APPENDIX: MATHEMATICAL DERIVATION OF THE EFFICIENT FRONTIER
8.1 SINGLE-INDEX MODELS
Single-index models are the simplest and most popular type of generating process used to explain what economic forces create investors’ returns. They are also used to estimate a firm’s undiversifiable systematic risk.
8.1.1 Return-Generating Functions
Preceding chapters suggested the single-period rate of return is the basic random variable of portfolio theory. As explained previously, for a share of common or preferred stock, a bond, a real estate investment, or other investments, thisholding period return(HPR) is computed as follows.
Holding period return,rt
=
Price change during the holding period
+
Cash income received during the holding period, if any
Purchase price at the beginning of
the holding period
or equivalently,
rt=
Pt−Pt−1 +dt
Pt−1 (1.1)
where rt, is the rate of return, or holding period return (HPR); Pt represents the market price at time t; and dt stands for cash dividend income, or other source of income received during the holding period.
This chapter reviews the historical development of two similar, yet significantly different, return-generating functions that underlie equation (1.1). These two return- generating functions are fundamental components of portfolio theory.
165
rit
eit
rmt rit = ai + birmt
FIGURE 8.1 One Possible Form for Equation (8.1)
In 1959 Harry Markowitz published graphs illustrating the return-generating functiondefined in equation (8.1). This equation is referred to as the single-index model,1
rit=αi+βirmt+εit (8.1) whereαi andβi are the intercept and slope coefficients that result from regressing the rate of return from asset i in period t, denoted rit, onto the simultaneous rate of return on some market index in period t, denoted rmt; and εit is the unexplained residual error term for assetiin periodt. This model is based on the assumption that the joint probability distribution between rit and rmt is stationary and bivariate normal.2 Figure 8.1 illustrates equation (8.1). Markowitz’s 1959 monograph provides the earliest presentation of this well-known model of which we are aware.
In the early 1960s William F. Sharpe finished his PhD in economics at UCLA while working at the RAND Corporation in Los Angeles. Sharpe collaborated with Harry Markowitz, who was also a RAND employee at that time. As a result of this collaboration, in 1963 Sharpe published a classic analysis of equation (8.1), referring to the equation as the diagonal model in that paper.3In later papers Sharpe variously referred to equation (8.1) as the single-index model, the one-factor model, and the single-factor market model.
Working independently, in 1965 Jack L. Treynor published a paper about equation (8.1), which he called the characteristic line.4 Other people like to call equation (8.1) the market model. The return-generating function in equation (8.1) obviously has several different names. Treynor’s classic paper delved into the portfolio management and risk management implications the classic Markowitz- Sharpe-Treynor characteristic line has for mutual fund managers.
Assuming that the error terms average out to zero, E εit
=0, theconditional expectationof equation (8.1) is shown as equation (8.2).
E rit|rmt
=αi+βi
rmt
(8.2)
rit = ai + bi(rmt) rit
rmt rm0
ri0
FIGURE 8.2 Another Possible Form for Equation (8.1)
Using this conditional expectation facilitates prediction. For example, if a financial economist predicts that the rate of return on the market index
rmt at time periodt=0 assumes the valuerm0, then equation (8.2) implies thatri0is the expected value of thei-th asset’s rate of return, whereri0=αi+βi
rm0
. Figure 8.2 graphically depicts this process. Note, however, that this form of forecasting is dependent on the constancy over time of the underlying regression model of equation (8.1).
The regression model such as equation (8.1) makes five assumptions about the random error term.
1. The expected value of the random error term is zero—that is,E εit
=0.
2. The variance of εit, denoted σεi2, is constant—that is, the error terms are homoscedastic.
3. The random error terms are uncorrelated withrmt—that is, Cov εit,rmt
=0.
4. The random error terms are serially uncorrelated—that is, Cov εit,εis
=0 for t=s.
5. The random error terms of an asset are uncorrelated with those of any other asset—that is, Cov
εit,εjt
=0 fori=j.
If the sample data conforms to the assumptions above, the least square estimates of αi and βi will be unbiased, minimum-variance, linear estimates of the true regression parameters.
The return-generating equation (8.1) can be partitioned into two mutually exclusive components: a systematic part and an unsystematic part:
rit
Total return
=αi+βirmt
Systematic part
+ εit
Unsystematic part
(8.1a)
The systematic part of the total return is systematically explained by the explanatory variable, the return on the market index (rmt). The unsystematic part is the remaining portion of the total return that is left unexplained by the explanatory variable.
