A THEORETICAL AND EMPIRICAL ANALYSIS OF THE SUITABILITY OF
SOUTH AFRICAN RISK-FREE RATE PROXIES
Barry Strydom*
School of Economics & Finance, University of Kwazulu-Natal
&
Ailie Charteris
School of Economics & Finance, University of Kwazulu-Natal
ABSTRACT
This study examines the theoretical requirements for a suitable risk-free rate proxy,
reviewing the most common proxies identified in the literature, and empirically
investigating the suitability of these proxies. It finds a lack of consensus regarding the
appropriate proxy.
While neither of the suggested proxies fully conform to the
theoretical requirements of a risk-free asset, the Treasury Bill rate, is found to be
particularly subject to excessive volatility and to contain a significant volatility premium.
It is concluded that the use of longer-term government bonds, matching the relevant
investment horizon, is preferable.
* Corresponding author: strydomb@ukzn.ac.za; room 332, School of Economics &
Finance, New Arts Building, P.Bag X01, 3209; , 033--2605794
1.
INTRODUCTION
The use of the CAPM in South Africa (S.A) to estimate the cost of equity has increased
considerably in recent years. Correia and Cramer (2008: 41) established that 71 percent
of firms surveyed used the model or a derivation thereof compared to the results of
Pocock et al (1991), in which only 35 percent of the respondents employed the CAPM
for this purpose. This greater reliance of S.A. firms on the model for determining the cost
of equity mirrors the changing practices observed in the United States of America (U.S.).
Graham and Harvey (2001) found that 74 percent of respondents in the U.S. utilised the
CAPM, in contrast to only 30 percent identified in the survey of Gitman and Mercurio
(1982). The model has thus grown in popularity in the last two decades despite a
substantial body of evidence (such as Roll, 1977; Banz, 1981; Fama and French, 1992;
Fama and French, 2004) disputing its validity.
The CAPM parameters (beta, the risk-free rate and the return on the market) are all
theoretical constructs that are not easily observable in the market, and consequently
proxies are used for these variables in practice (Firer, 1993: 25). The accuracy of the
CAPM in determining the cost of equity is therefore, at least partly, a function of the
suitability of the proxies employed. Further to this, as Correia and Ulius (2004: 66)
explained it is the difficulty in selecting these surrogates that has hampered the
implementation of the model.
The choice of the appropriate market portfolio proxy has been well-documented in S.A.
literature, with some degree of consensus that the ALSI provides a reasonable estimate
for the market portfolio, although it may overestimate the cost of equity for financial and
1
industrial shares (Correia and Ulius, 2004: 78). The ALSI is typically also used in the
estimation of market betas using monthly data (Firer et al; 2004: 411).
Almost no explicit research however, has been conducted on the most suitable proxy to
estimate the risk-free rate in S.A. Firer (1993: 37), in concluding his discussion on the
matter stated “…at this point it should be clear that the issue of estimating the risk-free
rate … is by no means resolved, and CAPM users have not been given any firm and
theoretically sound guidelines on which to base their estimates”. This study thus seeks to
address this gap in the literature by formulating a set of theoretical requirements that an
asset must closely satisfy to be considered a suitable proxy for the risk-free rate in any
market. Thereafter, the most commonly used S.A. surrogates will be compared to these
requirements to assess their appropriateness.
The layout of this study continues as follows. Section 2 reviews the most widespread
practices with regards to the estimation of the risk-free rate in both the U.S. and S.A., and
establishes the criteria for the correct specification of the risk-free rate. In Section 3,
short-term and long-term S.A. government securities are theoretically and empirically
compared to the said requirements. Conclusions and recommendations for further
research are discussed in Section 4.
2
2.
LITERATURE REVIEW
2.1.
The Choice of Proxy
Most U.S. undergraduate finance textbooks advocate government securities as the most
suitable proxies for the risk-free rate (see for example Harrington, 1987; Copeland, et al,
2000; Bodie et al, 2003; Brigham and Ehrhardt, 2005; Reilly and Brown, 2006). These
instruments have been utilised for this purpose in research for the past forty years (Black et
al, 1972; Malkiel, 1995; Fama and French, 2004), and a study of corporate practices in the
U.S. by Bruner et al (1998: 16) revealed that this method of estimating the risk-free rate is
as prevalent in industry, as 85 percent of corporations and 90 percent of financial advisors
surveyed used the yields on government securities as a proxy for the risk-free rate.
The difficulty with regards to the estimation of this parameter is the choice of the
appropriate maturity of the instrument as there is little consensus as to whether short-term
treasury bills (T-Bills), with maturity of less than a year, or longer-term treasury bonds
(T-Bonds), with maturities in excess of three years, are more suitable. Three month TBills have been favoured historically (Harrington, 1987: 149), but T-Bonds have become
increasingly popular as evinced in the findings of the Bruner et al (1998: 19) survey, in
which almost all of the corporations and financial advisors surveyed (77 percent and 80
percent respectively) favoured the use of securities with maturities greater than three
years for the risk-free asset proxy.
T-Bonds usually have a higher yield than T-Bills to compensate investors for the
inconvenience of being without their funds for a longer period of time. Thus the growing
3
practice of employing these securities undoubtedly represents an attempt to reconcile
theory with practice, as several empirical studies have indicated that the minimum return
required by investors often exceeds the T-Bill yield (Black et al, 1972; Fama and
MacBeth, 1973; Stambaugh, 1982; Fama and French, 2004). However, a review of U.S.
research revealed almost no evidence of longer-term securities being applied and, several
texts continue to advocate T-Bills as the most suitable proxy (Bodie et al, 2003; Reilly
and Brown, 2006). It is therefore apparent that academics and practitioners in the U.S.
appear to concur that a government security is the appropriate proxy to use, but that there
is no agreement as to whether or not this proxy should be a short-term or long-term
government security.
Similar practices to the U.S. for the estimation of the risk-free rate have generally been
observed in S.A. Firer (1993: 29) conducted a review of research papers published in
S.A. in which the author was required to estimate the risk-free rate as part of the
methodology. The articles examined all employed a proxy as the means to estimate this
rate (Firer, 1993: 29). Affleck-Graves et al (1988), and Page and Palmer (1991) both
estimated the rate as the three month T-Bill yield; whilst De Villiers et al (1988) used the
one year equivalent (Firer, 1993: 29). In contrast, Bradfield et al (1988) employed the
twelve month fixed deposit yield (Firer, 1993: 29).
