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Generalized solutions of higher-order duration measures
Journal
of Banking
and Finance
14 (1990)
1143-1150.
North-Holland
GENERALIZED SOLUTIONS OF HIGHER-ORDER DURATION MEASURES
Sanjay K. NAWALKHA University OJ Massachusetts, Received November
and Nelson J. LACEY* Amherst, MA 01003, USA
1989, final version
received January
1990
Higher-order duration measures have been shown to permit near perfect immunization. This paper presents formulae for these measures which are valid not only at coupon payment dates but between these dates as well. We show that errors caused by using formulae valid only at the coupon payment dates increase as one advances from lower-order to higher-order duration measures. A portfolio immunization example comparing both traditional and generalized closedform solutions is given.
1. Introduction
Duration research has witnessed a shift away from single-factor models and their restrictive term structure assumptions towards multiple factor models. Single-factor duration models [see, e.g., Bierwag (1977)] assume a particular stochastic process that governs the term structure movements. Because the assumed stochastic process does not often confirm to the actual movements in the term structure, even sophisticated single-factor duration models have failed to demonstrate significant improvement vis-a-vis the simplest single-factor model [see, e.g., the evidence presented by Bierwag et al. (1983)]. In contrast, higher order duration models allow for changes in spot yields across the term structure to be captured by more than one factor such that these changes are not perfectly correlated. Examples of higher-order duration models include Fong and Vasicek’s (1983) M-Square measure, the two-factor model of Bierwag et al. (1988), the duration vector of Chambers et al. (1988), and the generalized polynomial model of Prisman and Shores (1988). In terms of performance, higher-order duration measures have demonstrated significant improvement over the traditional scalar models, and more importantly, have demonstrated near perfect immunization [see, e.g., Chambers et al. (1988)]. In this paper we derive closed-form solutions of higher-order duration *Correspondence address: Department of General Business and Finance, ment, University of Massachusetts, Amherst, MA 01003, U.S.A. 0378-4266/90/%03.50
0
1990-Elsevier
Science Publishers
B.V. (North-Holland)
School
of Manage-
1144
SK. Nawalkha and N.J. Lmey, Higher-order
duration measures
measures which are valid at and between coupon payment dates. We show that the formulae presented by Nawalkha and Lacey (1988), valid only at coupon payment dates, introduce errors that are quite significant and which magnify when moving from lower-order to higher-order duration measures. In addition, we illustrate that the composition of an immunized portfolio is altered significantly when optimal portfolio positions are generated through the more correct measures presented here.
2. Derivation of generalized formulae Proper bond valuation between coupon payments must include interest accrued between the last coupon payment and the current time period. Let N be the total number of coupons due until maturity. Defining P as the bond’s price, f as the time elapsed since the date of last coupon payment relative to time between two coupon payments, fC as the amount of interest accrued, C as the dollar value of the coupon payment per period, F as the face value of the bond, and i as the bond’s adjusted yield to maturity, the value of a bond between coupon payments can be given by:
P+fC=
t
C/(l+i)‘-J+F/(l+i)N-‘.
(1)
1=1
The adjusted yield to maturity contrasts with the reported yield to maturity by including any accrued interest. Generalized solutions for various immunization measures are derived from a generalized mth order duration measure given as
Substituting F =C/c in eq. (2), where c is the coupon rate per period, and rewriting eq. (2) defines an equivalent duration measure: D(m) = [i[cS,( 1 + i)N-f + (N -f)“]]/[c[(
1 + i)N- l] + i],
(3)
where S,= $J [(r-f)“/(l 1=1
+i)‘-f].
The generalized formula given for D(m) in eq. (3) is equivalent
to the
SK. Nawalkha
and NJ.
Lacey, Higher-order
1145
duration measures
traditional closed-form solution of D(m) given by Nawalkha and Lacey (1988) for the special case of f =O. The traditional measure is given by:’ D(m) = [i(cS,(l + i)N+ N”)]/[c(( 1 + i)N- 1) + i],
(3’)
where t”/( 1 + i)‘.
