xk. di The aggregate preset value of the collectio of all series of cashflows is m X = X, ad k= k m k= d X = d X k. The Macaulay duratio of the di di combiatio of all the series of cashflows is D m m ( ) d d X + i Xk Dk Xk di k = di k = = ( + i) = =. X X X Xk If we defie the factor v k to be v k =, the X m m D = v D, ad k k= v =. We see that the Macaulay duratio of the overall portfolio of k k = k