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Applied Economics Letters
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Analysing the c-minus-age strategy for life-cycle investing
Christine W. Lai a; Tsung-Chyan Lai b
a Department of Finance, Yuan-Ze University, Chung-Li 320, Taoyuan 320, Taiwan b Department of Business
Administration, National Taiwan University, Taipei 106, Taiwan First Published on: 16 April 2008
To cite this Article Lai, Christine W. and Lai, Tsung-Chyan(2008)'Analysing the c-minus-age strategy for life-cycle investing',Applied Economics Letters,
To link to this Article: DOI: 10.1080/13504850701221758
URL: http://dx.doi.org/10.1080/13504850701221758
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Analysing the c-minus-age strategy
for life-cycle investing
Christine W. Lai
a,* and Tsung-Chyan Lai
ba
Department of Finance, Yuan-Ze University, 135, Yuan-Tung Road, Chung-Li 320, Taoyuan 320, Taiwan
b
Department of Business Administration, National Taiwan University, 1, Section 4, Roosevelt Road, Taipei 106, Taiwan
The c-minus-age strategy is a popular strategy for life-cycle investing. When applying the c-minus-age strategy, an investor first chooses an indirect preference parameter c and at age t will hold a percentage of c minus t in equity assets. In this article, we use a linear and a multiplicative mean-variance utility function to quantitatively analyse the term structure of the mean-variance tradeoffs implied by the c-minus-age strategy. We also provide an optimal procedure to determine c, based on the two direct preference parameters, elicited from an investor, of a multiplicative mean-variance utility function.
I. Motivation and Background
The guideline of 100-minus-age strategy for life-cycle investing suggests that the percentage of a holding portfolio in equities for an investor at age t is equal to 100 t (e.g. Bodie and Crane, 1997). The general form for this guideline is c t, where c is a constant typically between 120 and 80. In fact, in practice many target maturity life-cycle funds implement the spirit of the c-minus-age strategy (e.g. Fidelity Freedom 2030). The portfolio managers of these type of funds adjust the asset allocation over time, with the portfolio consisting of less equities the nearer the target date. This type of fund has been popularly used as a default option in retirement plans. According to Holden and VanDerhei (2005), this default option can significantly improve the replace-ment rate of workers.
This popular strategy receives empirical support from the long-run predictability behaviour in equity. Fama and French (1988) find that stock returns exhibit strongly negative autocorrelation for holding periods over 1 year, implying mean-reverting
behaviour of equity returns. Poterba and Summers (1988), employing variance ratio tests, and Bekaert and Hodrick (1992), Campbell (1996), Barberis (2000) and Campbell and Viceira (2005), using time-varying variance ratio statistics from vector autore-gressive (VAR) models report the evidence that the variance of stock returns does not grow in propor-tional with the investment horizon, as predicted by the random walk model, but grows more slowly as investment horizon increases, implying negative serial correlation of stock returns in the long-run. This pattern of long-run predictability of stock returns implies the rejection of random walk hypoth-esis of stock returns and several rationales are provided. Asset pricing models with time-variation asset returns (e.g. Cox et al., 1985, among other financial economists), over-reaction hypothesis (De Bondt and Thaler, 1985) and fads hypothesis (Shiller, 1990) are among the theoretical explana-tions. In addition, in the portfolio construction, contrarian strategies are investigated to capture this mean-reverting behaviour of equity in the long-run (e.g. De Bondt and Thaler, 1985; Lakonishok et al.,
*Corresponding author. E-mail: [email protected]
Applied Economics LettersISSN 1350–4851 print/ISSN 1466–4291 onlineß 2008 Taylor & Francis 1 http://www.tandf.co.uk/journals
DOI: 10.1080/13504850701221758
1994), where theses strategies are constructed by holding previous ‘losers’ and shorting previous ‘winners’ 3–5 years prior in order to generate positive excess returns in the following 3–5 holding periods. Prior (1999) also gives a profitable trading rule using mean reverting investment trusts discounts. Menkhoff and Schmidt (2005) use survey to describe the characteristics of fund managers relying on the contrarian trading. Previous studies in this context of mean reversion in equity focus on US market or developed markets without experiencing signifi-cant interruptions in the stock markets. The most recent empirical studies on this issue focus on smaller markets or emerging capital markets. For example, Barkoulas et al. (2000) use spectral regression method to investigate the long-run dependence in the Greek stock market. By applying the variance ratio tests of Lo and MacKinlay (1988), Chang and Ting (2000) examine the Taiwan’s stock market, one of the most volatile markets in the Pacific region, and Lima and Tabak (2004) investigate the Hong Kong, Singapore and China markets, especially the A and B class shares within the China markets, to study the special effects of information transmission. Narayan and Smyth (2004) investigate the stock returns in Korea by using the two break unit root tests to accommodate the structural breaks in market crash and Asian financial crisis (1997–1998). The long-term persistence is found in some of these studies, and the results exhibit characteristics different from those in developed countries in terms of information trans-mission, liquidity and nonsynchronous trading.