S, = i 1=1
In eq. (3), if a solution of S, can be found, then the closed-form solution of D(m) can be obtained through substitution of S, in eq. (3). A closed-form solution for S, exists’ and is given by
S_=+
(l-f)“(l [
(1 +N--f)” (1 +i)N-f
+i)f+mil &S,-r=o
1
for all
m&l,
(4)
where m!
and
mC’=(m-t)!t!
S,,,co=(l +i)f -[l-&l i
for
m=O.
The generalized formula given for S(m) in eq. (4) is equivalent traditional closed-form solution of S(m) for the special case off=@
s_+[ l+m$
,C,S,-
s
1=0
1
for all
rnz 1,
to the
(4’)
where &=i[l-
&]
for
m=O.
2.1. Generalized D( 1) solution To obtain the generalized closed-form solution of D(l), we first obtain the closed-form of S, by substituting m= 1 in eq. (4). Substituting the value of S1 and m= 1 in eq. (3) and further simplifying gives: ‘For j=O, the number of coupon payments due prior to maturity is equal to the number of periods until maturity, and the adjusted yield to maturity is equal to the reported yield to maturity. 2The proof of the closed-form solution for S,, m= 1,2,. . , n, can be obtained from the authors.
1146
S.K. Nawalkha
D(l),c[(l
and N.J. Locey,
+i)“(l +i-fi)-(1
Higher-order
duration measures
+i)]+i(N--f)(i-cc)
ci[( 1 + i)N- l] + i2
(5)
The generalized D(1) solution above is equivalent to the solution for Macaulay duration given by Chua (1988). Also, the generalized solution given in eq. (5) is equivalent to the traditional closed-form solution of D(1) for the special case of f = 0: D(l)=c(l+i)[(l+i)N-l]+(Ni)(i-c) ci[(1+i)N-l]+i2
’
(5’)
2.2. Generalized D(2) solution A widely applied duration measure is D(2), defined as the weighted average time to maturity squared of bond’s cash flows. One interesting aspect of this measure is that it is subject to at least three different interpretations; the second element in Chambers et al. (1988) duration vector that captures nonparallel slope shift in the term structure, the second factor of Bierwag et al. (1987, 1988) model, and a component of Fong and Vasicek’s (1983) M-square risk measure, given by M-square = D(2) - 2HD( 1) + H2,
(6)
where H is the portfolio’s planning horizon. The solution for D(2) is obtained by following steps similar to those used in obtaining D( 1): D(2) = c{(l+i)N[(1+i-fi)2+1+i]-(1+i)[2(1+Ni-fi)+i]}+i2(N-f)2(i-c) ci2[(1+i)N-l]+i3 (7)
The generalized closed-form solution of D(2) given in eq. (7) is equivalent to the traditional closed-form solution of D(2) reported by Nawalkha and Lacey ( 1988)3 for the special case off =0: D(2),~{(1+i)(2+i)[(1+i)N-l]-(2Ni)(l+i)}+(N2i2)(i-c) ci2[( 1 + i)N- l] + i3
(7’)
‘A slight error in the specification of D(2) by Nawalkha and Lacey [1988, eq. (lo)] has been corrected in this paper in eq. (7’). A similar error appears in their specification of D(3).
SK
Nawalkha and N.J. Lacey, Higher-order duration measures
1147
Table 1 Calculating
generalized
SpWS
higher-order
&,=8.3 1=
S 01Scq.‘4)S= 1
S,, S,, S,, S,, S,=S,
41.3*0(l)= 286.8=%(2)
So, S,, S2cg.(4LS: = S,, S,, S,, S3*Sq
duration
=
2,289.2-D(3) 19,682.5*0(4)=
= 177,222.5*0(5)
solutions.
