Topic 8: Risk and return
Factor models
A required return on any risky asset is defined as the minimum rate of return that investors will accept on acquiring the asset. The required return is comprised of three components: • real risk-free rate of interest on holding cash • expected inflation rate (of the general price level) • compensation for risk (the possibility of losing the investment capital). The first two components, when summed, equal the nominal risk-free rate, which is often proxied by the Treasury bill or bond yield. Investors expect to receive the risk-free rate plus compensation for the risk of the asset losing value. This compensation is often called the risk premium for the asset. Investors are faced with the challenge of investing without fully understanding the risks inherent in different assets and asset classes. Also, they do not know what the rewards will be for assuming these unknown risks. Finance theory sheds light on these issues through a large body of research known as asset pricing theory. There are many asset pricing models, perhaps the best known of which is the CAPM. The goal of asset pricing models is to relate the risk of an asset to its expected return. But how does one define risk? Some models, such as CAPM, argue that there is only one relevant source of risk: market risk. Other models such as the APT argue that there are several risk factors — but what they are exactly, is unknown.
Time-weighted versus money-weighted rates of return
A time-weighted rate of return measures the compound growth rate of one unit of money, say $1, over a given timeframe. In other words, this approach measures the compound growth rate of the initial value of the portfolio over the course of the evaluation period. This approach is suited to measuring the performance of different investment managers and eliminates the effects of new cash flows into and out of the fund or portfolio (which are factors over which the manager has no control). In contrast, a money-weighted rate of return measures the average compound rate of return of funds invested and incorporates the timing and size of cash flows into and out of the portfolio. This approach is suited to measuring portfolio performance.
Dimension reduction
An asset pricing factor model condenses a large number of variables into a smaller number of explanatory factors. The factors described by the model, to one degree or another, should be common to all funds or assets whose future returns are being analysed, and the factors should explain most of the return, variance and covariance. Culling this large number of variables down into a smaller number of explanatory factors is known as dimension reduction. Dimension reduction is important to efficient analysis of data. Take for example a universe of 2000 hedge funds. The covariance matrix would have N × N or 4,000,000 terms. As a covariance matrix is symmetrical, half the matrix is redundant from a calculation standpoint. Even taking this into consideration, there are still N × (N + 1)/2 or 2,001,000 terms (1,999,000 covariances plus 2000 variances). This is too much data to comprehend efficiently. Factor analysis condenses this vast amount of information into a smaller number of relevant risk factors. Assume that three relevant risk factors can be identified in the data. In the universe of 2000 funds this would reduce the data from 2,001,000 terms to 6000 terms (i.e. 2000 funds × 3 factors) as well as the covariance structure of the factors (i.e. six terms). Such a decrease in the data set is the essence of dimension reduction. Based on the 6006 terms and the variance of the residual error, all the 2,001,000 variances and covariances can be derived. Dimension reduction is one benefit among several of using factor models.
Asset-weighted returns
An asset-weighting approach involves weighting each fund's return by the ratio of the assets in the individual fund to the sum of all the assets in all of the funds being analysed (i.e. the relative portfolio weight of the fund based on asset values). The aggregate return is then calculated by multiplying the fund returns by their respective weights and summing the weighted returns.
Information ratio
Both Sharp ratio and IR are very useful for evaluating portfolios with returns that are normally distributed. However, they are not applicable for asymmetric return strategies which are frequently used by hedge funds. Furthermore, both ratios do not consider the correlation between asset classes over time.