In order to ascertain how practices have evolved in S.A., a similar examination of
published research was conducted over the period 2001 to 2008. Friis and Smit (2004),
and Samouilhan (2007) chose simply to estimate the risk-free rate in their respective
studies as the three month T-Bill yield. On the contrary, de Wet (2005), Moolman and du
Toit (2005), de Wet (2006), de Wet and Hall (2006), and de Wet and du Toit (2007) all
4
utilised the yield on government bonds. None of these authors provided any justification
for their decision, but notably, shorter-term government bonds, such as the R150 and
R153 (maturities of less than five years) were favoured in all of these studies except that
of Moolman and du Toit (2005), who employed the R157 bond which represented a ten
year instrument.
Several of the research papers examined questioned the validity of government securities
as proxies for the risk-free rate in the S.A. market, and consequently chose to estimate the
parameter using short-term private sector securities which they deemed to be more
appropriate. Graham and Uliana (2001), Cloete et al (2002), Oldham and Kroeger (2005),
and Msweli-Mbanga and Mkhize (2007) employed the rate on Bankers’ Acceptances
(BAs), which are perceived to be the safest private sector short-term instrument, as the
proxy for the risk-free rate. The other private sector security that has become increasingly
popular is the three month Negotiable Certificate of Deposit (NCD), as this was used by
Akonjolirie and Smit (2003), Correia and Uliana (2004) and Viviers et al (2008).
Firer and McLeod (1999) discuss the direct and indirect influence of the government’s
policies on T-Bills, BAs, and NCDs. One of the issues highlighted was that of the
introduction of the Bank’s Act of 1965, which required financial institutions to hold
various types of assets that matched the liquidity of their liabilities (Firer and McLeod,
1999: 6). Both T-Bills and BAs were ranked as a liquid asset in terms of this legislation
resulting in artificial demand for these instruments from banks. Thus these rates do not
necessarily provide an accurate representation of a market-determined rate, and hence
Firer and McLeod (1999: 7) argue that NCDs represent a more realistic marketdetermined rate and should be used as the proxy for the risk-free rate in S.A. However,
5
Firer and McLeod’s (1999: 6) argument intimates that it is the less liquid nature of the
NCD which has prevented it from being listed as a liquid asset in terms of the required
holdings of banks and other institutions. Yet, a risk-free asset should be a highly liquid
instrument as will be discussed further in Section 2.2.7.
Corporations and financial advisors in S.A., however, have not mirrored the evolving
practices in research of using private sector securities rather than government instruments
to estimate the risk-free rate, as evinced in the results of several recent surveys. The
PriceWaterHouseCoopers Surveys of financial institutions in 2003 and 2005 indicated
that almost all of the respondents employed either the R153 or R157 government bond
yields (as quoted in Correia and Cramer, 2008: 42). Correia and Cramer (2008: 42), who
surveyed a wider spectrum of firms, found that the majority (55 percent) of those
surveyed employed the R153 government bond yield, with only 15 percent using the
R157 bond. The other firms used the R186, R196, R201, or the All Bond Index (ALBI)
(Correia and Cramer, 2008: 42).
It is certainly apparent that there is little or no clarity in S.A. as to the appropriate proxy
to use for the risk-free rate in the CAPM and yet the choice can have a material impact on
the cost of equity value obtained. Therefore, to best assess which security provides the
most suitable proxy for this parameter, a set of requirements that an asset must satisfy so
that the risk-free rate is correctly specified will be formulated.
6
2.2.
Theoretical Criteria for the Correct Specification of the Risk-Free Rate
By definition a riskless asset should be devoid of risk. However, given the wide range of
risks securities are exposed to, this requirement has a number of implications.
2.2.1
A Pure Interest Rate
The first requirement that Sharpe (1964: 431) identified for a security to be classified as a
risk-free instrument was that the return on a risk-free security must be equal to the pure
interest rate. Reilly and Brown (2006: 15) confirm this, suggesting that the real return
should only depend on the time preference of individuals for the consumption of income
and the investment opportunities available in the economy. This implies that the return
earned should not be influenced by external factors, such as government policies or
economic events (Harrington, 1987: 153). In this way, the relative independence of the
yield on a risk-free security in comparison to other assets should assist in ensuring both
that the returns are stable over time (Reilly and Brown, 2006: 15), and that the returns do
not move in tandem with the returns on the market.
2.2.2
Inflation Risk
Investors require a return which includes an inflation premium to provide compensation
for the expected change in price levels over the investment horizon (Harrington, 1987:
40, 155; Reilly and Brown, 2006: 21). A risk-free asset should thus provide a yield
greater than the real minimum return to compensate the holder for inflation (Firer, 1993:
28); that is, the return should comprise the pure interest rate plus a premium for inflation.
7
However, a risk-free security must still satisfy the criterion that it is free of inflation risk,
which arises when actual inflation differs from expectations and the premium included in
the yield on the security does not accurately reflect the observed price level changes
during the period of the investment (Blake, 2000: 86). Consequently, for an asset to be
free of this risk, the expectations of and the actual inflation level must be consistent.
2.2.3
Variance in Returns
Risk is measured in portfolio and capital market theory as the variation of the actual
returns around the mean or expected value, as determined by variance (Rees, 1995: 223).
For an asset to be risk-free in an uncertain environment the variance must equal zero for
the duration of the investment, implying that the actual returns earned over the period are
always equal to the expected return (Sharpe, 1964: 431).
2.2.4
Interest Rate Risk
Interest rate risk comprises both reinvestment and price risk (Reilly and Brown, 2006:
775), which is a function of volatility in interest rates. The yield on the risk-free rate
proxy should not include compensation for the risk of interest rates fluctuating
unfavourably during the life of the security. However, if the returns of the proxy
instrument vary significantly over time, then the possibility exists that the return may
include a premium for interest rate risk.
8
2.2.5
Covariance with the Market
A riskless asset, whose returns are always equal to the expected return, will have zero
variation around its mean and thus it cannot be related to the movement of a risky asset’s
returns around its mean value (Reilly and Brown, 2006: 232). Hence the covariance of a
risk-free security with any risky instrument must be zero (Sharpe, 1964: 431). In addition,
the covariance of a riskless asset with the market portfolio, which includes all risky
assets, should be zero.