8.04 = =
76.54 764.96 7,798.92
= 80,311.96
2.3. Generalized solutions beyond D(2)
Duration measures beyond D(2) are not subject to easy interpretation. Suffice it to say that these measures have been successful in controlling interest-rate risk for virtually any non-parallel term structure shift. Although these measures can be obtained through the method described above, they become complicated to report. For illustrative purposes, the approach suggested by Nawalkha and Lacey (1988) is next illustrated. Consider a bond which matures in 5 years and 3 months, pays a semiannual coupon rate of 5.57& and has a reported semi-annual yield to maturity of 5.5%. The bond is priced at its face value of $100. The bond will pay 11 coupons before maturity, with the first coupon due in 3 months. The variables needed to compute the generalized higher-order duration measures are: c = the semi-annual coupon rate = 0.055, i = adjusted yield to maturity computed through eq. (1) =0.05492, N = total number of coupon payments = 11, f =elapsed time between coupon payments=(3 months/6 months) =0.5. First, S, for any m>O is computed from eq. (4) by substituting previously determined lower-order values of Se, S,, . .., S,_r. Next, the value of S, is substituted in eq. (3) to obtain D(m). The generalized closed-form solutions of the first five elements of the duration vector are given in table 1. For comparative purposes, traditional closed-form solutions are computed for the same bond. Because the traditional formulae are valid only at coupon payment dates, they are computed from the previous coupon payment date. The computations use c equal to 0.055, i equal to 0.055, N equal to 11, and employ a similar method used in table 1 except eqs. (3’) and (4’) are used in the place of eqs. (3) and (4). Table 2 compares the generalized solutions (column 1) with traditional solutions (column 2). Differences between the two sets of solutions are defined as percentage error. As can be seen from table 2, the percentage error increases as one advances from lower order to higher order duration
1148
S.K. Nawalkha
and N.J. Lucey, Higher-order duration measures
Table 2 A comparison
of generalized and traditional order duration solutions.
Generalized solutions
DC1-J D(2) D(3) D(4) D(5)
8.04 16.54 164.96 1,798.92 10,311.96
higber-
Traditional solutions
Percentage error (%j
8.54 84.82 885.92 9,441.72 101,819.80
- 6.22 - 10.82 - 15.81 -21.14 - 26.78
Table 3 A portfolio immunization example. Traditional versus generalized duration measures. Bond 1
Bond 2
Panel A: The bond portfolio Number of coupon payments prior 4 2 to maturity 5.5 5.0 Semi-annual coupon rate (%)
Bond 3
6 6.0
Bond 4
10 6.5
Panel B: Traditional and generalized D( 1) and D(2) solutions Through traditional formulae 3.10 5.21 1.66 1.95 D(1) 14.30 29.51 68.66 3.86 D(2) Through generalized formulae 3.20 4.11 7.16 1.45 D(1) 10.85 24.54 61.25 2.15 D(2) Panel C: Comparison of optimum portfolio weights 0.15 - 0.66 -0.71 1.68 Traditional solutions 0.19 -0.48 -0.56 1.45 Generalized solutions
Bond 5
14 1.0
9.36 110.13 8.86 101.63
0.54 0.40
measures. Given the non-trivial magnitudes of these errors, generalized formulae should be used when applying the more complex immunization strategies that depend on the use of higher-order duration measures for bonds whose price includes accrued interest.
3. Implications for portfolio construction The percentage errors shown in table 2 can alter significantly the relative holdings of bonds in immunized portfolios. As an illustration, consider the five bond portfolio given in Panel A of table 3. All bonds are assumed to be priced at par, and the first coupon payment of each bond is assumed to occur in exactly 3 months.