Desirable properties of factor models
For a factor model to be useful, there are several desirable properties, including the following: • the model should be intuitively easy to understand • the model should be based on careful research • the parameter estimates can be made in a reasonable amount of time • the number of factors must be small enough to avoid overfitting (i.e. avoid extra or redundant independent variables) but large enough to adequately explain fund performance • the model must reflect commonalities in returns and allow analysts to make systematic benchmarking, performance analysis and segmented analysis decisions. Topic 9
Credit risk
For credit risk, exposure is the size or value of loss that would be realised if a credit event occurred. The recovery rate is the percentage of assets that could be recovered from a counterparty after a credit event (such as default) occurs. A credit event relates to a change in a counterparty's ability to perform its previously agreed financial obligations. Market prices incorporate changes to credit ratings or changes to default probabilities, which can be looked at as both market risk and credit risk. Therefore, instances can exist where a change in price is due to market and credit risk. Sovereign risk refers to the risks resulting from a country's actions. Sovereign risk differs from the other forms of credit risk in that it is country specific and is beyond legal arbitration. A country's willingness and ability to repay its obligations are often factors looked at when evaluating the sovereign risk of foreign government debt. The sources of sovereign risk stem from a country's political and legal systems. Settlement is the exchange of two payments or the exchange of an asset for payment. Settlement risk is the risk that a counterparty will fail to deliver its obligation after the party has made its delivery. Pre-settlement risk is lower than settlement risk because, with this measure, payments will offset (i.e. are netted). On the other hand, settlement risk exposure deals with the full value of each payment.
Overview
In the financial services context, risk has several, usually complementary, meanings. Speaking generically, risk is anything and everything that constrains or prevents the achievement of anticipated outcomes (e.g. investment returns). Generic risk aside, there are also specific risks with the potential to impact negatively on one or more of the factors that affect financial outcomes including, but not limited to, investment returns. One of the key concepts in this topic is that a 'good' investment return is not necessarily (or even usually) equal to the expected return. The application of quantitative techniques facilitates the required rigour and specificity in terms of defining the problem (issue or task), objectives, variables, metrics and measurement approach. The topic concludes with an examination of statistical techniques to assess the skills of investment managers and/or analysts.
Benchmarking
Investment decisions often depend on risk and return characteristics and investor preferences. The major challenge surrounds the choice of risk and return statistics most appropriate for the manager or investor. Funds are routinely evaluated based on risk and return performance relative to passive benchmarks, customised targets, or pre-specified benchmarks including capture indicator, up capture indicator, down capture indicator, up number ratio, down number ratio, up percentage ratio, down percentage ratio, per cent gain ratio and ratio of negative months over total months. Note that benchmarks in general should be: • intuitive • investible • representative of the assets the client wants to invest in. Note also that one can consider benchmarks from (at least) two perspectives: • measurement benchmark — where one simply views the benchmark as a measuring stick (i.e. compares the portfolio return to the benchmark) • management benchmark — where one looks more deeply at the underlying structure of the benchmark and structures a portfolio on the basis of the benchmark structure, deviating from it where one believes one can generate active return.
Market risk
Market risk can be categorised as follows. Directional risks are linear risk exposures in economic or financial variables (e.g. interest rates and stock indices). Non-directional risks are risks that have non-linear exposures or neutral exposures to changes in economic or financial variables. Basis risk is the risk that the price of a hedging instrument and the price of the asset being hedged are not perfectly correlated. An example of basis risk is using a put option to hedge an equity exposure. In this case, the option position will have to be monitored and adjusted appropriately since the change in the put option will likely not be exactly equal to the change in the equity price. Volatility risk is the risk of loss from changes in actual or implied volatility of market prices. The volatility of equity indices or interest rates may change due to market events, significant investor uncertainty, political instability or structural changes in the economy. Firms with exposure to equity markets may see significant losses if there is an unexpected change in volatility.