Economic insights can be gained from equation (8.1a)’s return-generating function by partitioning its total variance into two economically meaningful components.
Taking the variance of both the left- and right-hand sides of equation (8.1a) produces equation (8.3)
Var rit
=Var
αi+βirmt+εit
=E
αi+βirmt+εit
−E
αi+βirmt+εit
2
=E βi2
rmt−μm2
+εit2+2βi
rmt−μm εit
=βi2σm2+σεi2 (8.3)
The total variance of equation (8.3) measures the total variability of return, or total risk, of the basic random variablerit. The first term of the equation (8.3),βi2σm2, measures systematic (or undiversifiable) risk, and the second term,σεi2, measures the unsystematic (or diversifiable) risk of asset i. Since all assets experience the same market variance, σm2, the beta emerges as an index of systematic (or diversifiable) risk. This relation may clarify partitioning the total variance:
σi2
Total risk
= β i2σm2
Systematic risk
+ σεi2
Unsystematic risk
The insights just introduced are the topics of several of the following chapters.
Because the return from stock j in periodt is also described by the following single-index model,
rjt=αj+βjrmt+εjt (8.1b) the covariance between two different securitiesiandjcan be expressed as
Cov rit,rjt
=E rit−E
rit rjt−E rjt
=E
αi+βirmt+εit
−E
αi+βirmt+εit
×
αi+βirmt+εit
−E
αi+βirmt+εit
=E βi
rmt−E rmt
+εit βj
rmt−E rmt
+εjt
= βiβjE
rmt−E rmt2
Thus,σij=βiβjσm2.
8.1.2 Estimating the Parameters
Assuming that the historical data have been generated by a stationary distribution, the statistical technique of simple regression [that is, ordinary least squares (OLS)]
can be used to estimate the security-specific regression parametersαi,βi, andσεi2. This statistical technique takes the set of paired sample values of
rmt,rit
and attempts to find the values ofαiandβithat create a line of best fit. Here ‘‘best fit’’ means that the sum of the squaredvertical distancesof the actual values ofritfrom the regression line is minimized. Figure 8.1 indicates that it is the squared vertical distances, not the squared perpendicular distances, and not the squared horizontal distances, that
is minimized in an OLS regression. The OLSαi,βi, andσεi2can be estimated fromT sample observations using the following formulas:
βˆi= T
t=1
rit−ri rmt−rm T
t=1
rit−ri2
αˆi=ri−βˆirm σˆεi2=
1 T−2
T t=1
rit−
αˆi+βˆirmt2
Hereriandrmare the estimated sample mean values ofritandrmt; these means are computed during the estimation period using equation (2.7).
EXAMPLE 8.1
Table 8.1 lists the monthly rates of return from stockiand the market portfolio mover the period from January through December of a recent year.
TABLE 8.1 Hypothetical Monthly Holding Period Returns5
Month t Return on
stocki rit
Return on the market
rmt
Fitted return on
stocki ˆ αi+βˆirmt
rit−ri ×
rmt−rm Residuals ˆ εit
(1) (2) (3) (4) (1)−(3)
January 1 0.0206 0.0194 0.0242 0.0001 −0.0036
February 2 −0.0596 −0.0140 −0.0087 0.0014 −0.0510
March 3 0.0142 0.0129 0.0178 0.0000 −0.0036
April 4 0.0843 0.0399 0.0443 0.0025 0.0400
May 5 0.0469 0.0389 0.0434 0.0012 0.0035
June 6 −0.0127 −0.0148 −0.0094 0.0005 −0.0033
July 7 0.0513 −0.0317 −0.0261 −0.0015 0.0774
August 8 0.0582 0.0116 0.0165 0.0003 0.0417
September 9 0.0095 0.0409 0.0453 −0.0001 −0.0358
October 10 −0.0143 0.0259 0.0305 −0.0005 −0.0448
November 11 −0.0908 −0.0493 −0.0434 0.0057 −0.0474
December 12 0.0278 −0.0043 0.0009 −0.0002 0.0269
Total 0.1355 0.0754 0.1355 0.00936 0.0000
Average (r) 0.0113 0.0063 0.0113 — 0.0000
Sum of squares* 0.02744 0.00952 0.00921 — 0.01824
Stddev ( ˆσr) 0.0499 0.0294 0.0289 — 0.0407
∗Sum of squares=T t=1
rt−r2
(Continued)
EXAMPLE 8.1
(Continued)The scatter plot for the returns from stock iand the market portfolio is drawn in Figure 8.3.