Of particular implication of these findings is that the long-run predictability of equity implies that the variance of equity returns does not grow in proportion with the investment horizon and there-fore, equity will appear relatively safe for young investors. By incorporating the human wealth into the total wealth for the portfolio choice, it also receives a theoretical support from Bodie et al. (1992), which justifies that the young can invest more in risky financial assets than the old. However, no studies of quantitative analysis are available on analysing the risk-return tradeoffs that are implied by this popular investing strategy and on how to best determine c for an investor based on the true preferences of that investor.
Some interesting results in portfolio choice can be obtained, both theoretically and empirically, by adopting a linear mean-variance utility function based on some restrictive assumptions on the utility forms or/and the return distributions (see, for example, Campbell and Viceira, 2002). In the presence of a risk-free asset, a linear mean-variance utility function is equal to the expected return
of a holding portfolio minus a multiplication of a constant (A/2) and the variance of the return on the holding portfolio, where A is the Arrow-Pratt coefficient of relative risk aversion (RRA). A linear mean-variance utility function was applied empiri-cally by Friend and Blume (1975), Gressis et al. (1976), Siegel and Hoban (1982), Morin and Suarez (1983), Szpiro (1986) to estimate the coefficient of the RRA(A). Jansen (1998) extends the linear mean-variance model with international capital market regulations and estimates the RRA and capital control effects based on the asset holdings of the German private sector. Using time-varying mean and variance estimations derived from a variant VAR model, the author find that a reasonable degree of risk aversion and actual magnitude of return risk appear to be incompatible within the framework of mean-variance model.
One could generalize a linear mean-variance utility function into an additive mean-variance utility function, where an additive mean-variance utility function is a nonnegatively weighed average of the expected return (mean) utility and the variance utility. For example, Van Eaton and Conover (1998) investigate a special form for an investigation of time diversification with the mean utility being equal to the mean and the variance utility being equal to the SD raised to the power of a positive number. Several works are available (e.g., Selvanathan, 1987; Clement et al., 1997; Selvanathan and Selvanathan, 2005) on the procedures for testing the adequacy of an additive mean-variance utility function.
However, for most investors there are matching effects between mean and variance, and an additive mean-variance utility function fails to accommodate these effects. For an illustration of the matching effects, consider the example with two expected return levels, r1 and r2, and two variances, v1 and
v2. The occurring of r1and r2has half and half chance
and the occurring of v1and v2has also half and half
chance. There are matching effects between mean and variance if and only if the lottery with a half chance of (r1, v1) and a half chance of (r2, v2) is not
indifferent with the lottery with a half chance of (r1, v2) and a half chance of (r2, v1). This motivates
us to generalize an additive mean-variance utility function into a multiplicative one, which can accom-modate the matching effects.
A multiplicative mean-variance utility function is a nonlinear function of the mean utility and the variance utility consisting of a mean utility term, a variance utility term and a nonlinear term with the product of the mean utility and the variance utility. Such a mean-variance utility function is not only able to accommodate the matching effects but is
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C. W. Lai and T.-C. Lai
also simple enough for practical use. It should be noted that there is one preference parameter (A) associated with a linear mean-variance utility function, one preference parameter is associated with an additive mean-variance utility function and two preference parameters are associated with a multiplicative mean-variance utility function. In the case of all risky assets, the preference parameter k associated with a linear mean-variance utility function is an approximation to A, the coefficient of the RRA of an investor. The preference parameter k1, 0 k11, associated with
an additive mean-variance utility function indicates the relative weight k1of the mean utility to the total
utility and the relative weight 1 k1of the variance
utility to the total utility. Each ki, 0 ki1, i ¼ 1, 2,
of the two preference parameters associated with a multiplicative mean-variance utility function, respectively, indicates the relative weight k1 of the
mean utility to the total utility and the relative weight k2of the variance utility to the total utility. Readers
who are interested in a theoretical explanation of a general multiplicative utility function may refer to the work of Keeney and Raiffa (1976). The multiplicative utility functions have been applied to a wide variety of decision contexts. For example, Levy et al. (2003) used a multiplicative utility function to show that the optimal saving may be positive even at a negative rate of return, where the nonlinear term is the product of the utility of current consumption and the utility of the consumption one period ahead. Guerrero and Herrero (2005) provided conditions for modelling individual preferences for lotteries on health profiles by using a multiplicative utility function.