2.2.6
Default Risk
The security employed as the proxy for the risk-free rate must be devoid of default risk
meaning that there is no possibility that the issuer of the security will default on the
repayment of the investment to the holder (Damodaran, 2001: 3). Government securities
are therefore identified as suitable proxies as a government can, under most
circumstances, print money or raise taxes to avoid default on its commitments
(Damodaran, 2001: 4; Reilly and Brown, 2006: 19).
2.2.7
Liquidity Risk
Liquidity refers to the ease with which an asset can be sold without a significant
reduction in value; liquidity risk thus arises from the difficulty in selling an asset in the
secondary market with respect to both the time taken to sell the security and the price to
be received (Reilly and Brown, 2006: 22). The risk-free rate in the CAPM should only
provide investors with compensation for the temporary illiquidity of being without their
9
funds for a single investment horizon (Firer, 1993: 28), and thus as a consequence of the
fact that a risk-free asset must not exhibit any liquidity risk, no liquidity premium should
be included in the yield.
In order for a proxy to correctly specify the riskless rate in the CAPM, it must closely
satisfy all of these conditions. Given the academic support for using government
securities as the risk-free rate proxy, together with their extensive use in practise, this
study will limit itself to examining the extent to which short-term and long-term
government securities (S.A T-Bills and T-Bonds ) conform to these requirements.
3.
SOUTH AFRICAN PROXIES: ANALYSIS AND EVIDENCE
3.1
Inflation Risk
To ascertain the exposure of S.A. and U.S. treasury securities to inflation risk, a
comparison of the Consumer Price Index for both countries was conducted for the period
1998 to 2008 based on monthly data from the South African Reserve Bank (SARB) and
Econstats. The results, which are shown in Figure 1, illustrate clearly that inflation in
S.A. is far more volatile than in the U.S.
10
Figure 1: U.S. and S.A. Consumer Price Index 1998-2008
.14
.12
.10
.08
.06
.04
.02
.00
98
99
00
01
02
03
SA_CPI
04
05
06
07
08
US_CPI
To confirm this, an F-test of the equality of the variance of the two series was performed
(see Keller and Warrack, 2000: 429). The test statistic obtained of 10.21 is considerably
larger than the critical values based on 132 degrees of freedom (1.33, 1.41, and 1.57 at
the ten, five and one percent significance levels respectively for a two-tailed test) and
therefore it can be concluded that inflation in S.A. is significantly more volatile than in
the U.S. It therefore follows that while inflation risk for U.S. securities is likely to be
relatively low; S.A. securities are exposed to greater inflation risk, thereby reducing their
suitability to act as a risk-free rate proxy.
In addition to this, it is also necessary to compare the inflation risk of longer-term TBonds as opposed to short-term T-Bills. With regards to inflation risk, there is no debate
as to which instrument more closely resembles the risk-free asset requirement: T-Bills
minimise the impact of changes in price levels (Harrington, 1987: 156-157; Blake, 2000:
86; Reilly and Brown, 2006: 19; Pike and Neale, 2005: 250). The reason for this is that it
is considerably harder for investors to anticipate long-term inflation movements with any
accuracy and hence determine its likely impact on returns (Harrington, 1987: 155);
11
consequently, yields on long-term bonds include a greater inflation risk premium to
compensate investors for the possibility that the promised return may be eroded by
unexpected changes in purchasing power (Pike and Neale, 2006: 250). For short-term
investments inflation risk is less prevalent as the shorter duration enables more accurate
expectations to be incorporated into the returns offered on the securities.
3.2
Variance in Returns
To assess whether T-Bills or T-Bonds in S.A. and the U.S. adhere to the requirement of
zero variation over the investment horizon, an analysis of historical fluctuations in the
returns of these instruments was performed. Yield data was collected from the SARB for
the three-month T-Bill and the R157, 13.5 percent coupon government bond. U.S. threemonth T-Bill and 10-year T-Bond yields were obtained from the Federal Reserve for the
period January 1996 to December 2008.
The asset prices of the U.S. T-Bills were obtained as per Bodie et al (2002: 30), with the
identical approach used for the S.A. securities, but with a period adjustment based on 365
rather than 360 days (Botha, 2006: 240). The calculation of the prices of the U.S. TBonds were based on a pure discount of the principal as these securities are valued as
zero-coupon bearing bonds, the yield determined at auction (Bodie et al, 2002: 31). S.A.
long-term government bonds in contrast, are coupon-bearing bonds and thus, the bond
valuation formula was applied, adjusting the yield and time to maturity based on the
semi-annual coupon payments of the R157 bond (as per Firer et al, 2004: 188). The
holding period returns on all four securities were computed as the difference in the price
of the assets across each day.
12
The variance and standard deviation values for the thirteen year period were calculated;
however, in order to accurately assess these results it was crucial to determine whether
the variance estimates were statistically significantly different from zero. To test the
significance of the variance estimates of the proxies, a value close to zero was needed and
hence an estimate of 0.001 was chosen. A one-tailed chi-squared test (Keller and
Warrack, 2000: 370-372) was conducted to determine whether the variance estimates of
the return series were equal to or greater than 0.001.
The values obtained for this analysis are shown in Figure 2 and indicate clearly that the
variance and standard deviation of both S.A. T-Bills and T-Bonds exceeds that of the
U.S. equivalents. The test statistic for U.S. T-Bills was smaller than the critical values
suggesting that these securities do not exhibit significant volatility over time. However,
the variability of S.A. T-Bills was significant over the period 1996-2008 as the test
statistic exceeded the critical values.
Figure 2: Variance Analysis for S.A. and U.S. Treasury Securities
SA T-Bills
SA T-Bonds
U.S. T-Bills
U.S. T-Bonds
0.00069
0.587
0.00025
0.003
Standard Deviation
0.026
0.766
0.016
0.057
Chi Squared Statistic
2344
1991449
863
10870
Variance
Chi Squared Critical Values
(3393 degrees of freedom)
10% = 1162;
5% = 1179;
1% = 1213
13
Examining the graph of the stationary returns of both the U.S. and S.A. T-Bills in Figure
3 confirms the greater volatility in the S.A. instruments; most notably during the 1998
financial crisis. Interestingly however, whilst both series show considerable clustering
towards the end of the period (2007-2008), it is the U.S. returns that are more prominent
highlighting the greater initial impact of the financial crisis on the U.S. than S.A. U.S. TBills thus satisfy the condition of zero variance and can be applied as the proxy for the
risk-free rate in the CAPM without introducing significant risk caused by variability in
interest rates. S.A. T-Bills in contrast, do not fulfil this requirement and therefore rolling
over this short-term instrument will create risk in the model because of the uncertainty of
future returns to be earned on the instrument.