rit
rmt 0.1000
0.0800 0.0600 0.0400 0.0200 0.0000 –0.0200
–0.0200 0.0000 0.0200 0.0400 0.0600
–0.0400 –0.0400
–0.0600 –0.0600
–0.0800 –0.1000
FIGURE 8.3 Plot of the Returns from a Stock and the Market Portfolio The estimates of the regression parameters are calculated as
βˆi= 12
t=1
rit−ri rmt−rm T
t=1
rit−ri2
= 0.00936
0.00952 =0.983 αˆi=ri−βˆirm
=0.0113−(0.983) (0.0063)=0.00511 σˆεi2=
1 T−2
12 t=1
εˆ2it
= 1
T−2 12
t=1
rit−
αˆi+βˆirmt2
= 1
12−2 (−0.0036)2+(−0.0510)2+ · · · +0.02692
= 1
12−2
(0.01824)=0.001824
8.1.3 The Single-Index Model Using Excess Returns
During the late 1960s Michael C. Jensen, a PhD student at the University of Chicago, suggested advantages to using the excess return (risk premium) defined in equation (8.4) as the basic random variable.6
Rit
Excess return
= rit
HPR
− rf
Riskless return
(8.4)
It can be argued that the excess return defined in equation (8.4) differs from the holding period return (HPR) defined in equation (1.1) by a positive constant and, therefore, the two return measures are not substantially different. Section 18.4.3 of Chapter 18 explains how Jensen uses the excess return defined in equation (8.4) to make a valuable contribution to investment performance evaluation. The riskless rate of return,rf, may be a meaningful random variable in its own right.
When the riskless rate of return fluctuates over time, it is given a time subscript, denotedrft.
Jensen showed that excess returns may also come from a return-generating function that is similar to, but different from, the Markowitz-Sharpe-Treynor return-generating function of equation (8.1). Jensen reformulated the single-index market model in HPRs of equation (8.1) to become the single-index market model stated in terms of excess returns shown in equations (8.5) and (8.5a).
rit−rf =αi+βi
rmt−rf
+εit (8.5)
or
Rit=αi+βiRmt+εit (8.5a) whereRit=rit−rfandRmt=rmt−rfare the excess return of assetiand the market index, respectively, for time periodt. The original version of the single-index model of equation (8.1) and its excess return version, equation (8.5), are very similar. The beta slope coefficient, βi, and the error term, εit, of the two versions are the same, because the dependent and independent variables are reduced by the same constant, rf. Only the intercept terms differ. The relationship between the two intercepts is
αi =αi−rf 1−βi
(8.6) The intercept of Jensen’s excess return version of the single-index model is called Jensen’s alpha. A variance decomposition resembling earlier equation (8.3) also holds for the excess return version of the single-index model.
Var Rit
=Var
αi+βiRmt+εit
=E
αi+βiRmt+εit
−E
αi+βiRmt+εit
2
=E βi2
Rmt−μm
2
+εit2+2βi
Rmt−μm
εit
=βi2σm2+σεi2
EXAMPLE 8.2
Assume an invariant riskless interest rate of three percent per year exists.
This annual rate of interest is converted into monthly returns by dividing rf =3.0 percent by the number of months in one year to obtain an equivalent riskless interest rate of (3% / 12 months =) 0.25% = 0.0025 per month.
Table 8.2 reproduces the holding period excess returns from Table 8.1 by subtracting the monthly riskless rate of returnrf =0.0025 from the monthly HPRs from the stock and from the monthly returns on the market.
