Risk Management Review PDF

PRELIM a significant concession in price because of the market’s potential inability to efficiently Risk means the uncertainty that an investment will earn...

PRELIM a significant concession in price because of the market’s potential inability to efficiently Risk means the uncertainty that an investment will earn accommodate the desired trading size. Liquidity its expected rate of return. Investment is a risky activity risk is present in both initiating and liquidating so we have to treat risk management as an important transactions, for both long and short positions, component of the investment process. For both but can be particularly acute for liquidating individual investments and portfolios, we have to transactions especially when such liquidation is examine and compare the full extent of risks and motivated by the need to reduce exposures in expected returns to ensure that the exposures we large losses. In trading securities, the size of the assume are justified by the rewards we expect to gain. bid-ask spread (the spread between bid and ask The proper identification, measurement, and control of prices), stated as a proportion of security price, risks are key to the process of investing. is frequently used as an indicator of liquidity. Risk management therefore is a process involving the 4. Operational risk is the risk of loss from failures identification of exposure to risk, the establishment of in a computer’s systems and procedures or from appropriate ranges for exposures with understanding of external events. an entity’s objectives and constraints, the continuous measurement of these exposures, and the execution of 5. Model risk is the risk that a model is incorrect or appropriate adjustments whenever exposure levels fall misapplied. In investments, it often refers to outside of target ranges. The process is continuous and valuation models. Model risk exists to some may require alterations in any of these activities to extent in any model that attempts to identify reflect new policies, preferences, and information. the fair value of financial instruments, but it is most prevalent in models used in derivatives An effective risk management identifies, assesses, and markets. Since the development s of the controls numerous sources of risk, both financial and seminal Black-Scholes-Merton option pricing non-market related to achieve the highest possible level model, both derivatives and derivative pricing of reward for the risks incurred. models have proliferated. Risk categories can be grouped into 6. Settlement (Herstatt) risk is the risk that one 1. Financial risks include liquidity risk, credit risk, party could be in the process of paying the commodity prices risk, equity prices risk, counterparty while the counterparty is declaring exchange rate risk, and interest rate risk. bankruptcy. Settlements are the payments associated with the purchase and sale of cash 2. Non-financial risks include tax risk, accounting securities such as equities and bonds, along risk, legal risk, regulations risk, settlement risk, with cash transfers executed for swaps, forward, model risk, and operations risk. options and other types of derivatives. The Identifying risks process of settling a contract involves on or both parties making payments and /or transferring 1. Market risk is the risk associated with interest assets to the other. Most regulated futures and rate, exchange rates, stock prices, and options market exchanges are organized in such commodity prices and it links to supply and a way that they themselves act as the central demand in various marketplaces. Much of the counterparty to all transactions. evolution that has taken place in the field of risk management begun from a desire to 7. Regulatory risk is the risk associated with the understand and control market risks. uncertainty of how transaction will be regulated or with the potential regulations to change. 2. Credit risk is the risk of loss caused by a Equities, bonds, futures, and exchange-traded counterparty or debtor’s failure to make a derivatives markets are usually regulated. promised payment. Regulations is a source of uncertainty, regulated 3. Liquidity risk is the risk that a financial markets are subject to the risk that the existing instrument cannot be purchased or sold without regulatory regime will become more onerous,  more restrictive, or more costly. In case of Other risks derivatives, companies that are regulated in 1. ESG risks- is the risks to a company’s market other ways may have their derivative business valuation resulting from environmental, social indirectly regulated. and governance factors. 