The test statistic for U.S. T-Bonds was greater than the critical values and therefore it can
be concluded that the variance of U.S. T-Bonds was statistically larger than zero.
Similarly, the test statistic for S.A. T-Bonds was far greater than the critical values and
the U.S. security, as clearly evident in the second panel in Figure 3.
Figure 3: The Volatility of U.S. vs. S.A. T-Bills and T-Bonds
.003
.08
.002
.04
.001
.000
.00
-.001
-.002
-.04
-.003
-.004
-.08
-.005
-.006
-.12
96 97 98 99 00 01 02 03 04 05 06 07 08
DLOGSATBILLP
DLOGUSTBILLP
96 97 98 99 00 01 02 03 04 05 06 07 08
DLOGSATBONDP
DLOGUSTBONDP
14
This conclusion of greater volatility in longer-term instruments, which is confirmed in
Figure 4 for both S.A. and the U.S, contradicts several texts which propose the use of TBonds as proxies for the risk-free rate because of their lower variation (Harrington, 1987;
Brigham and Ehrhardt, 2005), but is in keeping with the findings of early studies of
Ibbotson and Sinquefield (1979) and Carleton and Lakonishok (1985). The greater
variation in T-Bond returns compared to T-Bills is however only pertinent if the
investment horizon does not match the maturity of the project being assessed. That is, if
the maturity of the T-Bond mirrors that of the investment being analysed in the CAPM,
the movements in the interest rate have no influence on the returns of the T-Bond if held
to maturity. In the event, however, that the security has to be rolled over or liquidated
prior to maturity, the variance is a source of risk for the investor.
Figure 4: The Volatility of U.S. T-Bills vs. T-Bonds and S.A. T-Bills vs. T-Bonds
.003
.08
.002
.04
.001
.00
.000
-.001
-.04
-.002
-.08
-.003
-.004
-.12
96 97 98 99 00 01 02 03 04 05 06 07 08
DLOGUSTBONDP
3.3
DLOGUSTBILLP
96 97 98 99 00 01 02 03 04 05 06 07 08
DLOGSATBONDP
DLOGSATBILLP
Interest Rate Risk
In light of this evidence that both short-term and long-term securities in S.A. exhibit
variation in returns, it is clear that investors are exposed to interest rate risk in some form
15
or another. This raises the question of whether or not the return offered on the security
includes compensation for this volatility; that is, is a premium included in the returns for
interest rate risk. The Generalised Autoregressive Conditional Heteroscedasticity in Mean
(GARCH-M) model was employed to address this issue. The general GARCH (p,q)
model attempts to model variance by specifying two equations: the conditional mean and
conditional variance. The former follows the specification determined by the research
problem, whilst the later models the variance based on the lags of the previous squared
error terms of the conditional mean (q) and on lags of the conditional variance (p)
(Brookes, 2008: 392-393). These are illustrated in equations 1 and 2.
yt = β1 + β2x2t + β3x3t + ut
(1)
σt2 = α + α1u2t-1 + α2u2t-2 + … + αqu2t-q+ γ1σ2t-1 + γ2σ2t-2 + …. + γpσ2t-p
(2)
Where:
-
σt2 is the conditional variance
-
σ2t-1 is the one period lagged conditional variance
-
u2t-1 is the one period lagged squared residuals of the mean equation
(Brooks, 2008:393)
It is widely observed across financial literature that a GARCH (1,1) specification is
sufficient to model the variability observed in the data (Bollerslev, Chou and Kroner,
1992), and hence this order was applied for the examination of T-Bills and T-Bonds.
The GARCH-M model expands the conditional mean equation to include a lagged term
of the conditional variance (Brooks, 2008: 409-410). By introducing this term into the
16
conditional mean equation it is possible to determine whether the returns on the particular
financial asset series are influenced by the volatility in the returns; hence the coefficient
on the conditional variance term represents a volatility risk premium. In the case of TBills and T-Bonds, the estimation of a GARCH-M model therefore essentially involves
determining the influence of volatility on the returns earned on the asset in the form of a
risk premium. The GARCH-M (1,1) model estimated for all four series is presented in
equations 3 and 4.
yt = β1 + δσ2t-1 + ut
(3)
σt2 = α + α1u2t-1 + γ1σ2t-1
(4)
(Brooks, 2008: 410)
Following the recommendation of Brooks (2008: 390) daily data was used for the same
period as examined previously (1996-2008). All four asset prices were non-stationary and
thus had to be transformed into first differences. The differenced log terms were used to
estimate the GARCH-M equations and the results for the four models are shown in
Figures 5-8. In the conditional mean equation for U.S. T-Bills, the risk premium is
statistically insignificant as the null hypothesis that this value is equal to zero could only
be rejected with 10 percent confidence. This proxy thus conforms to the requirements of
zero variance over the investment period and does not provide compensation for investors
for the risk observed in the returns. For S.A. T-Bills, the coefficient on the GARCH term
is highly statistically significant (p value equal to zero); thereby indicating that the returns
offered on S.A. T-Bills include a premium to compensate investors for the risk inherent
in these instruments caused by volatility over time.
17
Figure 5: GARCH-M Model for U.S. T-Bills
Dependent Variable: DLOGUSTBILLP
Variable
Coefficient
Std. Error
z-Statistic
Prob.
GARCH
C
13.96832
-4.89E-06
107.8545
1.43E-06
0.129511
-3.419174
0.8970
0.0006
9.078010
29.64465
132.4890
0.0000
0.0000
0.0000
Variance Equation
C
RESID(-1)^2
GARCH(-1)
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
1.86E-10
0.238777
0.800223
-0.003190
-0.004375
0.000160
8.65E-05
26455.92
1.759064
2.04E-11
0.008055
0.006040
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Hannan-Quinn criter.
3.82E-06
0.000159
-15.59606
-15.58703
-15.59283
Figure 6: GARCH-M Model for S.A. T-Bills
Dependent Variable: DLOGSATBILLP
Variable
Coefficient
Std. Error
z-Statistic
Prob.
