2

The basics of performance measurement

2.1 INTRODUCTION

This chapter provides a high-level overview of performance calculation for both individual securities and portfolios of securities.

Routine performance calculations are almost always carried out by specialist software packages that can handle the large volumes of data involved. However, even if you never need to calculate performance manually, an appreciation of how performance numbers work and are used is necessary to understand attribution.

Security and portfolio performance is well described elsewhere (Bacon, 2008), and any reader of this book is already likely to have a good understanding of the basics of the subject. The material is therefore presented at a relatively high level.

2.2 DEFINING RETURN

The concept of return is suggested by questions such as:

• If I had invested $100 in a particular stock, what would my investment be worth after a year?
• Would I have been better off investing in another security?
• If I had invested twice as much, what would my profit have been?

Return is defined as the rate of increase r in the value of an investment. Alternatively, return is the ratio of money gained or lost (whether realised or unrealised) on an investment, relative to the amount of money initially invested. In algebraic terms, r is given by

where P is the value of the investment at the outset, and δP its change in value.

The advantages of expressing profit as a dimensionless quantity r, rather than as a P/L (profit and loss) in currency terms, are twofold:

• Return allows different investments to be compared on a common basis. Investible portfolios typically range in size from individual securities to managed funds containing many thousands of securities from completely different markets, but both generate a return that can be measured and compared.
• The use of a return measure allows quality of investments to be compared when amounts invested are quite different. In particular, the use of return makes it possible for the portfolio to be compared to a reference portfolio (or benchmark), as well as against its peers, which would not be possible if only currency P/L were available.¹

2.3 COMPOUNDED RETURNS

Consider an investment that pays 12% annually. To calculate its rate of return when payments are reinvested, we also need to know its frequency of compounding, since the value of the amount invested will change over each compounding interval.

• If the investment is held for a year and pays a single lump sum of 12% at the end of this interval, its annual return is 12.0%.
• If interest is paid at six-monthly intervals, then the return is calculated over two intervals.
  At the end of the first six months, the value of an invested dollar investment is 1 + (1 × 0.06) = 1.06. Therefore, the starting value for the investment over the second six-month interval is 1.06, and the overall return is 6% compounded twice: r = (1 + 0.06)² − 1 = 12.36%.
• If interest is paid at one-monthly intervals, the return is 12% ÷ 12 = 1% compounded 12 times: r = (1 + 0.01)¹² − 1 = 12.68%.

In the limit where the compounding interval goes to zero, the return r is given by

where e is the constant 2.71828… In this case the return is e^0.12 − 1 = 12.75%.

Investments often have their aggregated returns calculated using continuously compounded return. The reason is that its use makes return very straightforward to calculate over multiple intervals:

In other words, the continuously compounded return of an investment over multiple intervals is simply the sum of the continuously compounded returns over each interval.²

2.4 TIME-WEIGHTED AND MONEY-WEIGHTED RETURNS

2.4.1 Time-weighted return

The most commonly used way to calculate return is based on the approach shown in (2.1). If there are no cash flows over the calculation interval, return r is given by

This expression is typically expanded to handle the effects of cash inflows or outflows over the calculation interval {t, t + 1}, as well as the effects of income from dividends or bond coupons:

or, equivalently

where

• BMV = P_t = P is the value of the investment at the start of the calculation interval;
• EMV = P_{t + 1} is the value of the investment at the end of the calculation interval;
• δP = EMV − BMV is the change in the value of the investment over the calculation interval;
• CF = CF_{t,t + 1} is the aggregated value of any cash flows occurring over the interval;
• Income = I_{t,t + 1} is any income accruing from the investment over the interval.

All these expressions are approximations to the true return, which requires complete revaluation of every portfolio asset each time a transaction occurs.

In practice it is difficult to measure the exact return of a portfolio, since prices are typically only sampled at particular fixed times, typically the start and end of each day. For transactions at other times of day, some form of approximation is required. The dollar P/L is never in question, but the rate of return can be, due to the resulting uncertainty in the portfolio’s revaluation.

These expressions represent time-weighted return, which is the industry standard for performance measurement. Time-weighted return is not sensitive to contributions or withdrawals.