In this article, we use a linear and a multiplicative mean-variance utility function to quantitatively analyse the term structure of the mean-variance tradeoffs implied by the c-minus-age strategy. We also provide an optimal procedure to determine c for the c-minus-age strategy based on the two preference parameters, k1 and k2, expressed by an
investor, of a multiplicative mean-variance utility function.
II. Theoretical Model
To analyse the mean-variance tradeoffs implied by the c-minus-age strategy, we express the mean-variance efficient portfolio weight functions and efficient variance functions in terms of expected return levels. In addition, we express the expected return levels in terms of c and t. These functions are
all linked to preference parameters in a linear and a multiplicative utility functions.
Efficient portfolio weight functions and efficient variance function
Suppose that there are a set {Xi, i ¼ 1, . . . , n} of n
risky asset classes available for investing. For a given investment horizon, let r ¼ (r1, . . . , rn) denote the
expected (mean) return vector, where ri¼E(Xi),
i ¼1, . . . , n and let V ¼ [vij] denote the n n symmetric
covariance matrix of the n asset classes, where vij¼cov(Xi, Xj) for i ¼1, . . . , n and j ¼1, . . . , n
during this horizon. Without loss of generality, we assume that r1 rn. Based on the classic
portfolio model of Markowitz (1952), we can express the efficient portfolio weight function w
iðrÞ of each asset class i, i ¼ 1, . . . , n as a linear function of r and express the efficient variance function f(r) as a convexly quadratic function of r, where we let
wiðrÞ ¼d1ir þd0i and
fðrÞ ¼ a2r2þa1r þ a0
The formulas for computing the n pair of coefficients (dli, d0i), i ¼ 1, . . . , n and the three
coefficients of a2, a1 and a0 in terms of
r ¼ (r1, . . . , rn) and V ¼ [vij] can be written as follows.
Let: (a) Yi¼XiX1ui(XnX1), i ¼2, . . . , n 1 and Y0¼X1þui(XnX1); (b) yii¼var(Yi), yi0¼cov(Yi, Y0), i ¼ 2, . . . , n 1, yij¼cov(Yi, Yj), i < j, i, j ¼ 2, . . . , n 1 and y00¼var(Y0); (c) b ¼ ( y20, . . . , y(n 1)0) ¼ b1r þ b0 (since b is a
linear function r), V ¼ ½y ij, i, j ¼ 2, . . . , n1, d1¼ V1b1 and d0¼ V1b0. We have that d1¼(d12, d13, . . . , d1(n1)), d0¼ (d02, d03, . . . , d0(n 1)), d11¼ Pn1 i¼2½ðrnriÞ= ðrnr1Þd1i, d01 ¼ ðrnrÞ=ðrnr1Þ Pn1i¼2 ðrnriÞ=ðrnr1Þd1i, d1n¼ Pn1i¼2 ðrir1Þ= ðrnr1Þd1i, and d0n¼ ðr r1Þ=ðrnr1Þ Pn1
i¼2½ðrir1Þ=ðrnr1Þd0i.
Since f(r) ¼ (d1r þ d0)V(d1r þ d0)0, we have that
a2¼d1Vd 0 1, a1¼2d1Vd 0 0 and a0 ¼d0Vd 0 0, where superscript t denotes transpose.
Multiplicative mean-variance utility function
A multiplicative mean-variance utility function is as follows:
Uðx, yÞ ¼ k1U1ðxÞ þ k2U2ðyÞ þ ð1 k1k2ÞU1ðxÞU2ðyÞ
where
U1(x) is the expected return utility function with x
being scaled to be in [0, 1],
U2( y) is the variance utility function with y being
scaled to be in [0, 1].
