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5
Performance measurement
Herve´ Bourquin and Roman Marton
1. Introduction
Performance analysis can be considered the final stage in the portfolio
management process as it provides an overall evaluation of the success of
the investment management in reaching its expected performance objective.
Furthermore, it identifies the individual contributions of each of its components and underlying strategies to the overall performance result. The
term ‘performance analysis’ covers all the techniques that are implemented
to study the financial results obtained in the portfolio management process,
ranging from simple performance measurement to performance attribution.
This chapter deals with performance measurement, which in turn can be
loosely defined as the analytical framework underlying the calculation and
assessment of investment returns. Chapter 6 introduces performance attribution as the second leg of a performance analysis.
Where Markowitz (1952) is often considered to be the founder of modern
portfolio theory (the analysis of rational portfolio choices based on the
efficient use of risk), Dietz (1966) may be seen as the father of investment
performance analysis. The theoretical foundations of performance analysis
can be traced back to classic economic equilibrium and corporate finance
theory. Over the years, numerous new concepts that describe the interdependencies between return (ex ante and ex post) and risk measures by the
application of specific factor models have been incorporated into the
evaluation of investment performance (e.g. Treynor 1965; Sharpe 1966;
Jensen 1968). Most of these models can be implemented directly into the
evaluation framework, whereby the choice of a method should match the
investment style of the portfolio management.
A critical component of any performance analysis framework is given by
the definition of a benchmark portfolio. A benchmark portfolio is a reference portfolio that the portfolio manager will try to outperform by taking
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Bourquin, H. and Marton, R.
‘active’ positions against it. These active positions are the expression of an
investment strategy of the portfolio managers, who – depending on their
expectations of changes in market prices and on their risk aversion – decides
to deviate from the risk factor exposures of the benchmark. In contrast,
purely passive strategies mean that portfolio managers simply aim at replicating the chosen benchmark, focusing e.g. on transaction cost issues.
Central banks usually run a rather passive management of their investment
portfolios, although still some elements of active portfolio management
are adopted and out-performance versus benchmarks is sought without
exposing the portfolio to a significantly higher market risk than that of the
benchmark (sometimes called ‘semi-passive portfolio management’). Passive investment strategies may be viewed as being directly derived from
equilibrium concepts and the Capital Asset Pricing Model (which is
described in Section 3.1). The practical applications of this investment
approach in the world of performance analysis are covered in Section 2 of
this chapter.
While the literature on the key concepts of performance measurement is
wide, such as e.g. Spaulding (1997), Wittrock (2000), or Feibel (2003), it
does not concentrate on the specific requirements of a public investor, like
a central bank, which typically conducts semi-passive management of fixedincome portfolios with limited spread and credit risk. The first part of this
chapter presents general techniques to properly determine investment
returns in practice, also with respect to leveraged instruments. The subsequent section then focuses on appropriate risk-adjusted performance
measures, also extending them to asymmetric financial products which are
in some central banks part of the eligible instrument set. The concluding
Section 4 presents the way of performance measurement at the ECB.
2. Rules for return calculation
The ‘global investment performance standards’ (GIPS) are a set of recommendations and requirements used to evaluate investment management
practice. It allows the comparison of investment performance internationally and provides a ‘best practice’ standard as regards transparency for the
recipients of performance reports. The GIPS were developed by the
‘Investment Performance Council’ and were adopted by the ‘CFA Institute
Board of Governors’ – the latest version is of 2005 (see Chartered Financial
Analyst Institute 2006).
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Performance measurement
2.1 Basic formulae
It is a requirement of the GIPS that the calculation of the single-period
return must be done using the time-weighted rate of return (TWRR) method
in order to neutralize the effect of cash flows (see e.g. Cotterill 1996 for
details on the TWWR). The term ‘time-weighted rate of return’ was chosen
to illustrate a measure of the compound rate of growth in a portfolio. Both
at instrument level and aggregate level (e.g. portfolio level) the TWRR based
on discrete points in time Dt ¼ [t – 1; t] is determined as follows:
TWRRdisc; Dt ¼
¼
MVt À CFin;t þ CFout;t À MVtÀ1
MVtÀ1
MVt À CFin;t þ CFout;t
À1
MVtÀ1
ð5:1Þ
where TWRRdisc,Dt is the time-weighted return in period Dt; MVt À 1 is the
market value (including accrued interest) at the end of time t À 1; MVt is
the market value (including accrued interest) at the end of time t; CFin,t and
CFout,t are the cash inflows and outflows during period Dt.