Example

A bond has a price of 104.5 at the start of a month, 102.0 at the end of the month, and generates a coupon payment of $3 during the month. There are no other cash flows. Its return over the month is

2.4.2 Money-weighted return

Money-weighted return is also occasionally encountered. Money-weighted return is the discount rate for a portfolio that makes the present value of its inflows equal to its outflows. Like time-weighted return, money-weighted return can measure the return of a security, a sector or a portfolio over an arbitrary period in which cashflows occur.

This is a rather opaque definition, so consider an example. We buy one share of a stock for $100 that pays a $5 dividend six months after it was purchased and another $5 dividend 12 months after it was purchased. The share is then sold just after the second dividend was paid at a price of $95. What was its money-weighted rate of return?

Here, the outflow for the investor is $100, and the inflows are the two $5 dividends and the sale price of $95. Therefore, the money-weighted rate of return will be the rate r that satisfies the following expression:

Equations of this type are seldom analytically tractable and must be solved numerically. In this case, r = 5.137%. Compare this to the time-weighted rate of return:

Apart from its calculation overhead, the main reason money-weighted return is not widely used is that it overweights the contribution to return made during periods when holdings are highest. Since holdings in a portfolio are typically outside the manager’s control, this can distort the overall returns and give a misleading view of the manager’s skill. Intuitively, one expects any measure of return to be unaffected by the size of the investment held. This is not necessarily the case for money-weighted returns.

2.5 PORTFOLIO RETURNS

Portfolios typically hold more than one security; this could be for a number of reasons, including:

• risk diversification;
• having multiple investment strategies in play;
• matching a set of known liabilities;
• matching the cashflows of a benchmark.

Each security i in a portfolio has a market value MV_i, a weight w_i and a return r_i. The weight is the security’s proportion of the overall portfolio value:

where the sum is over the entire portfolio.

This implies that the sum of the security weights over the portfolio is always 1:

Note that weights can be negative. This most commonly occurs when a portfolio holds derivatives such as currency forwards, futures or swaps, which may have negative market values. In this case, the sum of the weights of the other securities in the portfolio will be greater than one.

Given the weight and return of each security in the portfolio, the portfolio’s overall return R is the sum-product of its security weights and returns:

Defining the performance contribution c_i of security i as c_i = w_ir_i, we can rewrite (2.9) as

In other words, the return of a portfolio over a given interval is the sum of its performance contributions over that interval. This apparently trivial observation is more useful than it might seem, as is shown in Section 2.6.1.

2.6 TRANSACTIONS AND CASH FLOWS

Typically, the composition of a portfolio changes over time. This may be due to:

• security transactions (buying or selling stocks);
• cash flows (dividends and coupons);
• revaluation (stock splits);
• internal changes within a security (sinking bond paydowns);
• changes from one security type to another (bonds maturing, bonds being called, convertible bonds turning into equity).

All these cases can be handled by the equations above.

2.6.1 Performance contribution, or weight and return?

The ‘obvious’ way to provide information about the securities that make up a portfolio is to supply their weights and returns, since the two are independent. However, many performance systems work with performance contributions and returns instead. There are two main reasons:

• Accurate intraday contributions
  A continuous view of the holdings and valuation of the managed portfolio is seldom available. Instead, the analyst must work with snapshots of the portfolio, usually at the start or end of the day, and include the effects of trades and other cash flows between these points.
  At some point a trader will buy into a stock after the market opens and sell it out before the market closes. The stock will generate a contribution to the portfolio’s overall performance, but its weight at the beginning and the end of the calculation interval will be zero. In this case, a conventional use of weights and returns will report the security’s return contribution as zero when it is not.
• Highly leveraged securities
  The other reason to work with performance contributions is that they often convey a clearer picture of the source of portfolio returns. A highly leveraged swap may have a very large return but a small exposure. In this case it may not be clear whether the swap is making any impact on the portfolio’s overall return. The swap’s performance contribution provides a much more transparent way to assess its impact on the portfolio’s return.

A disadvantage of using return contributions is that weight must now be calculated as

The sum of the w_i terms may differ slightly from 1 on days where there is significant intraday trading, but this is always preferable to reporting the wrong overall return.