By letting x ¼ ðr rÞ=ð r rÞ and y ¼ ð fð rÞ fðrÞÞ= ðfð rÞ fðrÞÞ, where r is the return level associated with the global minimum variance of fðrÞ, i.e. r ¼ a1=2a2, and r is the best return level judiciously specified by a financial analyst or investor, we can scale x and y such that, U1(0) ¼U2(0) ¼
U(0,0) ¼ 0, U1(1) ¼ U2(1) ¼U(1, 1) ¼ 1, 0 x 1 and
0 y 1. In addition, it is shown in the Appendix that y ¼ 1 x2. The two preference parameters k1
and k2of an investor can be elicited as follows. Let
the ‘paradise portfolio’ denote the portfolio with both its expected return and variance attaining the best level and the ‘evil portfolio’ with both its expected return and variance attaining the worst level. Also, let ‘paradise-evil portfolio’ denote the portfolio with its expected return attaining the best level and its variance attaining the worst level and let ‘evil-paradise portfolio’ denote the portfolio with its expected return attaining the worst level and its variance attaining the best level. We let alternative A0 denote a two-outcome lottery with the prob-ability of the ‘paradise portfolio’ being p and the probability of the ‘evil portfolio’ being 1 p; let alternative A1 denote a sure outcome of the ‘paradise-evil portfolio’; let alternative A2 denote a sure outcome of the ‘evil-paradise portfolio’. The preference probability p of an investor making alternatives A0 and A1 indifferent is his/ her k1 and the preference probability p making
alternatives A0 and A2 indifferent is k2. Intuitively,
k1 is the relative weight of the whole mean utility
to the total utility, and k2 is the relative weight
of the whole variance utility to the total utility. We note that if k1þk2¼1, then the
multi-plicative mean-variance utility function reduces to an additive one.
Determination of the specified expected return level by the c-minus-age strategy
The c-minus-age strategy implies that, at a given age t, the total percentage of all the efficient portfolio weights in equities is equal to c t The n risky asset classes are partitioned into two categories, the category of equity asset class (denoted by S) and the category of fixed income asset class. Let D1¼Pi2Sd1i and D0¼Pi2Sd0i for every asset class i 2 S. Since D1r þ D0¼c t by the strategy, we
have that r ¼ (c t D0)/D1 and thus have the
following relation on the scaled expected return
level x (i.e. x ¼ ðr rÞ=ð r rÞ) determined by the c-minus-age strategy:
x ¼ðc t D0Þ ðr rÞD1
r
r r ð1Þ
Relationship with the preference parameters
Consider a linear mean-variance utility function U(r) ¼ r (k/2)f(r), where r is the expected return level, f(r) is the efficient variance function, i.e. f(r) ¼ a2r2þa1r þ a0, and k is an approximation of
the Arrow-Pratt RRA coefficient. We have, by the first-order-condition of U(r)with respect to r and by r ¼(c t D0)/D1, the relationship: 1 k¼ a2ðc t D0Þ D1 þa1 2 ð2Þ We now consider a multiplicative mean-variance utility function U(x) with U(x) ¼ k1U1(x) þ k2
U2( y) þ (1 k1k2)U1(x)U2( y), where y ¼1 x2.
Given a scaled expected return level x, we can obtain, by the first-order-condition of U(x) with respect to x, the set of the preference parameters k1
and k2, which makes the maximum of U(x) be
achieved at the given expected return level of x, as follows: k2¼e1ðxÞk1þe0ðxÞ ð3Þ where U01¼@ðU1U2Þ @x , U11¼ @U1 @x , U21 ¼ @U2 @x , e1ðxÞ ¼U11ðxÞ U01ðxÞ U01ðxÞ U21ðxÞ e0ðxÞ ¼ U01ðxÞ U01ðxÞ U21ðxÞ:
In the case of a linear mean utility and a linear variance utility, we have that U1(x) ¼ x and
U2( y) ¼ 1 x2. We now have that U(x) ¼ k1x þ
k2(1 x2) þ (1 k1k2)x(1 x2), or alternatively,
U(x) ¼ (k1þk21)x3k2x2þ(1 k2)x þ k2, and
have that k2¼e1(x)k1þe0(x), where e1(x) ¼ (3x2)/
3x2þ2x þ 1 and e1(x) ¼ (1 3x2)/(3x2þ2x þ 1).
III. Procedures for Determining the c Value
We present an optimal procedure to determine c value in the c-minus-age strategy. Suppose that the beginning age and the ending age of the
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C. W. Lai and T.-C. Lai
investment horizon for an investor is t1(say 30) and t2
(say 65), respectively. The optimal procedure for determining the c value can be described as follows:
Step 1: elicit from the investor the preference parameters of k1 and k2 for each age t
from t1to t2.