The cash flows adjustment is done both at instrument and portfolio level.
At instrument level, CFin,t and CFout,t occur within the portfolio and are
induced by trades (e.g. bond buy or sale) or by holdings (e.g. coupon or
maturity payments). At portfolio level CFin,t represent flows into the
portfolio and CFout,t are flows out of the portfolio. As this method eliminates the distorting effects created by exogenous cash flows,1 it is used to
compare the returns of investment managers.
The TWRR formula (5.1) assumes that the cash flows occur at the end of
time t. Following a different approach, the cash flow occurrence can be
treated as per beginning of t.2 The alternative TWRR*disc,Dt is then
TWRR Ãdisc;Dt ¼
1
MVt À MVtÀ1
MVtÀ1 þ CFin;t À CFout;t
ð5:2Þ
The following example illustrates the neutralization of the cash flows by the TWRR. Assume a market value of 100 on
both days t–1 and t; therefore, the return should be zero at the end of day t. If a negative cash flow of À10 occurs on
day t, the market value will be 100 – 10 ¼ 90 and the corresponding TWRR will be
MV end
100 À 10 þ 10
100
À1¼
À 1 ¼ 0:
À1¼
100
100
MV begin
2
In practice, the end-of-period rule is more often used than the start-of-period approach. A compromise would be
weighting each cash flow at a specific point during the period Dt as the original and the modified Dietz methods do –
see Dietz (1966).
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Bourquin, H. and Marton, R.
If the return calculation is done based on specific (finite) points in time (e.g.
on a daily basis) then the resulting returns are called discrete-time returns –
as shown in equations (5.1) and (5.2). For ex post performance measurement, in practice discrete-time returns are the appropriate instrument
to determine the growth of value from one relevant time unit (e.g. day t À 1)
to another (e.g. day t). A more theoretical method is the concept of continuously compounded returns (also called continuous-time returns).
Discrete-time returns are converted into their continuous-time equivalents
TWRRcont,Dt by applying equation (5.3):
À
Á
ð5:3Þ
TWRR cont;Dt ¼ ln 1 þ TWRR disc;Dt
As its name already indicates, the continuously compounded return is the
product of the geometrically linked (factorized) returns of every infinitesimal small time unit within the analysis period. The main disadvantage of
this approach is that underlying market values are not intuitively understandable and, hence, it is rather difficult to explain those performance
results to a broad audience, since they assume that the rate of growth of a
portfolio can be compounded continuously – over every infinitesimal (and
therefore theoretical) time interval. Practitioners can better size the sense of
a return when it captures the actual evolution of the market value and cash
flows at end versus the start of the observation day. Even if logarithmic
returns possess some convenient mathematical attributes (e.g. linkage over
time can be processed additively in contrast to discrete returns for which the
compounding must be done multiplicatively), discrete-time time-weighted
returns are usually favoured by practitioners. The returns in this chapter will
thus from now on represent discrete-time time-weighted returns.
Once one has generated the TWRR for every given instrument for each
given single period, the return on a portfolio for the total period is calculated
in two steps. First, for the specified discrete time unit Dt (e.g. the course of
a day), the returns on the different instruments that compose a portfolio are
arithmetically aggregated by the respective market value weights, i.e.
RP;Dt ¼
N
X
ðRP;i;Dt · wP;i;tÀ1 Þ
ð5:4Þ
i¼1
where RP,Dt is the portfolio return and RP,i,Dt is the return on the i-th
component of portfolio P in period Dt, respectively; wP,i,t–1 is the market
value weight of the i-th component of portfolio P as of day t – 1; and N is
the number of components within portfolio P.
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Performance measurement
In a second step, the portfolio return is quantified for the whole period
observed. The return RP,DT for the entire period DT is obtained by geometrically linking the interim (i.e. single-period) returns (this linkage
method is also a requirement from the GIPS):
Y
RP;DT ¼
ð1 þ RP;Dt Þ À 1
ð5:5Þ
8Dt2DT
So far, the case of the determination of return figures in a single-currency
world has been presented. But the foreign reserves portfolios of central
banks consist by definition of assets denominated in multiple currencies.