8. Legal/Contract risk is the possibility of loss a. Environmental risk is created by the arising from the legal systems failure to enforce operational decisions made by the company a contract in which an enterprise has a financial managers, including decisions concerning stake. Financial transaction is subject to some product and services to offer and the form of contract law and contracts involve two processes to use in producing those parties. The possibility of conflict can arise due products and services. to breach of contract, illegal contract, fraudulent act and others. b. Social risk derives from the company’s various policies and practices regarding 9. Tax risk arises because of uncertainty associated human resources, contractual arrangement, with tax laws. Tax covering the ownership and and the work place. transaction of financial instruments can be extremely complex, and the taxation of c. Governance risk is the flaws in corporate derivatives transactions is an area of even more governance policies and procedures with confusion and uncertainty. Tax rulings clarify direct and material effects on a company’s these matters on occasion, but on other value in the market place. occasions, they confuse them further. Also, tax policy often fails to keep pace with innovations 2. Performance netting risk or netting risk, which in financial instruments. applies to entities that fund more than one strategy, is the potential for loss resulting from 10. Accounting risk arises from uncertainty about the failure of fees based on net performance to how a transaction should be recorded and the fully cover contractual payout obligations to potential of accounting rules and regulations to individual portfolio managers that have positive change. Accounting statements are a key, if not performance when other portfolio managers primary, source of information on publicly have losses and when there are asymmetric traded firms. The international Accounting incentive fee arrangements with the portfolio Standard Board (IASB) sets global standards for managers. accounting. The IASB together with different accounting standard boards of other countries 3. Settlement netting risk- refers to the risk that a such US Financial Accounting Standards Board liquidator of a counterparty in default could (FASB) have been working together toward challenge a netting arrangement so that convergence of accounting standards profitable transactions are realized for the worldwide. benefit of creditors. 11. Sovereign and political risks- Sovereign risk is a The practice of Risk Management form of credit risk in which the borrower is the Risk management should be a process, not just government of a sovereign nation. Its an activity. A process is continuous and subject magnitude has two components: likelihood of to evaluation and revision. Effective risk default and the estimated recovery rate. management requires the constant and Political risk is associated with changes in the consistent monitoring of exposures, with an eye political environment. It could be both overt toward making an adjustment, whenever and (change of economic system) or subtle (change wherever the situation calls for them. Risk political party control) and it exist in every management in its totality is all at once a jurisdiction where financial instruments trade. proactive, anticipative, and reactive process that continuously monitors and control risk. Practical application of the process The company faces a range of financial and nonfinancial risks and response to these challenges by establishing a series of risk management policies and procedures. It defines its risk tolerance, which is the level of risk it is willing and able to bear. It then identifies the risks, drawing on all sources of information, and attempts to measure these risks using information or data related to all of its identified exposures. After having an effective risk identification and measurement mechanisms, it is now in a position to adjust its risk exposures, whenever and wherever exposures diverge from previously identified target ranges. These adjustments take the form of risk-modifying transactions (including risk transfer). The execution of risk management transactions is itself a distinct process; for portfolios, this step consists of trade identification, pricing, and execution. The process then loops around to the measurement of risk and continues in that manner, and to the constant monitoring and adjustments of the risk, to bring it into or maintain it within the desired range. Risk Management Process: practice of risk management Nonfinancial the company Financial risks risks Set policies and procedures Information Define risk and tolerance Information/data data Identify risk Measure risk derivatives Execute risk Adjust level of Mgt. transactions risk non-derivatives identify appropriate transactions price transactions execute transactions In applying risk management process to portfolio management, managers must devote considerable amount of attention to measuring and pricing the risks of financial transactions or positions particularly those involving derivatives. Risk Management Process: pricing and measuring of risk Identify the sources of risks Measure risks Select appropriate model Determine determined Market price or value model price or value Compare Attractively priced not attractively priced Execute transaction Seek alternative transaction Risk management involves adjusting levels of risk to Decentralization has the advantage of allowing the appropriate levels, not necessarily eliminating risk people closer to the