SK
Nawalkha and N.J. Lacey, Higher-order
duration measures
1149
The two factor [D(l) and D(2)] immuni~tion model of Bierwag et al. (1987, 19881, is used to determine optimal portfolio weights. Panel B of table 3 reports values for D(1) and D(2) using both traditional formulae and generalized formulae. A Lagrangian technique is used to minimize the sum of the squares of the weights given to the different bonds subject to two constraints. Assuming an instantaneous planning horizon, the first constraint requires the duration of the portfolio to be equal to zero, and the second constraint requires D(2) of the portfolio to be equal to zero. Solutions to this optimization problem are obtained for both the traditional and generalized closed-form solutions and are reported in Panel C of table 3. Panel C of table 3 illustrates that optimal portfolio weights can vary significantly depending on the formulae used. For example, the relative portfolio weights are altered for each of the five bonds, and change by as much as 35%. Further, because the analysis investigates a D(1) and D(2) model only, the optimal weights would be expected to vary more when measures beyond that of D(2) are used in the construction of the immunization scheme. 4. Conclusions
Higher-order duration measures derived previously have been restricted to hold at coupon payment dates only. This paper derives alternative measures generalized to hold between coupon payment dates as well as at coupon payment dates. The analysis shows that optimal bond portfolio weights are sensitive to the formulae utilized. Given that the payment of accrued interest is quite common in bonds, portfolio managers will find a non-trivial increase in precision when applying the generalized formulae developed here. The benefits of tine tuning become especially important when managing large portfolios. References Bierwag, GO., 1977, Immunization, duration, and the term structure of interest rates, Journal of Financtal and Quantitative Analysis, Dec., 725-742. Bierwag, G.O., G.G. Kaufman and C.M. Latta, 1987, Bond portfolio immunization: Tests of maturity, one-factor, and two-factor duration matching strategies, Financial Review, May, 203-219. Bierwag, G.O., G.G. Kaufman and C.M. Latta, 1988, Duration models: A taxonomy, Journal of Portfolio Management, Fall, 50-54. Bierwaa. G.O.. G.G. Kaufman. R. Schweitzer and A. Toevs. 1983. The art of risk management in bond portfolios, Journal Portfolio Management, Spring, 27-36. Chambers, D.R., W.T. Carleton and R.W. McEnally, 1988, Immunizing default free bond portfolios with a duration vector, Journal of Financial and Quantitative Analysis, March, 89-104. Chua, J., 1988, A generalized formula for calculating bond duration, Financial Analysts Journal, Sept./Ott., 65-67.
of
1150
S.K. Nawalkha and N.J. Lacey, Higher-order duration measures
Fong, H.G. and 0. Vasicek, 1983, Return maximization for immunized portfolios, in: G.G. Kaufman, G.O. Bierwag and A. Toevs, eds., Innovations of bond portfolio management (JAI Press, Greenwich, CT). Nawalkha, S.K. and N.J. Lacey, 1988, Closed-form solutions of higher order duration measures, Financial Analysts Journal, Nov./Dee., 82-84. Prisman, E.Z. and M.R. Shores, 1988, Duration measures for specific term structure estimations and applications to bond portfolio immunization, Journal of Banking and Finance 12, 493-W.
of Banking
and Finance
14 (1990)
1143-1150.
North-Holland
GENERALIZED SOLUTIONS OF HIGHER-ORDER DURATION MEASURES
Sanjay K. NAWALKHA University OJ Massachusetts, Received November
and Nelson J. LACEY* Amherst, MA 01003, USA
1989, final version
received January
1990
Higher-order duration measures have been shown to permit near perfect immunization. This paper presents formulae for these measures which are valid not only at coupon payment dates but between these dates as well. We show that errors caused by using formulae valid only at the coupon payment dates increase as one advances from lower-order to higher-order duration measures. A portfolio immunization example comparing both traditional and generalized closedform solutions is given.
1. Introduction
Duration research has witnessed a shift away from single-factor models and their restrictive term structure assumptions towards multiple factor models. Single-factor duration models [see, e.g., Bierwag (1977)] assume a particular stochastic process that governs the term structure movements. Because the assumed stochastic process does not often confirm to the actual movements in the term structure, even sophisticated single-factor duration models have failed to demonstrate significant improvement vis-a-vis the simplest single-factor model [see, e.g., the evidence presented by Bierwag et al. (1983)]. In contrast, higher order duration models allow for changes in spot yields across the term structure to be captured by more than one factor such that these changes are not perfectly correlated. Examples of higher-order duration models include Fong and Vasicek’s (1983) M-Square measure, the two-factor model of Bierwag et al. (1988), the duration vector of Chambers et al. (1988), and the generalized polynomial model of Prisman and Shores (1988). In terms of performance, higher-order duration measures have demonstrated significant improvement over the traditional scalar models, and more importantly, have demonstrated near perfect immunization [see, e.g., Chambers et al. (1988)]. In this paper we derive closed-form solutions of higher-order duration *Correspondence address: Department of General Business and Finance, ment, University of Massachusetts, Amherst, MA 01003, U.S.A. 0378-4266/90/%03.50
0
1990-Elsevier
Science Publishers
B.V. (North-Holland)
School
of Manage-
1144
SK. Nawalkha and N.J. Lmey, Higher-order
duration measures
measures which are valid at and between coupon payment dates. We show that the formulae presented by Nawalkha and Lacey (1988), valid only at coupon payment dates, introduce errors that are quite significant and which magnify when moving from lower-order to higher-order duration measures. In addition, we illustrate that the composition of an immunized portfolio is altered significantly when optimal portfolio positions are generated through the more correct measures presented here.