The conversion of currency-local returns into returns in base currency is
given by
RBase;Dt ¼ ð1 þ Rlocal;Dt Þ · ð1 þ RxchÀrate;Dt Þ À 1
ð5:6Þ
where for period Dt: RBase,Dt is the total return in base currency (e.g. EUR);
Rlocal,Dt is the total return in local currency; and Rxch-rate,Dt is the change of
the exchange rate of the base currency versus the local currency. As it can be
seen in Section 4.1 of Chapter 6 on performance attribution, this multiplicative relationship leads to intra-temporal interaction effects in additive
attribution models.
2.2 Trade-date versus value-date approach
The general requirement for a sound performance measurement is that the
mark-to-market values are applied, i.e. all future cash flow must be properly
discounted. For example, when a bond is bought, the market value of the
position related to this transaction at trade date is the net of the market
value of the bond at trade date and the discounted value of the settlement
amount. The general description of this approach to performance measurement is commonly referred to as the trade-date approach. It is often
compared with the so-called value-date approach, where in the case of a
purchase of a bond, this position would only appear in the market value of
the portfolio at value (i.e. settlement) date. Accordingly, the price of the
bond would not influence the market value of the portfolio in the period
between trade and value date. This is not satisfactory since market movements in the position from trade date (when the actual investment decision
is taken) are not taken into account as opposed to the trade-date approach
and as all modern standards for performance measurement recommend.
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Bourquin, H. and Marton, R.
The trade-date approach has actually three main advantages. Firstly, portfolio injections and divestments impact properly the portfolio market value
at trade date. Secondly, from trade date on it calculates correctly the return
on an instrument consecutive to its purchase or sale. Finally, payments (e.g.
credit interest, fees, etc.) are properly reflected in the performance figures
as of trading date.
2.3 Actual versus all-cash basis
In principle there are two ways for a portfolio to outperform a corresponding benchmark. First, by selecting bonds, i.e. by being over- (under-)
exposed relative to the benchmark in those bonds with a better (worse)
performance than the benchmark. This covers yield-curve and duration
position taking as well as pure bond selection. Second, by using leveraged
instruments, i.e. by using instrument with a different payoff structure than
the normal spot bonds. Leveraged instruments include forwards, futures,
options etc. In order to separate the two out-performance factors, the GIPS
require that the performance be measured both on actual basis and all-cash
basis. These concepts can be defined as follows. Actual basis measures the
growth of the actual invested capital; i.e. it is a combination of both fixedincome instrument picking and used leverage. This is the conventional
method of measuring (active) returns by looking at the growth in value of
the funds invested. All-cash basis tries to eliminate the effect of the used
leverage by restating the position into an equivalent spot position having the
same market exposure. The return is then stated under the following form:
(MVend – Interestmargin) / MVstart where Interestmargin corresponds to the
daily margin.3 This removes the effect of the leverage on the return. The allcash basis (active) return is consequently the (active) return measured on
the restated cash equivalent positions. The comparison of the actual and allcash basis returns allows calculating the return at the level of the leveraged
instruments or leveraged instruments types (e.g. daily return for a given
bond future or for all bond futures included in a portfolio).
3
After entering a futures contract the investor will have a contract with the clearer, while the clearer will have a
contract with the clearing house. The clearer requires the investor to deposit funds (known as initial margin) in a
margin account. Each day the futures contract is marked-to-market and the margin account is adjusted to reflect the
investor’s gain or loss. This adjustment corresponds to a daily margin that is noted Interestmargin in the formula of the
text above. At the close of trading, the exchange on which the futures contract trades, establishes a settlement price.
This settlement price is used to compute the gains or losses on the futures contract for that day.
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Performance measurement
When managers use leverage in their portfolio, then the GIPS require that
the returns be calculated on both the actual and the all-cash basis. Since the
benchmarks of most central banks portfolios are normally un-leveraged, the
comparison between benchmark and all-cash basis returns shows the
instrument selection ability of the fund manager, whereas the difference
between the actual and the all-cash basis returns indicates how efficient the
use of leverage in the fund management was, i.e. MVend / MVstart – (MVend –
Interestmargin) / MVstart ¼ Interestmargin / MVstart.
3. Two-dimensional analysis: risk-adjusted performance measures
3.1 Capital Asset Pricing Model as a basis
Investors have a given level of risk aversion, and the expected return on their
investment depends on the level of risk they are ready to bear. Therefore,
considering the return dimension in performance measurement would only
be half of the truth. The insertion of the risk dimension into the performance analysis has been formalized by the Capital Asset Pricing Model
(CAPM) and its diverse modifications and applications (Treynor 1962;
Sharpe 1964; Lintner 1965; Mossin 1966). The capital market line which
represents the ray connecting the profiles of the risk-free asset and the
market portfolio M in a risk–return diagram is given by
RM À RF
ð5:7Þ
RP ¼ RF þ
Á rðRP Þ
rðRM Þ
where RF is the risk-free rate; RP is the return on investment portfolio P; RM
is the return on market portfolio M; r(RP) is the standard deviation of
historical returns on investment portfolio P; and r(RM) is the standard
deviation of historical returns on market portfolio M.