actual risk taking to more directly altogether. Risk management is a general practice that manage it. Centralization permits economies of scale involves risk modification (risk reduction or risk and allows the company to recognize the offsetting expansion) as deemed necessary and appropriate by the nature of distinct exposure that an enterprise might custodians of capital and its beneficial owners. assume in a day-to-day operation. Risk Governance: management responsibility Example: Assuming one subsidiary of Corporation Kwekwek buys from Japan and another subsidiary sells Risk governance is a process of setting overall policies to Japan, with both engaged in yen-denominated and standards in risk management. It involves choices of transaction, each subsidiary has some foreign exchange governance structure, infrastructure, reporting, and exposure, from centralized viewpoint these risks have methodology. The quality of risk governance can be offsetting effects that reduces the overall need to judged by its transparency, accountability, effectiveness hedge. (achieving objectives), and efficiency (economy in the use of resources). Centralized risk management gives an overall picture of the company’s risk position, and ultimately overall is Governance Structure: centralized or decentralized risk what counts. The centralized risk management is known management system as Enterprise Risk Management (ERM). Its distinguishing 1. Centralized risk management system approach- feature is a firmwide or across-enterprise perspective company has a single risk management group that is sometimes called firmwide risk management. In that monitors and ultimately controls all of the ERM, the organization must consider each risk factor to organization’s risk-taking activities. which it is exposed – both in isolation and in terms of any interplay among them. For risk taking entities, it is 2. Decentralized risk management system contradictory to suggest that an organization has sound approach- places risk management corporate governance without maintain a clear and responsibility on individual business unit continuously updated understanding of its exposure as managers. Thus, each unit calculates and an enterprise level. reports its exposure independently. In case of decentralized risk management approach, it will require a mechanism by which senior managers can inform themselves about the enterprise’s overall risks exposures. At enterprise level, companies should control not only referred to as assets’ volatility. Volatility is an adequate the sensitivity of their earnings to fluctuations in the description of portfolio risk, particularly for those stock market, interest rate foreign exchange rate, and portfolios composed of instruments with linear payoffs. commodity prices but also their exposure to credit In some applications, such as indexing, volatility relative spreads and default risk, to gaps in the timing match of to a benchmark is paramount. In those cases, the focus their assets and liabilities and to operational/system should be on the volatility of the deviation of a failures, financial fraud, and other factors that can affect portfolio’s returns in excess of a stated benchmark corporate profitability and survival. portfolio’s returns known as active risks, tracking risk, tracking error volatility or simply as tracking error. An effective ERM system usually incorporates the Volatility associated with individual position beside following steps: being a useful risk management metric can be combined 1. Identify each risk factor to which the company is with other simple statistics such as correlations to form exposed the building blocks for the portfolio-based risk management system that has become an industry 2. Quantify each exposure’ size in money term. standard. 3. Map these inputs into a risk estimation A portfolio’s exposure to losses because of market risk calculation. takes one of two forms: (1) sensitivity to adverse 4. Identify overall risk exposures as well as the movements in the value of a key variable in valuation contribution to overall risk deriving from each (primary or first-order measures of risk); and (2) risks risk factor. measures associated with changes in sensitivities (secondary or second-order of measures of risk). 5. Set up a process to report on these risks periodically to senior management, who will set Primary measures of risk often reflect linear elements in up a committee of division heads and executives valuation relationship; secondary measures often take determine capital allocations, risk limits, and account of curvature in valuation relationships. Each risk management policies. asset class (e.g., bonds, foreign exchange, equities) has specific first and second-order measures. 