2. Derivation of generalized formulae Proper bond valuation between coupon payments must include interest accrued between the last coupon payment and the current time period. Let N be the total number of coupons due until maturity. Defining P as the bond’s price, f as the time elapsed since the date of last coupon payment relative to time between two coupon payments, fC as the amount of interest accrued, C as the dollar value of the coupon payment per period, F as the face value of the bond, and i as the bond’s adjusted yield to maturity, the value of a bond between coupon payments can be given by:
P+fC=
t
C/(l+i)‘-J+F/(l+i)N-‘.
(1)
1=1
The adjusted yield to maturity contrasts with the reported yield to maturity by including any accrued interest. Generalized solutions for various immunization measures are derived from a generalized mth order duration measure given as
Substituting F =C/c in eq. (2), where c is the coupon rate per period, and rewriting eq. (2) defines an equivalent duration measure: D(m) = [i[cS,( 1 + i)N-f + (N -f)“]]/[c[(
1 + i)N- l] + i],
(3)
where S,= $J [(r-f)“/(l 1=1
+i)‘-f].
The generalized formula given for D(m) in eq. (3) is equivalent
to the
SK. Nawalkha
and NJ.
Lacey, Higher-order
1145
duration measures
traditional closed-form solution of D(m) given by Nawalkha and Lacey (1988) for the special case of f =O. The traditional measure is given by:’ D(m) = [i(cS,(l + i)N+ N”)]/[c(( 1 + i)N- 1) + i],
(3’)
where t”/( 1 + i)‘.
S, = i 1=1
In eq. (3), if a solution of S, can be found, then the closed-form solution of D(m) can be obtained through substitution of S, in eq. (3). A closed-form solution for S, exists’ and is given by
S_=+
(l-f)“(l [
(1 +N--f)” (1 +i)N-f
+i)f+mil &S,-r=o
1
for all
m&l,
(4)
where m!
and
mC’=(m-t)!t!
S,,,co=(l +i)f -[l-&l i
for
m=O.
The generalized formula given for S(m) in eq. (4) is equivalent traditional closed-form solution of S(m) for the special case off=@
s_+[ l+m$
,C,S,-
s
1=0
1
for all
rnz 1,
to the
(4’)
where &=i[l-
&]
for
m=O.
2.1. Generalized D( 1) solution To obtain the generalized closed-form solution of D(l), we first obtain the closed-form of S, by substituting m= 1 in eq. (4). Substituting the value of S1 and m= 1 in eq. (3) and further simplifying gives: ‘For j=O, the number of coupon payments due prior to maturity is equal to the number of periods until maturity, and the adjusted yield to maturity is equal to the reported yield to maturity. 2The proof of the closed-form solution for S,, m= 1,2,. . , n, can be obtained from the authors.
1146
S.K. Nawalkha
D(l),c[(l
and N.J. Locey,
+i)“(l +i-fi)-(1
Higher-order
duration measures
+i)]+i(N--f)(i-cc)
ci[( 1 + i)N- l] + i2
(5)
The generalized D(1) solution above is equivalent to the solution for Macaulay duration given by Chua (1988). Also, the generalized solution given in eq. (5) is equivalent to the traditional closed-form solution of D(1) for the special case of f = 0: D(l)=c(l+i)[(l+i)N-l]+(Ni)(i-c) ci[(1+i)N-l]+i2
’
(5’)
2.2. Generalized D(2) solution A widely applied duration measure is D(2), defined as the weighted average time to maturity squared of bond’s cash flows. One interesting aspect of this measure is that it is subject to at least three different interpretations; the second element in Chambers et al. (1988) duration vector that captures nonparallel slope shift in the term structure, the second factor of Bierwag et al. (1987, 1988) model, and a component of Fong and Vasicek’s (1983) M-square risk measure, given by M-square = D(2) - 2HD( 1) + H2,
(6)
where H is the portfolio’s planning horizon. The solution for D(2) is obtained by following steps similar to those used in obtaining D( 1): D(2) = c{(l+i)N[(1+i-fi)2+1+i]-(1+i)[2(1+Ni-fi)+i]}+i2(N-f)2(i-c) ci2[(1+i)N-l]+i3 (7)
The generalized closed-form solution of D(2) given in eq. (7) is equivalent to the traditional closed-form solution of D(2) reported by Nawalkha and Lacey ( 1988)3 for the special case off =0: D(2),~{(1+i)(2+i)[(1+i)N-l]-(2Ni)(l+i)}+(N2i2)(i-c) ci2[( 1 + i)N- l] + i3
(7’)
‘A slight error in the specification of D(2) by Nawalkha and Lacey [1988, eq. (lo)] has been corrected in this paper in eq. (7’). A similar error appears in their specification of D(3).