GARCH
C
282.3648
-9.79E-06
64.63317
4.97E-06
4.368728
-1.968015
0.0000
0.0491
55.31760
56.80576
2246.375
0.0000
0.0000
0.0000
Variance Equation
C
RESID(-1)^2
GARCH(-1)
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
6.92E-10
0.016427
0.971636
-0.010133
-0.011326
0.000265
0.000237
23855.66
1.965955
1.25E-11
0.000289
0.000433
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Hannan-Quinn criter.
2.60E-06
0.000263
-14.06289
-14.05385
-14.05966
The model estimated for U.S. T-Bonds revealed a statistically insignificant coefficient for
the risk premium (p value of 95.52 percent); providing evidence that although these
instruments do exhibit fluctuations in returns over time, no compensation for this
volatility is provided to investors in long-term instruments. For S.A. T-Bonds, the
evidence with regards to the presence of a volatility premium is mixed, as the highest
18
level at which the null hypothesis that the risk premium coefficient is zero can be rejected
is 16.13 percent. Thus, whilst this p value lies above the conventional significance levels,
there is still some evidence to suggest that the inclusion of a volatility premium in the
R157 bond yields cannot be completely rejected.
Figure 7: GARCH-M Model for U.S. T-Bonds
Dependent Variable: DLOGUSTBONDP
Variable
Coefficient
Std. Error
z-Statistic
Prob.
GARCH
C
-4.276366
8.59E-06
76.07181
2.24E-05
-0.056215
0.382956
0.9552
0.7018
4.353492
10.34074
229.3098
0.0000
0.0000
0.0000
Variance Equation
C
RESID(-1)^2
GARCH(-1)
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
2.70E-09
0.035964
0.956390
-0.000049
-0.001230
0.000567
0.001089
20698.00
1.922804
6.21E-10
0.003478
0.004171
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Hannan-Quinn criter.
9.50E-06
0.000567
-12.20106
-12.19203
-12.19783
Figure 8: GARCH-M Model for S.A. T-Bonds
Dependent Variable: DLOGSATBONDP
Variable
Coefficient
Std. Error
z-Statistic
Prob.
GARCH
C
4.356673
7.21E-08
3.110609
0.000101
1.400585
0.000715
0.1613
0.9994
14.77548
26.35847
112.5048
0.0000
0.0000
0.0000
Variance Equation
C
RESID(-1)^2
GARCH(-1)
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
F-statistic
Prob(F-statistic)
2.13E-06
0.142915
0.825995
0.002684
0.001506
0.007671
0.199320
12426.86
2.278816
0.058511
1.44E-07
0.005422
0.007342
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Hannan-Quinn criter.
Durbin-Watson stat
0.000107
0.007677
-7.324209
-7.315175
-7.320980
1.938690
19
The fact that the volatility premium is smaller in T-Bond than T-Bill returns in S.A is
somewhat at odds with the results obtained in the analysis of variance in the preceding
section, which revealed the much greater volatility in the longer-term instrument. A
possible explanation for this is the different perspectives of investors purchasing
securities of varying maturities. Investors purchasing longer-term T-Bonds generally
have a long-term investment horizon and therefore short-term fluctuations in returns are
not likely to have an impact on their wealth position. Consequently, these investors will
not demand compensation for the volatility in the returns earned on the instrument.
Although the wealth position of short-term investors in T-Bills is guaranteed once the
security is purchased, their ability to reinvest in the following period at the same rate
depends on the movement in interest rates. In a volatile market these investors are thus
exposed to substantial risk from this source, for which they require compensation in the
form of a volatility risk premium.
The two conclusions from this analysis are therefore that firstly, S.A. government
securities are volatile and consequently investors are provided with a risk premium to
compensate them for the potential loss in value caused by unfavourable interest rate
fluctuations compared to the U.S. where this is not the case. Secondly, the risk premium
is much larger and more significant in the returns of short-term S.A. T-Bills compared to
longer-term T-Bonds despite the fact that T-Bonds are considerably more volatile. These
results suggest that provided that a T-Bond with a maturity equal to the investment
horizon is purchased, the T-Bond yield more closely resembles the risk-free asset in
terms of minimising the presence of a risk premium for volatility.
20
3.4
Covariance
As explained in Section 2.2.5, the covariance of the risk-free asset with the market
portfolio should be zero, and as with the preceding requirements, the extent to which S.A.
securities satisfy this condition can be assessed empirically. The market portfolios used to
represent the two markets for this analysis were chosen as the Standard and Poor’s top
500 Index (S&P 500) and the ALSI respectively. Daily index values were obtained for
the period 1996 to 2008 from McGregor’s Database and Econstats, and the returns earned
on each of these indices were computed. As with the variance values, it was deemed
necessary to assess the statistical significance of the covariance estimates so as to
determine the extent to which the proxies satisfy the zero covariance constraint. Rather
than testing the covariance however, the correlation between the risk-free and market
portfolio surrogates were examined because of the difficulty in determining the
distribution for the covariance measurement (Freund and Perles, 2003: 455-457). The ztest statistic was calculated for this purpose (see Freund and Perles, 2003: 456) based on
monthly data.
The results for the analysis are presented in Figure 9 and clearly indicate that the
correlation of U.S. treasury securities with the market is considerably smaller in absolute
terms than for the comparable S.A. securities. This is confirmed in the results of the
statistical tests. The null hypothesis of zero correlation could not be rejected for U.S. TBonds as the statistic computed was smaller in absolute terms than the critical value,
whilst for U.S. T-Bills, there was some evidence that the returns do move in tandem with
the returns on the market, but this relationship was only significant at the 10 percent
level. For both S.A. T-Bills and T-Bonds, the null hypothesis that the correlation is zero
21
was rejected and thus neither of these two securities conforms to the risk-free asset
requirement. However, contrary to the U.S., it is T-Bills that perform slightly better in
this regard in S.A. than T-Bonds.
Figure 9: Correlation Analysis for S.A. and U.S. Treasury Securities
SA T-Bills
SA T-Bonds
U.S. T-Bills
U.S. T-Bonds
Covariance
0.21
4.73
-0.038
-0.9
Correlation
0.25
0.32
-0.14
-0.09
Z Statistic
3.20
4.09
-1.77
-1.10
Critical Values
10% = 1.64;
5% = 1.96;
1% = 2.60
These statistical tests therefore suggest that in S.A. the co-movement of T-Bills with the
share market is potentially a source of risk for the investor as a consequence of the need
to reinvest over the life of the project. For an investor who purchases a long-term TBond, the statistically significant co-movement of this asset with the market will only be
of importance if the maturity of the security does not match the investment horizon and
the security therefore has to be liquidated prior to maturity, or rolled over.