6. Monitor compliance with policies and risk limits. Considering measures of primary sources of risks, for a Step 5 and 6 allows the organization to quantify the stock or stock portfolio, Beta (a measure of the magnitude and distribution of its exposures and sensitivity of a given investment or portfolio to enabling it to use the ERM system’s output to more movements in the overall market) measures sensitivity actively align its risk profile with its opportunities and to market movements and is linear risk measure. For constraints on a routine, periodic basis. Effective ERM bonds, Duration (a measure of the approximate system feature centralized data warehouse, where a sensitivity of a security to a change in interest rate i.e., a company stores all pertinent risk information, including measure of interest rate risk) measures the sensitivity of position and market data, in a technologically efficient a bond or bond portfolio to a small parallel shift in the manner. The process of identifying and correcting errors yield curve and is a linear measure. Delta (ratio change in a technologically efficient manner can be resource in the option price to a small change in the price of intensive especially when effort requires storing underlying asset) for options, and is a linear measure, historical information on complex financial instruments. which measures an option’s sensitivity into small change Measures of Risk in price of its underlying. These measures all reflect the expected change in price for a financial instrument for a Market risk refers to the exposure associated with unit change in the value of another instrument. actively traded financial instrument whose prices are exposed to the changes in interest rate, exchange rates, Second order measures of risks deal with the change in equity prices, commodity prices or a combination of the price sensitivity of a financial instrument. They them. The most widely used statistical tools to describe include Convexity (a measure of how interest rate market risk is the standard deviation of price outcomes sensitivity changes with a change in interest rate) for associated with an underlying asset, this measure is fixed-income portfolios and Gamma (a numerical measure for sensitivity of delta to a change in Elements of Measuring Value at Risk underlying ‘s value) options. Appropriate VAR requires the user to make a number of For options, two major factors determine price: (1) decisions about the calculation’s structure such as volatility and (2) time to expiration, both first order or (1) picking a probability level, primary effects. Sensitivity to volatility is reflected in Vega (a measure of the sensitivity of an option’s price to (2) selecting the time period over which to measure a change in the underlying assets’s volatility). Option VAR, and prices are also sensitive to changes in time to expiration as measured by Theta (the rate at which an option’s (3) choosing the specific approach to modeling the loss time value decays) which the change in price of an distribution. option associated with a one-day reduction in its time to The probability chosen is typically either.05 or.01 that expiration. Theta and Vega are risks associated corresponds to 95 percent and 99 percent confidence exclusively with options. level, respectively. The use of.01 leads to a more VALUE AT RISK (VAR) conservative VAR estimate, since it sets the figure at the level where there should be a 1 percent chance that a Value at Risk (VAR) is an estimate of the loss (in money given loss will be worse than the calculated VAR. terms) that we expect to be exceeded with a given level of probability over a specified time period. It is a The trade-off is that the VAR risk estimate will be much probability-based measure of loss potential for a larger with a 0.01 probability than it will be for a.05 company, a fund, a portfolio, a transaction, or a in units probability. There is no definitive rule exists preferring of currency. Any position that exposes one to loss is one probability from the other. For portfolios with potentially candidate for VAR measurement. VAR is largely linear risk characteristics, the two probability widely and easily used to measure the loss from market levels will provide essentially identical information. risk and to measure the loss from credit risk and other However, the tails of loss distribution may contain a types of exposure subject to more complexity. wealth of information for portfolios that have a good deal of optionality or non-linear risks, and in this case VAR has important elements which are risk managers may need to select the conservative probability threshold. (1) estimate of the loss that we expect to be exceeded, The second important decision for VAR users is choosing (2) VAR is associated with a given probability, the time period. VAR is often measured over a day, but (3) VAR has a time element and that as such, VARs others, the common are longer time periods. Regardless cannot be compared directly unless they share the same of the time interval selected, the longer the period the time interval. greater the VAR number will be because of the magnitude of potential losses varies directly with the Example of VAR for an investment portfolio: time span over which they are measured. The VAR for a portfolio is 1.5 million for one day with a Once these primary paremeters are set , one can probability of 0.05 which means there is 5 percent proceed to actually obtain the VAR estimate, the third chance that the