This relationship implies that in equilibrium the rate of return on every
asset is equal to the rate of return on the risk-free asset plus a risk premium.
The premium is equal to the price of the risk multiplied by the quantity of
risk, where the price of risk is the difference between the return on the
market portfolio and the return on the risk-free asset. The systematic risk,
i.e. the beta, is defined by
bP ¼
rðRP ; RM Þ
r2 ðRM Þ
ð5:8Þ
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Bourquin, H. and Marton, R.
where bP is the beta of investment portfolio P with respect to market
portfolio M; r(RP,RM) is the covariance between the historical returns on
investment portfolio P and market portfolio M; and r2(RM) is the variance
of historical returns on market portfolio M.
By using the beta expression, the CAPM relationship can be written as
follows:
RP ¼ RF þ b P Á ðRM À RF Þ
ð5:9Þ
In the following paragraphs of this section two groups of selected risk-adjusted
performance ratios are presented. The first one is applied to the absolute
return, comprising the Sharpe and Treynor ratios, while the second group
consists of extended versions, i.e. reward-to-VaR and information ratios, that
are focusing on relative return (i.e. performance). All of these measures are
considered to be relevant for the return and performance analysis of a central
bank’s portfolios. The second group identifies the risk-adjusted performance
due to the portfolio management relative to a benchmark – this allows us to
determine how successful the management activity (passive, semi-passive or
active) has been in the performance generation process.
3.2 Total performance: Sharpe ratio
The Sharpe ratio SRP of portfolio P (or reward-to-variability ratio as it was
originally named by Sharpe) is defined (in its ex post version) as follows (see
Sharpe 1966):
SRP ¼
RP À RF
rðRP Þ
ð5:10Þ
Comparing with formula (5.7) reveals the intuition behind this measure: if
the ratio of the excess return and the total risk of a portfolio lies above
(beneath) the capital market line, it will represent a positive (negative) riskadjusted performance versus the market portfolio. Since central banks are
naturally risk averse and manage their portfolios in a conservative manner
by taking limited active leeway against the benchmark, the core of the return
is generated by the benchmark, while the performance of the managed
portfolio against its benchmark represents a small fraction of the overall
return. An appropriate performance/risk ratio could therefore provide
information regarding the ‘efficiency’ of the reference benchmark. The
major problem by using the Sharpe ratio as a performance measure in
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Performance measurement
a central bank is that its interpretation can be difficult. To use it for an
assessment, one would ideally have to compare it with the Sharpe ratio of a
market index having exactly the same characteristics in terms of credit risk,
market risk and liquidity profile as the benchmark. However, by doing so,
one would obtain a reference Sharpe ratio that should be almost identical
to the one of the measured portfolio. For that reason, the ECB calculates
the Sharpe ratio of its benchmarks, but considers this ratio more as an
indicative ex post measure rather than an efficient performance indicator.
There is an alternative pragmatic approximation which allows making
use of the Sharpe ratio under these circumstances. Assuming that the
market index, which should be used for the comparison, took the same total
risk as the benchmark, the implied return for the market index could be
calculated via the capital market line of the market index. When comparing
the returns on the portfolio and the hypothetical market index (with the
same risk exposure as the benchmark) the risk-adjusted out-/underperformance, the portfolio alpha aP, can be determined:
aP ¼ RP À ðRF þ SR Index Á rðRB ÞÞ ¼ SRP Á rðRP Þ À SR Index Á rðRB Þ
ð5:11Þ
where SRIndex is the Sharpe ratio of the market index (any representative
market index in terms of asset classes and weightings) and r(RB) is the
standard deviation of historical returns on benchmark B.
To be able to rank different portfolios with different risk levels (i.e. to
compare the risk-adjusted out- or underperformances), it is in addition
necessary to normalize the alphas, i.e. to set them to the same total risk unit
level by dividing by the corresponding portfolio standard deviation. The
resulting performance measure is called the normalized portfolio alpha
anorm,P (see Akeda 2003):4
a norm;P ¼
aP
rP
ð5:12Þ
3.3 Passive performance: Treynor ratio
The ex post Treynor ratio TRP of portfolio P is given by (see Treynor 1965)
TRP ¼
4
RP À RF
bP
See also Treynor and Black (1973) for adjusting the Jensen alpha by the beta factor.