SK
Nawalkha and N.J. Lacey, Higher-order duration measures
1147
Table 1 Calculating
generalized
SpWS
higher-order
&,=8.3 1=
S 01Scq.‘4)S= 1
S,, S,, S,, S,, S,=S,
41.3*0(l)= 286.8=%(2)
So, S,, S2cg.(4LS: = S,, S,, S,, S3*Sq
duration
=
2,289.2-D(3) 19,682.5*0(4)=
= 177,222.5*0(5)
solutions.
8.04 = =
76.54 764.96 7,798.92
= 80,311.96
2.3. Generalized solutions beyond D(2)
Duration measures beyond D(2) are not subject to easy interpretation. Suffice it to say that these measures have been successful in controlling interest-rate risk for virtually any non-parallel term structure shift. Although these measures can be obtained through the method described above, they become complicated to report. For illustrative purposes, the approach suggested by Nawalkha and Lacey (1988) is next illustrated. Consider a bond which matures in 5 years and 3 months, pays a semiannual coupon rate of 5.57& and has a reported semi-annual yield to maturity of 5.5%. The bond is priced at its face value of $100. The bond will pay 11 coupons before maturity, with the first coupon due in 3 months. The variables needed to compute the generalized higher-order duration measures are: c = the semi-annual coupon rate = 0.055, i = adjusted yield to maturity computed through eq. (1) =0.05492, N = total number of coupon payments = 11, f =elapsed time between coupon payments=(3 months/6 months) =0.5. First, S, for any m>O is computed from eq. (4) by substituting previously determined lower-order values of Se, S,, . .., S,_r. Next, the value of S, is substituted in eq. (3) to obtain D(m). The generalized closed-form solutions of the first five elements of the duration vector are given in table 1. For comparative purposes, traditional closed-form solutions are computed for the same bond. Because the traditional formulae are valid only at coupon payment dates, they are computed from the previous coupon payment date. The computations use c equal to 0.055, i equal to 0.055, N equal to 11, and employ a similar method used in table 1 except eqs. (3’) and (4’) are used in the place of eqs. (3) and (4). Table 2 compares the generalized solutions (column 1) with traditional solutions (column 2). Differences between the two sets of solutions are defined as percentage error. As can be seen from table 2, the percentage error increases as one advances from lower order to higher order duration
1148
S.K. Nawalkha
and N.J. Lucey, Higher-order duration measures
Table 2 A comparison
of generalized and traditional order duration solutions.
Generalized solutions
DC1-J D(2) D(3) D(4) D(5)
8.04 16.54 164.96 1,798.92 10,311.96
higber-
Traditional solutions
Percentage error (%j
8.54 84.82 885.92 9,441.72 101,819.80
- 6.22 - 10.82 - 15.81 -21.14 - 26.78
Table 3 A portfolio immunization example. Traditional versus generalized duration measures. Bond 1
Bond 2
Panel A: The bond portfolio Number of coupon payments prior 4 2 to maturity 5.5 5.0 Semi-annual coupon rate (%)
Bond 3
6 6.0
Bond 4
10 6.5
Panel B: Traditional and generalized D( 1) and D(2) solutions Through traditional formulae 3.10 5.21 1.66 1.95 D(1) 14.30 29.51 68.66 3.86 D(2) Through generalized formulae 3.20 4.11 7.16 1.45 D(1) 10.85 24.54 61.25 2.15 D(2) Panel C: Comparison of optimum portfolio weights 0.15 - 0.66 -0.71 1.68 Traditional solutions 0.19 -0.48 -0.56 1.45 Generalized solutions
Bond 5
14 1.0
9.36 110.13 8.86 101.63
0.54 0.40
measures. Given the non-trivial magnitudes of these errors, generalized formulae should be used when applying the more complex immunization strategies that depend on the use of higher-order duration measures for bonds whose price includes accrued interest.