Sensitivity to benchmark return changes
Often, a fund's beta is estimated by regressing the fund's returns (dependent variable) against the benchmark's returns (independent variable). The slope of the regression is the historical beta. A regression slope equals the average change in the fund return relative to a 1-unit change in the benchmark return. Therefore, the beta measures the sensitivity or exposure of the fund to changes in the benchmark. For example, a fund beta equal to 2 indicates that the fund return changes 2 percentage points for every 1 percentage point change in the benchmark return, on average. A fund beta equal to 0.50 indicates that the fund return changes 0.50 percentage points for every 1 percentage point change in the benchmark return, on average. A market neutral hedge fund is expected to have a beta close to zero. Beta does not measure the total risk of the fund; it only measures that part of the fund's risk that is related to the benchmark, often called 'market' or 'systematic' risk (when the benchmark is the overall market). Therefore, a fund with a low beta does not necessarily have low total risk. The low beta fund may have significant amounts of specific risk, also called 'unsystematic risk'.
Topic learning outcomes
On completing this topic, students should be able to: • analyse the risk and return of a portfolio using the chi-squared test • describe the calculation of the Markowitz efficient frontier for investment • explain the difference between diversifiable and systematic risk, and calculate the systematic risk (beta) • assess investment manager/analyst skills using statistical analysis, including the use of a range of ratios • describe the use of contribution analysis and performance attribution in identifying generators of value and analysing fund performance • describe the use of benchmarks in measuring fund performance.
Operational risk
Operational, market and credit risk are interrelated. An operational failure may increase market and credit risks. A bank that engages in buying and selling derivatives without an adequate understanding of the derivatives market could suffer significant losses. Those losses could then result in a change in credit rating for the firm and a reduction in market price for its securities. Model risk is the risk of loss due to the use of misspecified or misapplied models. An institution buying or selling collateralised mortgage obligations (CMOs) may be exposed to model risk if the model used to price the CMOs does not adequately account for the probability of default in the underlying mortgages. People risk relates to the risk associated with fraud perpetuated by internal employees and/or external individuals. An example of people risk is a rogue trader within an institution that intentionally falsifies reports related to losses incurred. Legal risk is the risk of a loss in value due to legal issues including lawsuits, fines, penalties and/or damages. An example of legal risk is when a counterparty sues a bank to avoid meeting its obligations. Legal risks are managed through appropriate corporate policies developed by legal counsel in conjunction with a firm's financial risk managers.
Significance tests of the Sharpe ratio
The Sharpe ratio is most often used to rank portfolios from best to worst performing. However, one might also want to know if Sharpe ratios for two portfolios are significantly different. Various Sharpe ratio significance tests exist. The Jobson and Korkie test is used to test the equality of Sharpe ratios between two portfolios (P and Q). A hypothesis test can be defined for the equality as follows: H0: Sharpe ratio for Portfolio P = Sharpe ratio for Portfolio Q HA: Sharpe ratio for Portfolio P ¹ Sharpe ratio for Portfolio Q. The Jobson and Korkie test statistic assumes portfolio returns are normally distributed, which is often violated by hedge funds. The Gibbons, Ross and Shanken test is used to test the equality of the Sharpe ratio between a managed portfolio and the mean-variance efficient market portfolio. Alternatively, the Gibbons, Ross and Shanken test can be viewed as a test of the efficiency of the managed portfolio. The Lo test relaxes the distributional assumptions on the portfolio returns, incorporating characteristics such as serial correlation and mean reversion. Lo shows that the Sharpe ratio itself is very sensitive to violations of traditional distributional assumptions.
Liquidity risk
The two main types of liquidity risk are discussed below. Asset liquidity risk, which is sometimes called 'market' (or 'trading') liquidity risks, results from a large position size forcing transactions to influence the price of securities. To manage asset liquidity risk, limits can be established on assets that are not heavily traded. Funding liquidity risk, which is sometimes called' cash flow' risk, refers to the risk that a financial institution will be unable: • to raise the cash necessary to roll over its debt • to fulfil the cash, margin or collateral requirements of counterparties, or • to meet capital withdrawals.