ð5:13Þ
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Bourquin, H. and Marton, R.
This performance indicator measures the relationship of the portfolio
return in excess to the risk-free rate and the systematic risk – the beta – of
the portfolio. The ratio can be directly derived from the CAPM (with the
benchmark portfolio B replacing the market portfolio M):
RP À RF
¼ RB À RF
bP
ð5:14Þ
The left-hand-side term is the Treynor ratio of portfolio P and the expression
on the right-hand side can be seen as the Treynor ratio for the benchmark B,
because the beta against the benchmark itself is one. The Treynor ratio is a
ranking measure (in analogy to the Sharpe ratio). Therefore, for a similar
level of risk (e.g. if two portfolios replicate exactly the benchmark and thus
are managed passively against that benchmark) the portfolio that has the
higher Treynor ratio is also the one that generates the highest return of
the two.
For the purpose of measuring and ranking risk-adjusted performances of
well-diversified portfolios, the Treynor ratio would be a better measure than
the Sharpe ratio, because it only takes into account the systematic risk which
cannot be eliminated by diversification. For its calculation, a reference
benchmark must be chosen upon which the beta factor can be determined.
In case of skewed return distributions (that are mainly the case for low
modified duration portfolios) a distorted beta and Treynor ratio can occur
(see e.g. Bookstaber and Clarke 1984 on incorrect performance indicators
based on skewed distributions). The majority of the central bank currency
reserves portfolios and their representative benchmarks normally do not
consist of instruments with embedded optionalities (i.e. uncertain future
cash flows), and so the empirical return distributions should not deviate in
a significant manner from the normal distribution in terms of skewness and
curvature.
3.4 Extension to Value-at-Risk: reward-to-VaR ratio
The quantile-based VaR has evolved rapidly to one of the most popular and
widespread tools in financial risk measurement (see e.g. Jorion 2006 and
Holton 2003),5 not at least because of its enshrinement in capital adequacy
5
See also Marton, R. 1997, ‘Value at Risk – Risikomanagement gema¨ß der Basler Eigenkapitalvereinbarung zur
Einbeziehung der Marktrisiken’, unpublished diploma thesis, University of Vienna.
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Performance measurement
rules. If the VaR concept is used for risk control, it could also be incorporated into the risk-adjusted performance analysis. This could be realized
by applying the reward-to-VaR ratio proposed by Alexander and Baptista
(2003), which is based on the Sharpe ratio (i.e. reward-to-variability ratio).
The reward-to-VaR ratio measures the impact on ex post portfolio return
of an increase by one percentage of the VaR of the portfolio, by moving a
fraction of wealth from the risk-free security to that portfolio. The calculation process depends on the assumption whether asset returns are
considered as being normally distributed or not. In the first case the rewardto-VaR ratio RVP of portfolio P is given by
RVP ¼
tÃ
SRP
À SRP
ð5:15Þ
whereof SRP is the Sharpe ratio of portfolio P and
t Ã ¼ ÀUÀ1 ð1 À aÞ
ð5:16Þ
where U–1(.) is the inverse cumulative standard normal distribution function
and (1–a) is the VaR confidence level (e.g. a ¼ 99 per cent implies t* % 2.33).
In the case of normally distributed investment returns and if t* > SRP is
true for every portfolio then the reward-to-VaR ratio and the Sharpe ratio
will yield the same risk-adjusted performance ranking.
Assume for example that the reward-to-VaR ratios of portfolios A and B
are 0.40 per cent and 0.22 per cent, respectively, and that this ratio is equal
to 0.34 per cent for the index that is used as proxy for the market portfolio.
The investor in A would have earned on average an additional 0.40 per cent
per year, bearing an additional percentage point of VaR by moving a
fraction of wealth from risk-free security to A. In this example, A outperformed the market portfolio and portfolio B, and B underperformed both A
and the market portfolio. The return on these portfolios was assumed to be
normal – had it not been the case, formula (5.15) could not have been
applied.
3.5 Active performance: information ratio
The information ratio is the most common tool to measure the success or
failure of the active investment management of a portfolio versus its
benchmark. The information ratio (sometimes also called appraisal ratio) is
defined as the quotient of the active return (alpha) on a portfolio to its