2.7 SECTOR RETURNS

2.7.2 Sector weights and returns

The weight w_s and return r_s of a sector S over a single time interval are given by

2.8 CALCULATING PORTFOLIO RETURNS OVER SUCCESSIVE INTERVALS

While contribution to return in a portfolio aggregates additively, it aggregates geometrically over multiple time intervals.

For instance, suppose a stock shows a return of 10% over one month and 20% over the following month. Using (2.3), the stock’s aggregated return R over both months is given by geometric aggregation:

R = (1 + 10%) × (1 + 20%) − 1 = 32%

Conversely, suppose a portfolio holds 50% of a stock that returns 10% over a month, and the remaining 50% in another stock that returns −20%. From (2.9), the return of the portfolio R_p over the month is given by additive aggregation:

R_p = (50% × 10%) + (50% × − 20%) = −5%

The fact that returns combine differently, depending on whether the aggregation is over time or sector, can and does cause difficulties. Chapter 5 describes various ways in which performance managers address this issue.

2.9 FUTURES CASH OFFSETS

In addition to physical cash holdings, many portfolios show large cash balances due to futures offset holdings (or notional cash). Futures offsets exist because, unlike a physical security, the market exposure of a futures contract is always zero, and this must be taken into account when calculating the portfolio’s exposures and risks.³

To put this another way, a future is a derivative that promises the same dollar profit and loss of the underlying security, but without the requirement to actually hold the security in the portfolio. If the future is modelled as a security with a non-zero market exposure, an equal but opposite amount to this market exposure should also be held in cash to ensure that the futures holding does not change the overall market value of the portfolio.

Futures offsets are not the same as margin, which is cash put up by the owner of the contract and deposited at the futures exchange to cover counterparties against potential losses. The analyst should always remain aware that futures offsets are an accounting convenience rather than an actual asset, and that they should not be combined with actual cash assets.

To see how a futures offset works, consider a portfolio containing a single bond and a single future. The effective exposures, market exposures and returns of both securities are shown in Table 2.1. The market exposure of each security is its actual value (which is zero for futures), and the effective exposure is its value for the purpose of calculating profit and loss.

Table 2.1 Exposures, risks and returns of sample portfolio without cash offsets

Consider the return R made by this portfolio. Using the holdings in Table 2.1,

where MV_i is the market exposure of security i, EE_i its effective exposure and r_i its return.

An alternative approach ignores the fact that the market value of the futures contract is zero. Instead, the future is treated as a bond with the same characteristics, and a cash offset with an equal and opposite market exposure to that of the bond is added to the portfolio. In this case we work solely with effective exposures:

Using the holdings in Table 2.2, the portfolio’s return is now given by the alternative expression

where EE_i is the effective exposure of security i. The result is identical to the return of the portfolio without cash offsets.

The advantage of using cash offsets to model returns for futures is that no special consideration need be given to the fact that the market exposure of futures contracts is always zero, which simplifies the calculation of returns.

The disadvantage is that the accounting system has to generate extra cash offset transactions that will have various sizes as the market value of the futures position varies. In addition, the cash offsets must always be grouped with the futures holdings so that holdings of physical cash are not incorrectly stated.

Arguably, a better and simpler way to model the zero-exposure feature of futures is to treat them as special types of security with zero market exposure, and to use this market exposure when calculating portfolio return. In this case, no futures offset holdings need ever be used.

For better or worse, futures offsets are now an established feature of many commercial performance systems.

2.10 EDGE CASES

The difficulty in running performance does not usually lie in applying these equations. The main issues tend to be edge cases, such as very highly leveraged portfolios or portfolio valuations that are zero or negative. In addition, there are the perennial workflow problems of bad pricing data, reversed trades and the like.

2.11 EXTERNAL RETURNS

The three sources of return described in the next sections (trading, pricing, swing) are all often shown on performance reports, but do not strictly belong in an attribution report as they are not generated by the interaction between market movements and investment strategies.

2.11.1 Trading return

Trading return measures any additional return made by buying or selling a security at a different rate to the end-of-day revaluation rate.

For instance, suppose that a dealer buys a bond in the middle of the day at 99.50. The price rises over the rest of the day to close at 100.00, which is its revaluation price. In this case he has added

to the subsequent return of the security by buying at a lower price than its official value for that day.