Step 2: for each investment horizon facing at the age of t, where t 2 [t1, t2] , we use (Equation
1) to estimate the expected return level specified by the c-minus-age strategy. Step 3: for each investment horizon facing at the
age of t, where t 2 [t1, t2], plug the
corre-sponding k1 elicited in Step 1 and the
expected return level obtained in Step 2 into (Equation 3) to obtain an estimated value of k2.
Step 4: the c value is determined by minimizing the total of the squared error of the elicited k2
and the estimated k2 summing across all
the periods facing at the age of t, where t 2[t1, t2].
IV. Data, Numerical Example and Results In this article, all the numerical results are based on the following dataset. The indices for asset classes1 based on Fabozzi et al. (2002) come from Stocks, Bonds, Bills and Inflation published by Ibbotson Associates. Data include real total rates of return on US 30-day T-Bills for the asset class of US cash, Lehman Brothers Aggregate Bonds for US bonds, S&P 500 for US large-cap equity, Russell 2000 for US small-cap equity, MSCI EAFE for Europe/Japan equity and MSCI Pacific for emerging markets equity.2 The annual data from 1979 to 2004 is adopted in this article since the c-minus-age strategy for life-cycle investing implies a modification of the portfolio decision each year. Based on historical asset return data, the bootstrapping method is used for generating the returns of the future. For each re-sampling there are 500 times of sequential random selections with replacement for each of the investment horizons n year long, n ¼ 1, 2, . . . , 35, where each random selection is a selection of the year in the time span of the historical return data. Once a year is selected, the returns of all the indices associated with the year are sampled.
Figure 1 plots the k values of (Equation 2) of the linear mean-variance utility function which serves as an approximation of the coefficients of the RRA over the years-to-retirement from 1 to 35 for the c-minus-age strategy with c ¼ 80, 90, 100, 110 and 120. From Fig. 1, we know that the k values decrease but their slopes increase as the years-to-retirement increase for each curve of c. For a user of a c-minus-age strategy, the implications are: (1) his/her RRA decreases but the rates increase across the years-to-retirement; (2) the coeffi-cient of the RRA for c ¼ 80 is, on average, 1.37 times that of c ¼ 90, 1.73 times that of c ¼ 100, 2.10 times that of c ¼ 110 and 2.46 times that of c ¼ 120.
Figure 2(a) and (b) show the iso-utility curve of (Equation 3) of the two preference parameters (k1 and k2) associated with a multiplicative
mean-variance utility function corresponding to each of the four age groups (ages 30–34, 40–44, 50–54 and 60–64) for c ¼ 100 and 120 (plots for c ¼80, 90 and 110 indicate similar results and are available upon request). Since an additive mean-variance utility function is a special case when k1þk2¼1, a user of an additive mean-variance
utility function refers to the line segment with negative 45 degrees. The plots indicate that for any level of c: (1) the iso-utility curve corresponding to an older age group is above the iso-utility curve corresponding to a younger age group; (2) the intercept of the iso-utility curve corresponding to an older age group is larger than that corresponding to a younger age group; (3) the slope of the iso-utility
Fig. 1. Coefficients of the relative risk aversion in a linear mean-variance utility function across investment horizons
1
Fabozzi et al.(2002) presented the commonly used indexes for asset classes and justified these indexes not only with longer histories, but also with more accuracy.
2
Our results may be sensitive to the number of funds (asset classes) in the analysis. Fortunately, Cromwell et al. (2000) found that approximately 70% of the reduction in the portfolio risk can be obtained with a four-fund portfolio.
curve corresponding to an older age group is smaller than that corresponding to a younger age group; (4) all slopes corresponding to all age groups are nonnegative; (5) each of all the iso-utility curves is a line segment. For a user of a c-minus-age strategy, the implications are: (1) the maximum amount of risk (variance) willing to trade for one unit of the expected return is nonnegative and is decreasing as one gets older; (2) if the weight of the expected return utility for one investor remains constant across age, then his/her weight of the risk (variance) utility increases as s/he gets older. In addition, many financial advisors suggest taking 120 minus age instead of 100 minus age, so as to increase equity holdings for supporting the longer life expectancies in retirement. When Fig. 2(a) is compared to Fig. 2(b), we find that, for any level of k1, the relative weight of the risk
utility, k2, will decrease when c ¼ 100 is compared to
c ¼120.