Value at risk
VAR is interpreted as the worst possible loss under normal conditions over a specified period of time for a given confidence level/probability. If an analyst says, 'for a given month, the VAR is $1 million at a 95% level of confidence', then this translates to mean 'under normal conditions in 95% of the months (19 out of 20 months), we expect the fund to lose no more than $1 million'. Analysts may also use other standard confidence levels (e.g. 90% and 99%). VAR can be denominated in a percentage return or in a currency such as dollars or yen. Using VAR, the decision maker can determine if the size of the potential loss is acceptable for the given level of probability. There are several techniques for measuring VAR: • parametric VAR approach • historical VAR approach • Monte Carlo VAR approach. VAR is an ex-ante (i.e. before the fact) measure and can be difficult to calculate. However, it does capture exposures to risk factors and accounts for variation and covariation in risk factors. VAR is comparable across different business units in a company with different assets and risk characteristics; that is, VAR is interpreted the same, regardless of the assets in question. VAR is also frequently used in the risk budgeting process, where senior management allocates a risk level to each asset class. The main limitation of VAR is that it gives a value of potential loss under normal conditions and not when economic and market factors take on extreme values and the relationship between variables change. A secondary limitation is that it is only a single measure, and a manager should not rely too heavily on just one value. Although VAR is easily understood and usually widely accepted, all methods for calculating VAR first require accurate inputs, and this issue becomes more and more daunting as the number of assets in a portfolio gets larger. Just identifying all the possible types of risks (without actually predicting their impacts on portfolio value) may not be feasible.
Valuation and risk management using VAR
Valuation is the process of discounting the expected future value of an asset to determine the current price of the asset. The expected value for an asset is the mean value for the distribution of possible values. The valuation of derivatives requires risk-neutral pricing so that arbitrage situations will not persist. VAR as a risk management tool attempts to explain the future possible distribution of asset values with specific focus on the lower tail of return distribution. VAR looks at the future value of an asset, not the present value, and utilises the distribution of returns that is often assumed to be equivalent to the historical distribution. Less precision is required in VAR analysis than in valuation because as long as the model is not biased, errors will tend to offset each other.
Chi-squared test for variance
chi-squared test is a hypothesis test that examines if the true population variance is significantly different from the assumed variance. A higher variance implies a larger estimation error. For instance, an index manager's primary objective is to minimise the variance of the differential return between their fund and the benchmark. An index manager has failed to meet the objective if the standard deviation of their active return (i.e. differential return between the index and their fund) is 5%, even if the mean active return is 0%.
Benefits of factor models in managing portfolios
With regard to portfolio management, factor models have several benefits including the ability to: • decrease the data set through dimension reduction • provide a framework for understanding the components of a fund's returns, including the risks taken by the investment manager • provide insight into the risks taken by the fund's manager in terms of overweighting and underweighting exposures to various risk factors • enable portfolio managers to accurately attribute risk to various sources. This is especially important for funds such as indexing funds and market neutral portfolios that must exactly identify risk exposures • provide a more accurate forecast of future risks. Factor models allow portfolio managers to predict return, correlation and volatility with greater accuracy • enable investors to identify an active portfolio manager's return contribution relative to a passive strategy.
Risk analysis using factor models
more generally. Some factors apply at the asset class level — the following are common to all Australian equities: • interest rate movements • return on the Australian sharemarket overall (e.g. All Ordinaries Index) • dividend yield on a stock. Other factors are specific to individual stocks. Management skill and plant breakdowns are examples of such factors. CAPM claims that much of the return on a particular stock can be explained by movements in the overall sharemarket. This model then attributes any unexplained residual variation in the return on the stock to stock-specific factors. CAPM is an example of a single-factor model because the only factor it examines is the overall sharemarket. Models based on CAPM, but which postulate more than one common factor as driving stock returns, are known as arbitrage pricing theory (APT) models. Unidentified stock-specific factors also have a role in these multi-factor models. Note that both the CAPM and APT models are partial equilibrium pricing models; that is, they provide an explanation of the demand for individual shares but not for the supply side.