3. Implications for portfolio construction The percentage errors shown in table 2 can alter significantly the relative holdings of bonds in immunized portfolios. As an illustration, consider the five bond portfolio given in Panel A of table 3. All bonds are assumed to be priced at par, and the first coupon payment of each bond is assumed to occur in exactly 3 months.
SK
Nawalkha and N.J. Lacey, Higher-order
duration measures
1149
The two factor [D(l) and D(2)] immuni~tion model of Bierwag et al. (1987, 19881, is used to determine optimal portfolio weights. Panel B of table 3 reports values for D(1) and D(2) using both traditional formulae and generalized formulae. A Lagrangian technique is used to minimize the sum of the squares of the weights given to the different bonds subject to two constraints. Assuming an instantaneous planning horizon, the first constraint requires the duration of the portfolio to be equal to zero, and the second constraint requires D(2) of the portfolio to be equal to zero. Solutions to this optimization problem are obtained for both the traditional and generalized closed-form solutions and are reported in Panel C of table 3. Panel C of table 3 illustrates that optimal portfolio weights can vary significantly depending on the formulae used. For example, the relative portfolio weights are altered for each of the five bonds, and change by as much as 35%. Further, because the analysis investigates a D(1) and D(2) model only, the optimal weights would be expected to vary more when measures beyond that of D(2) are used in the construction of the immunization scheme. 4. Conclusions
Higher-order duration measures derived previously have been restricted to hold at coupon payment dates only. This paper derives alternative measures generalized to hold between coupon payment dates as well as at coupon payment dates. The analysis shows that optimal bond portfolio weights are sensitive to the formulae utilized. Given that the payment of accrued interest is quite common in bonds, portfolio managers will find a non-trivial increase in precision when applying the generalized formulae developed here. The benefits of tine tuning become especially important when managing large portfolios. References Bierwag, GO., 1977, Immunization, duration, and the term structure of interest rates, Journal of Financtal and Quantitative Analysis, Dec., 725-742. Bierwag, G.O., G.G. Kaufman and C.M. Latta, 1987, Bond portfolio immunization: Tests of maturity, one-factor, and two-factor duration matching strategies, Financial Review, May, 203-219. Bierwag, G.O., G.G. Kaufman and C.M. Latta, 1988, Duration models: A taxonomy, Journal of Portfolio Management, Fall, 50-54. Bierwaa. G.O.. G.G. Kaufman. R. Schweitzer and A. Toevs. 1983. The art of risk management in bond portfolios, Journal Portfolio Management, Spring, 27-36. Chambers, D.R., W.T. Carleton and R.W. McEnally, 1988, Immunizing default free bond portfolios with a duration vector, Journal of Financial and Quantitative Analysis, March, 89-104. Chua, J., 1988, A generalized formula for calculating bond duration, Financial Analysts Journal, Sept./Ott., 65-67.
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S.K. Nawalkha and N.J. Lacey, Higher-order duration measures
Fong, H.G. and 0. Vasicek, 1983, Return maximization for immunized portfolios, in: G.G. Kaufman, G.O. Bierwag and A. Toevs, eds., Innovations of bond portfolio management (JAI Press, Greenwich, CT). Nawalkha, S.K. and N.J. Lacey, 1988, Closed-form solutions of higher order duration measures, Financial Analysts Journal, Nov./Dee., 82-84. Prisman, E.Z. and M.R. Shores, 1988, Duration measures for specific term structure estimations and applications to bond portfolio immunization, Journal of Banking and Finance 12, 493-W.