TABLE 8.2 Hypothetical Monthly Holding Period Excess Returns from a Stock and the Market
Month t Excess
return on stocki Rit (=rit−rf)
Excess return on the market
Rmt (=rmt−rf)
Fitted excess return on
stocki ˆ
αi+βˆiRmt
Product (Rit−Ri)×
(Rmt−Rm)
Unexplained residuals
ˆ εit
(1) (2) (3) (4) (1)–(3)
January 1 0.0181 0.0169 0.0217 0.0001 −0.0036
February 2 −0.0621 −0.0165 −0.0112 0.0014 −0.0510
March 3 0.0117 0.0104 0.0153 0.0000 −0.0036
April 4 0.0818 0.0374 0.0418 0.0025 0.0400
May 5 0.0444 0.0364 0.0409 0.0012 0.0035
June 6 −0.0152 −0.0173 −0.0119 0.0005 −0.0033
July 7 0.0488 −0.0342 −0.0286 −0.0015 0.0774
August 8 0.0557 0.0091 0.0140 0.0003 0.0417
September 9 0.0070 0.0384 0.0428 −0.0001 −0.0358
October 10 −0.0168 0.0234 0.0280 −0.0005 −0.0448
November 11 −0.0933 −0.0518 −0.0459 0.0057 −0.0474
December 12 0.0253 −0.0068 −0.0016 −0.0002 −0.0269
Total 0.1055 0.0454 0.1055 0.00936 0.0000
Average (R) 0.0088 0.0038 0.0088 — 0.0000
Sum of squares* 0.02744 0.00952 0.00921 — 0.01824
Stddev ( ˆσR) 0.0499 0.0294 0.0289 — 0.0407
*Sum of squares=T t=1
rt−r2
As in Example 8.1, statistical estimates of the regression’s population parameters are calculated next.
βˆi= 12
t=1
Rit−Ri Rmt−Rm T
t=1
Rit−Ri2
= 12
t=1
rit−ri rmt−rm T
t=1
rit−ri2 =0.00936
0.00952 =0.983
EXAMPLE 8.2
(Continued) αˆi=Ri−βˆiRm=0.0088−(0.983) (0.0038)=0.00506 σˆεi2=
1 T−2
12 t=1
εˆ2it= 1
T−2 12
t=1
Rit−
αˆi+βˆiRmt2
= 1
12−2
(0.01824)=0.001824
The estimates of the slope coefficient (βi) and the error variance (σεi2) in this example are exactly the same as those in Example 8.1. Only the intercept estimate is different. In fact, as mentioned previously, the intercept estimate in this example ( ˆαi) can be obtained through equation (8.6).
If Jensen’s single-index market model stated in terms of excess returns in Table 8.2 were graphed, that graph would differ slightly from Figure 8.3. All the little differences that occur can be attributed to the two alpha intercept terms that differ slightly. The difference in the two illustrations would be so small it would be difficult to see in this particular numerical example. If the underlying sample of market data were different, sometimes it would be possible to find a graph of Jensen’s single-index market model stated in terms of excess returns that would differ noticeably from a graph of the Markowitz-Sharpe-Treynor single-index market model computed with holding period returns.
8.1.4 The Riskless Rate Can Fluctuate
Portfolio theory equates risk with variability of return. Thus, it would seem that the riskless interest rate should be invariant. However, over a century ago a well-known economics professor named Irving Fisher taught us that the level of market interest rates rise and fall with the rate of inflation.7Figure 8.4 illustrates the way the one- year U.S. Treasury bill interest rate (solid line) and the annualized inflation rate in the consumer price index (CPI) (dashed line) covary through time; their correlation is 0.70.
When financial analysts perform empirical work, they often use U.S. Treasury bill interest rates as surrogates for the riskless interest rate. Although U.S. Treasury bills are U.S. government bonds that are default free, their prices and yields nevertheless fluctuate because of changing inflationary expectations and other factors. In other words, empirical researchers sometimes use a riskless interest rate that involves risky variations, Var
rft
>0. All U.S. Treasury bills mature within one year. The prices of these short-term bonds fluctuate less than the prices of the longer-term U.S.
Treasury bonds. And, default-free U.S. Treasury bonds fluctuate less than corporate bonds with equal maturities, because the corporate bonds also involve default risk.
But, as illustrated in Figure 8.4, the one-year T-bill rate fluctuates significantly over time.
0 2 4 6 8 10 12 14 16 18
1959/… 1961/… 1962/… 1964/… 1965/… 1967/… 1969/… 1970/… 1972/… 1973/… 1975/… 1977/… 1978/… 1980/… 1981/… 1983/… 1985/… 1986/… 1988/… 1989/… 1991/… 1993/… 1994/… 1996/… 1997/… 1999/… 2001/… 2002/… 2004/… 2005/… 2007/…