Trading return is generated by the skill, market knowledge and contacts of the trader.

2.12 BENCHMARKS

A benchmark is a reference portfolio that has its composition and return available to investors and managers, often for a substantial fee.

Benchmarks are usually published by large financial institutions or data vendors, and can vary in size from a few tens of securities to many thousands.

Benchmark holdings are usually set up in terms of the amount of a security on issue. For instance, the FTSE 100 is an index of the 100 companies listed on the LSE (London Stock Exchange) that have the largest market capitalisations.

Some investors also use composite benchmarks, which are constructed by combining several benchmarks together in varying proportions to meet a predefined risk objective.

Benchmark data requirements for the equity performance analyst are often straightforward. For a simple comparison of overall return, one only needs a single time series measuring the benchmark’s return. Even for a Brinson analysis, only the sector weights and returns are required.

In contrast, a detailed fixed income attribution analysis requires security-level weights and returns for both the portfolio and the benchmark. This vastly increased data requirement is often one of the major costs and causes of difficulty in running attribution.

The situation may be eased if the market reaches agreement on the types of fixed income attribution to be used. In this case, benchmark vendors may decide to publish summary attribution analyses for benchmarks as a supplement to their full holdings. To date no benchmark vendor has made such data available, perhaps because there are as yet no agreed, standardised approaches to calculating quantities such as parallel curve movements.

Benchmarks may also be hedged to reduce foreign exchange exposures while retaining exposures to selected overseas markets. This topic is covered in Chapter 4.

Which benchmark, or combination of benchmarks, to use is conventionally decided when a fund is set up. The selected benchmark should always match the aims and investment strategies of the fund. For instance, if the fund manager has a mandate to invest in small-cap stocks in emerging markets, a blue-chip index would be inappropriate, as the performance of stocks in the latter category is not necessarily a measure of how the former market is performing.

A comprehensive guide to fixed income benchmarks can be found in Brown (1994).

2.13 ACTIVE RETURN

The return that a portfolio generates over and above its benchmark is called its active return.

Many managers like to publish their positive active returns, as these are held to be evidence of a superior investment strategy. However, a random investment strategy can also generate positive active returns, so this measure on its own should not be taken as prima facie evidence of investment skill. An active return can still be positive even if a portfolio lost money.

2.14 STOCHASTIC ATTRIBUTION

An alternative way of measuring skill shown in running a portfolio is as follows:

• Build the set of all possible portfolios available to the manager.
• Calculate the return of each.
• Form a cumulative returns graph showing the frequency of each return. This will usually have the familiar shape of a normal distribution.
• Observe where the return of the managed portfolio lies on this graph.

If 99% of all portfolios showed return between 0% and 1%, then a return of 2% is clearly exceptional as it is several standard deviations away from a return that might have been generated by chance alone. Conversely, a return of 0.1% is more likely to have been a random occurrence, and is much less indicative of investor skill.

Naturally, it is seldom practical to build every possible portfolio. However, statistical sampling and Monte Carlo simulation can give excellent approximations to the distribution of expected returns.

In practice the main obstacles to this type of attribution are ensuring that the portfolios generated are realistic. For instance, investment compliance rules may place strict limits on the types, amounts and turnover of investments.

A particularly appealing feature of this type of attribution is that it does not require a benchmark.

2.15 LIABILITY-DRIVEN INVESTMENT (LDI)

Many portfolios are managed with the aim of maintaining a stream of cash flows to fund future liabilities. For instance, a pension fund will forecast the regular payments required to its members over the remainder of their expected lifetimes. In this case the portfolio’s benchmark can be regarded as a portfolio of future payment streams.

In contrast to managing a portfolio against a benchmark, where the aim is to outperform, LDI aims to minimise risks by matching the portfolio’s future cash flows to its liabilities as closely as possible.

For cases in which cash flows are known precisely, fixed income securities such as bonds (which pay known cash flows) may be used for matching liabilities. In other cases the manager may prefer to use over-the-counter (OTC) instruments such as inflation-linked swaps, which can be precisely tailored to the fund’s liability requirements.

Attribution analysis is particularly useful for LDI, as it provides detailed insight into how well the portfolio of assets matched its liabilities over varying market conditions.