We now consider a numerical example for the optimal procedure to determine the c value. We suppose that the investor is now 30 years old and plan to retire at the age of 65. We also assume the growth rates of k1 and k2 for an investor over his/her life
cycle, where the decreasing rate of k1is set at the level
of 1.5% and the increasing rate of k2is set at the level
of 1%.3The region of c is mapped out in Fig. 3 based on the procedure mentioned in the section of ‘Procedures for Determining the c Value’. Fig. 3 is useful for a financial analyst to provide professional advices to an investor whose growth rates of preference parameters k1 and k2 are 1.5 and 1%
over 30–65 years old, respectively. Based on Fig. 3, one can determine a suitable c value as follows: if the initial values of k1and k2, elicited from the investor at
age 30, are 0.55 and 0.35, respectively, then the recommended c value is 100 and so on.
V. Conclusions
We use a linear and a multiplicative mean-variance utility function to quantitatively analyse the term structure of the mean-variance tradeoffs implied by the c-minus-age strategy. We also provide an optimal procedure to determine c based on the two preference parameters, elicited from an investor, of a multi-plicative mean-variance utility function.
When a linear mean-variance utility function is used, our investigation of the term structure of the mean-variance tradeoffs implied by the c-minus-age strategy reveals that the k values decreases as the years-to-retirement increases for each curve of c. For a user of the c-minus-age strategy, the implications
Fig. 3. The Region Graph of c: k1decreasing rate ^ 1.5%
and k2increasing rate ^ 1%
Fig. 2. Preference spaces (c ^ 100 versus c ^ 120)
3
Other figures mapping out the region of c with c ¼ 70, 80, 90, 100, 110 and 120 for different combinations of the growth rates of k1and k2are available upon request.
6
C. W. Lai and T.-C. Lai
are: (1) his/her RRA decreases but the rates increase across years-to-retirement; (2) the coefficient of the RRA for c ¼ 80 is, on average, 1.37 times that of c ¼90, 1.73 times that of c ¼ 100, 2.10 times that of c ¼110 and 2.46 times that of c ¼ 120.
When a multiplicative mean-variance utility function is used, our investigation of the term structure of the mean-variance tradeoffs implied by the c-minus-age strategy reveals that for any level of c: (1) the iso-utility curve corresponding to an older age group is above the iso-utility curve corresponding to a younger age group; (2) the intercept of the iso-utility curve corresponding to an older age group is larger than that correspond-ing to a younger age group; (3) the slope of the iso-utility curve corresponding to an older age group is smaller than that corresponding to a younger age group; (4) all slopes corresponding to all age groups are nonnegative; (5) each of all the iso-utility curves is a line segment. For a user of the c-minus-age strategy, the implications are: (1) the maximum amount of risk (variance) willing to trade for one unit of the expected return is nonnegative and is decreasing as one gets older; (2) if the weight of the expected return utility for one investor remains constant across age, then his/her weight of the risk (variance) utility increases as s/he gets older. Figure 3 provides a numerical example on choosing a suitable c value according to one’s preference parameters.
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Appendix
Proof of y(x) ¼ 1 x2 in the multiplicative mean-variance utility function
¼ ðr þ a1=2a2Þ r þ a1=2a2 Recall that f(r) ¼ a2r 2 þa1r þ a0 and r ¼ a1=2a2. Since x ¼r r r r ¼ðr þ a1=2a2Þ r þ a1 2a2 ¼2a2r þ a1 2a2r þ a1 we have that x ¼2a2r þ a1 2a2r þ a1 We note that 1 x ¼ 1 r r r r¼ r r r r and 1 þ x ¼ 1 þ2a2r þ a1 2a2r þ a1 ¼2a2r þ 2a2r þ2a1 2a2r þ a1 ¼a2r þ a2r þ a1 a2r þ ða1=2Þ ¼a2ðr þ rÞ þ a1 a2ðr þ rÞ þ a1 : Since yðrÞ ¼fðrÞ fðrÞ fðrÞ fðrÞ¼ a2r2þa1r a2r2a1r a2r2þa1r a2r2a1r ¼a2ðr 2r2Þ þa1ðr rÞ a2ðr2r2Þ þa1ðr rÞ ¼r r r r a2ðr þ rÞ þ a1 a2ðr þ rÞ þ a1 ¼ ð1 xÞð1 þ xÞ ¼1 x2,
we have that y(x) ¼ 1 x2.