It is interesting to note that the correlations observed between government securities and
the share market in the U.S. and S.A. are opposite in sign; that is, in S.A., both shares and
government securities move in the same direction over time, whilst in the US, the two
move in opposite directions. The reversal of this simple relationship in S.A. highlights the
dissimilarity between the two markets and in so doing, confirms that practices appropriate
in the U.S. are not necessarily appropriate in S.A. because of the differences between the
countries.
22
3.5
Default Risk
As discussed in Section 2.2.6, theoretically, a risk-free asset should be free of default risk.
Generally it is assumed that an asset backed by the government meets this requirement
(Damodaran, 2001: 3) but there is some evidence that not even U.S. T-Bills are entirely
devoid of default risk (Nippani et al, 2001; Zivney and Marcus, 1989). In the case of
emerging markets a sizable default risk premium exists (Robinson; 2007: 5); the S.A.
default risk premium (as measured by the sovereign spread between S.A. and U.S.
treasury instrument yields) has represented a significant portion of domestic long-term
interest rates (Nowak and Ricci; 2005: 212). As can be seen in Figure 10 the sovereign
risk premium has steadily decreased over time, although in Figure 11 it increased, from a
low of 70 basis points in April 2007 to 285 basis points in March 2008 in response to
declining yields in the U.S. and a ratings downgrade for S.A. from positive to stable
(SARB, 2008: 60-61).
Figure 10: A Comparison of U.S. Dollar Denominated Bond Yields
(Source: SARB, 2007: 36)
23
Figure 11: South Africa’s Sovereign Risk Premium
(Source: SARB, 2008: 61)
To compare the sensitivity of T-Bills and T-Bonds to sovereign risk a simple comparison
was conducted for the period 1998-2008. The short-term sovereign risk premium was
estimated by subtracting the real return on U.S. T-Bills from the real return on S.A. T-Bills,
while the long-term premium was estimated by calculating the difference between the real
return on U.S. Treasury Bonds and the R157 bond. Figure 12 shows the result.
Figure 12: Sovereign Spread for T-Bills and T-Bonds 1998-2008
.12
.08
.04
.00
-.04
-.08
98
99
00
01
02
TBILL_SPREAD
03
04
05
06
07
08
TBOND_SPREAD
It can be seen that the sovereign spreads on short-term T-Bills and long-term T-Bonds have
tended to move together although short-run deviations do occur. It is also noticeable that in
24
2008 the long-term spread fell substantially below that of the short-term spread. A standard
z-test was conducted in order to test whether or not there was a significant difference in the
means of these two spreads. Using a null hypothesis that the mean T-Bill spread is equal to
the mean T-Bond spread a z-statistic of 2.1208 was calculated (see Wegner, 1993: 230).
This is less than the critical value at the 1 percent level of 2.58, but greater than the critical
value at the 5 percent level of 1.96. Therefore, at the 5 percent level the T-Bill spread is
statistically significantly larger than the T-Bond spread; hence S.A. T-Bills appear to
incorporate a more substantial sovereign risk premium compared to longer-term
instruments. This finding is consistent with the maturity crisis hypothesis developed by
Johnson (1967), in which he argued that short-term securities may be exposed to greater
default risk than longer-term securities because investors believe that the Treasury’s
inability to pay may be resolved in the future.
At face value the decrease in the sovereign risk premium observed in Figures 10 and 11 is
favourable in terms of the suitability of government securities as a proxy for the risk-free
asset. In addition, if the S.A. investment environment is inherently riskier than a market
such as the U.S. then the inclusion of an appropriate risk premium in the minimum
required rate of return for S.A. investments is arguably appropriate. However, the
fluctuations in the premium are problematic as they imply that the appropriate return on
investments in S.A. is subject to change. Less reliance can therefore be placed on CAPM
valuations based on a proxy subject to such fluctuations.
3.6
Liquidity Risk
U.S. treasury securities are, in general, highly liquid instruments because of their
perceived risk-free nature and are thus easily brought and sold in the secondary market at
25
close to their true values (Reilly and Brown, 2006: 22). Short-term S.A. government
securities are considered to be liquid instruments, but longer-term T-Bonds have only
become reasonably liquid in the past few years, as the domestic bond market has
expanded; however, it is now considered to be one of the most liquid emerging bond
markets in the world (Botha, 2006: 303). Thus, with respect to liquidity risk, S.A.
securities on average do not appear to deviate substantially from the theoretical
requirement. The issue with respect to liquidity risk therefore is whether or not longerterm securities include a liquidity risk premium and if so, whether this premium is at odds
with the theoretical criterion.
The liquidity preference hypothesis holds that investors require a premium to induce
them to invest in volatile long-term bonds (Reilly and Brown; 2006: 637). Amira (2004:
809) demonstrates empirically using sovereign Eurobonds that longer maturity fixed-rate
bonds display greater price volatility and that as a result there is a positive relationship
between maturity and yield spread. It therefore follows that longer-term government
bonds will include a liquidity risk premium in their yields in contravention of the
requirements for a risk-free asset. However, given that investors using the CAPM to
make long-term capital investment decisions are likely to require compensation for the
liquidity risk of tying up their funds for an extended period of time it is arguable that
incorporating an appropriate liquidity risk premium in the minimum required rate of
return is justifiable.
A further implication, however, is that it would then become
essential to match the maturity of the risk-free asset with that of the proposed investment
horizon in order to ensure the use of a suitable liquidity premium.
26
4. SUMMARY AND CONCLUSIONS
The CAPM remains the pre-eminent tool for estimating required rates of return used in
financial decision making and yet it is reliant on the use of proxies for most of its key
inputs. This paper has sought to fill a gap in the S.A. literature by examining in detail one
of these key variables, namely the risk-free rate. A review of academic literature, and
surveys of industry practises was performed and it was found that a wide range of
securities are used as a proxy for the risk-free asset.
The theoretical requirements for the risk-free asset were examined and the following
were identified: zero volatility; zero interest rate risk; zero covariance with the market
portfolio; zero default risk; zero exposure to inflation risk; and zero liquidity risk. In
principle then, the appropriate risk-free rate of return should represent the pure interest
rate plus a premium for expected inflation. The most preferred proxies for the risk-free
rate asset in S.A., T-Bill and T-Bond rates were then analysed and compared according to
these criteria.
The likelihood of inflation risk was found to be significantly greater for S.A. T-Bills and
T-Bonds than for corresponding U.S. assets. In addition, the longer the maturity of a
fixed interest security the greater its exposure to changes in inflation making T-Bonds
more sensitive to inflation risk.
With regard to volatility, U.S. T-Bills, but not T-Bonds were found to conform to the
theoretical requirement, while both S.A. T-Bills and T-Bonds exhibited significant
fluctuations over the period 1996-2008. In addition, a GARCH-M analysis was
27
performed and S.A. T-Bills were found to include a volatility premium in their returns
whilst the evidence was unclear with regards to the T-Bond.
The returns on U.S. T-Bills and Bonds were found to not be correlated with the market
index with the T-Bond showing lower correlation than the T-Bill. Returns for S.A. TBills and T-Bonds in contrast were found to be significantly correlated with the market
proxy but T-Bonds, in this instance, displayed greater correlation than T-Bills.
U.S. securities are essentially free of default risk but a significant sovereign risk premium
is evident in the yields on S.A. government securities, particularly short-term securities.
S.A. T-Bills and T-Bonds are thus found to violate the requirement of zero default risk.
However, if this premium represents compensation for structural risk associated with
investing in S.A. then it can be argued that including such a premium in the minimum
required rate of return for evaluating S.A. assets is not inappropriate.
Finally, both theory and empirical evidence indicate that longer-term bonds are subject to
greater liquidity risk. However, it was suggested that if a long-term investment involves
liquidity risk then using a proxy for the risk-free rate that incorporates a liquidity
premium may be reasonable providing the maturity of the proxy matches the maturity of
the investment horizon.
In light of the greater volatility of S.A. T-Bill returns, their greater correlation with the
market proxy, the presence of a volatility premium in the pricing of T-Bill returns, and
the more significant default premium its suitability as a risk-free proxy must be
questioned. While the use of a longer-term T-Bond does introduce issues concerning
28
inflation and liquidity risk it can be argued that their presence is not inappropriate in the
assessment of long-term investments that will equally be subject to inflation and liquidity
risks. This argument would, however require matching the maturity of the risk-free proxy
with that of the investment horizon to ensure that appropriate premia are applied. It
therefore appears that using a long-term government security with a maturity equivalent
to the investment horizon as the risk-free rate proxy is preferable to the use of a shortterm T-Bill.
These findings suggest areas for further research on the interaction between the choice of
risk-free rate proxy and the forecasting ability of the CAPM. Specifically examining the
interaction between the presence of volatility, liquidity and inflation in risk-free proxies
and CAPM forecasts would be appropriate. The shortcomings of T-Bills and T-Bonds as
risk-free rate proxies identified in this study also point to the need to investigate the
suitability of alternative approaches such as the use of inflation-linked government bonds.
29
5.
REFERENCES
Akonjolirie, A. and Smit, E (2003). South African unit trust performance and strategy in a
changing climate (1989 – 2002), Investment Analysts Journal, 58: 41-50.
Amira , K. (2004). Determinants of sovereign Eurobonds yield spread, Journal of Business
Finance & Accounting, 31(5) & (6): 795-821.
Banz, R. (1981). The relationship between return and market value of common stocks,
Journal of Financial Economics, 9(1): 3-18.
Black, F., Jensen, M. and Scholes, M. (1972). The capital asset pricing model: some
empirical tests, in Studies in the Theory of Capital Markets, edited by Jensen, M. New
York: Praeger.
Blake, (2000) Financial Market Analysis, Second Edition. United States of America: Wiley.
Bodie, Z., Kane, A. and Marcus, A. (2002). Investments, Fifth Edition. New York: McGrawHill Irwin.
Bodie, Z., Kane, A. and Marcus, A. (2003) Essentials of Investments, International Edition,
New York: McGraw-Hill Irwin
Bollerslev, T., Chou, T. and Kroner, K. (1992). Arch modelling in finance: A review of the
literature, Review of Econometrics, 52: 5-59.
Botha, Z. in Van Zyl, C.; Botha, Z. and Skerrit, P. (Eds.) (2006). Understanding South
African Financial Markets, Second Edition. Pretoria: Van Schaik.
Brigham, E. (1985). Financial Management: Theory and Practice, Fourth Edition. New
York: Dryden Press.
Brigham, E. and Ehrhardt, M. (2005). Financial Management: Theory and Practice, Twelfth
Edition. United States of America: Thomson South-Western.
Brooks, C. (2008). Introductory Econometrics for Finance, Second Edition. United
Kingdom: Cambridge University Press.
Bruner, R., Eades, K., Harris, R. and Higgins, R. (1998). Best practices in estimating the cost
of capital: survey and synthesis, Financial Practice and Education, (8): 13-28.
Carleton, W., and Lakonishok, J. (1985). Risk and return on equity: The use and misuse of
historical estimates, Financial Analysts Journal, 41(1): 38-47.
Cloete, G., de Jonah, P. and de Wet, T. (2002). Combining Vasicek and robust estimators for
forecasting systematic risk, Investment Analysts Journal, 55(3)
Copeland, T., Koller, T. and Murrin, J. (2000). Valuation – Measuring and Managing the
Value of Companies, Third Edition. New York: Wiley.
Cornell, B., Hirschleifer, J., and James, E. (1997). Estimating the cost of equity,
Contemporary Finance Digest, 5-26.
Correia, C. and Ulius, E. (2004). Market segmentation and the cost of equity of companies
listed on the Johannesburg Stock Exchange, South African Journal of Accounting
Research, 18(1): 65-81.
Correia, C. and Cramer, P. (2008). An analysis of cost of capital, capital structure and capital
budgeting practices: a survey of South African listed companies, Meditari: Research
Journal of the School of Accounting Science, 16(2): 31-52.
Damodaran, A. (2001). Estimating Risk-Free Rates, Stern School of Business.
de Wet, J. (2005). EVA versus traditional accounting measures of performance as drivers of
shareholders value – a comparative analysis, Meditari: Research Journal of the School of
Accounting Sciences, 13(2): 1-16.
de Wet, J. (2006). Determining the optimal capital structure: a practical contemporary
approach, Meditari: Research Journal of the School of Accounting Sciences, 14(2): 1-16.
30
de Wet, J. and du Toit, E. (2007). Return on equity: A popular, but flawed measure of
corporate financial performance, South African Journal of Business Management, 38(1):
59-69.
de Wet, J. and Hall, J. (2006). An analysis of strategic performance measures of companies
listed on the JSE securities exchange South Africa, South African Journal of Economic and
Management Sciences, 9(1): 57-71.
Engle, R., Lillian, D. and Robbins, R. (1987). Estimating time-varying risk premia in the
term structure: The ARCH-M model, Econometrica, 55: 391-407.
Fama, E. (1975). Short-term interest rates as predictors of inflation, American Economic
Review, 65(3): 269-282.
Fama, E. and French, K. (1992). The cross section of expected stock returns, Journal of
Finance, 52(1): 427-465.
Fama, E. and French, K. (2004). The capital asset pricing model: theory and evidence,
Journal of Economic Perspectives, 18(3): 25-46.
Fama, E. and MacBeth, J. (1973). Risk, return, and equilibrium: empirical tests, Journal of
Political Economy, 81(3): 607-636.
Firer, C. (1993). Estimating the Return Parameters of the Capital Asset Pricing Model, South
African Journal of Accounting Research (De Ratione), 7(1): 23-39.
Firer, C. and McLeod, H. (1999). Equities, bonds, cash and inflation: Historical Performance
in South Africa 1925 to 1998, Investment Analysts Journal, 50: 7-28.
Firer, C., Ross, S., Westerfield, R. and Jordan, B. (2004). Fundamentals of Corporate
Finance, Third South African Edition. Bershire: McGraw-Hill.
Freund, J. and Perles, B. (2003) Statistics: A First Course, Eighth Edition. United States of
America: Prentice-Hall.
Friis, L. and Smit, E. (2004). Are some fund managers better than others? Manager
characteristics and fund performance, South African Journal of Business Management,
35(3): 31-40.
Graham, J.R. and Harvey, C.R. (2001). The theory and practice of corporate finance:
evidence from the field, Journal of Financial Economics, 60: 187-243.
Graham, M. and Uliana, E. (2001). Evidence of a value-growth phenomenon on the
Johannesburg Stock Exchange, Investment Analysts Journal, 53(1)
Gitman, L.J. and Mercurio, V. (1982). Cost of capital techniques used by major U.S. firms:
survey and analysis of Fortune’s 1000, Financial Management, 14: 21-29.
Harrington, D. (1987). Modern Portfolio Theory, the Capital Asset Pricing Model, and
Arbitrage Pricing Theory: A User’s Guide, Second Edition. New Jersey: Prentice Hall.
Ibbotson, R., and Sinquefield, R. (1979). Stocks, bonds, bills and inflation: Updates,
Financial Analysts Journal, 35(4): 40-44.
Johnson, R. (1967). Term structure of corporate bond yields as a function of risk of default,
Journal of Finance, 22 (2): 313-351.
Keller, G. and Warrack, B. (2000). Statistics for Management and Economics, Fifth Edition.
United States of America: Duxbury Thomson Learning.
Malkiel, B. (1995). Returns from investing in equity mutual funds 1971-1991, Journal of
Finance, 50(2): 549-572.
Moolman, E. and du Toit, C. (2005). An econometric model of the South African stock
market, South African Journal of Economic and Management Sciences, 8(1): 77-91.
Msweli-Mbanga, P. and Mkhize, H. (2007). The risk-adjusted performance of companies
with female directors: a South African case, South African Journal of Economic and
Management Sciences, 10(2): 207-213.
Nippani, S., Liu, P. and Schulman, C. (2001). Are Treasury securities free of default?
Journal of Financial and Quantitative Analysis, 36 (2): 251-265.
31
Nowak, M. and Ricci, L. (2005). Post-Apartheid South Africa: The First Ten Years,
International Monetary Fund, Washington, D.C..
Oldham, G. and Kroeger, J. (2005). Performance, persistence and benchmarks of selected
South African unit trusts for the period 1998-2002, South African Journal of Business
Management, 36(4): 81-90.
Pike, R. and Neale, B. (2006). Corporate Finance and Investment, Fifth Edition. Essex:
Prentice Hall.
Pocock, A.S., Correia, C. and Wormald, M. (1991). An analysis of the approaches used by
industrial companies listed on the JSE to identify their cost of capital, South African
Journal of Accounting Research (De Ratione), 5(1): 27-40.
Rees, B. (1995). Financial Analysis, Second Edition. Hertfordshire: Prentice-Hall.
Reilly, F. and Brown, K. (2006). Investment Analysis and Portfolio Management, Eighth
Edition. United States of America: Thomson South-Western.
Robinson, Z. (2007). The sovereign risk premium in South Africa: An empirical approach.
[Online]. Available at: https://editorialexpress.com/cgibin/conference/download.cgi?db_name=IIPF63&paper_id=187. Accessed: 10th April 2009.
Roll, R. (1977). A critique of the asset pricing theory’s tests, Journal of Financial
Economics, 4 (4): 129-176.
Samouilhan, N. (2007). The price of risk in the South African equity market, South African
Journal of Economics, 75(3): 442-458.
Sharpe, W. (1964). Capital asset prices: A theory of market equilibrium under conditions of
risk, Journal of Finance, 19(3): 425-442.
South African Reserve Bank (SARB) (2007). Quarterly bulletin: March. Pretoria: SARB.
South African Reserve Bank (SARB) (2008). Annual Report 2007/2008: March. Pretoria:
SARB.
Stambaugh, R. (1982). On the exclusion of assets from tests of the two-parameter model,
Journal of Financial Economics, 10: 237-268.
Van Rensburg, P. and Slaney, K. (1997). Market Segmentation on the Johannesburg Stock
Exchange, Journal of Studies in Economics and Econometrics, 21(3): 1-23.
Viviers, S., Bosch, J., Smit, E. and Buijs, A. (2008). The risk-adjusted performance of
responsible investment funds in South Africa, Investment Analysts Journal, 68: 39-55.
Wegner, T. (1993) Applied Business Statistics: Methods and Applications. Cape Town: Juta
& Co Ltd.
Zivney, T. and Marcus, R. (1989). The day the United States defaulted on Treasury Bills,
Financial Review, 24(3): 475-489.
32