portfolio will lose at least 1.5 million in a decision will be the choice of technique. The basic idea single day. The 1.5 million loss is a minimum. To behind estimating VAR is to identify the probability describe VAR as a maximum: The probability is 95 distribution characteristics of portfolio returns. percent that the portfolio will lose no more than 1.5 million in a single day. This perspective states that VAR uses confidence level that says, with 95 percent confidence (or for 95 percent confidence level), the VAR for a portfolio is 1.5 million for a single day. Return on portfolio probability 5. VAR fails to incorporate positive results into its risk profile, and as such, it arguably provides an Less than -40%.01 incomplete picture of overall exposures. -40% to -30%.01 The process of comparing the number of violations of -30% to -20%.03 VAR thresholds with the figure implied by the user- selected probability level is part of a process known as -20% to -10%.05 Back-testing. It is important to go through this exercise, -10% to -5 %.100 ideally across multiple time intervals, to ensure that the VAR estimation method adopted is reasonably accurate. -5% to – 2.5%.125 Example: if the VAR estimate is based on daily -2.5% to 0%.175 observations and targets a 0.05 probability, then in 0% to 2.5%.175 addition to ensuring that approximately a dozen threshold violation occur during given year, it is also 2.5% to 5%.125 useful to check other, shorter time intervals, including the most recent quarter (for which, given 60-odd 5% to 10%.100 trading days, we would expect approximately three VAR 10% to 20%.01 exceptions – i.e., losses greater than the calculated VAR), and the most recent month (20 observations, 20% to 30%.01 implying a single VAR exception). The results should not 30% to 40%.03 be expected to precisely match the probability level predictions but should be at a minimum of similar Greater than 40%.01 magnitude. If the results vary much from those that the Three standardized methods for estimating VAR model predicts, then users must examine the reasons and make appropriate adjustments. 1. Analytical or Variance-Covariance Method An accurate VAR estimate can also be extremely difficult 2. The Historical Method or Historical Simulation to obtain for complex organizations. Consolidating the Method effects of risk exposures into a single risk measure can 3. Monte Carlo Simulation Method be difficult. However, most large bank, though expose to thousands of risks, manage to do so, and do an excellent The Advantage and Limitations of Value at Risk (VAR) job of managing their risk. Although Value at risk has become the industry standard Advantages of VAR for risk assessment, it also has documented imperfections: 1. VAR has the attraction of quantifying the potential loss in simple terms and can be easily 1. VAR can be difficult to estimate, and different understood by senior management. estimation methods can give quite different values; 2. Regulatory bodies have taken note of VAR as a risk measure, and some require that institutions 2. VAR can also lull one into a false sense of provide it in their reports; security by giving the impression that the risk is properly measured and under control; 3. VAR is one acceptable method of reporting how companies manage their financial risk; 3. VAR often underestimates the magnitude and frequency of the worst returns, although this 4. VAR is versatile: companies use VAR as a problem often derives from erroneous measure of their capital at risk, they estimate assumptions and models; VAR associated with a particular activity such as a line of business, a subsidiary, or a division. 4. VAR for individual positions does not generally They evaluate performance, taking into account aggregate in a simple way to portfolio VAR;  the VAR associated with this risky activity, and loss that we expect to be exceeded with a given companies allocate capital based on VAR. probability over a specified period of time. It can be used when a company (or portfolio of Surplus at risk: VAR as it applies to pension funds assets) generates earnings but cannot readily The difference between the value of the pension fund’s valued in a publicly traded market, or when the assets and liabilities is referred to as the surplus. It is the analyst’s focus is on the risk to earnings in a value that pension fund managers seek to enhance and valuation. protect. If this surplus falls into negative territory, the 4. Tail value at risk (TVAR) also known as plan sponsor must contribute funds to make up the conditional tail expectation – TVAR is VAR plus deficit over a period of time that is specified as a part of the expected loss in excess of VAR, when such the fund’s plan. Pension fund managers apply VAR loss occurs. methodologies not to their portfolio of assets but to the surplus. They express their liability portfolio as a set of Stress Testing short securities and calculate VAR on the net position, VAR analysis is to quantify potential losses under normal once this adjustment is made the three VAR conditions. Stress testing, by comparison, seeks to methodologies can be applied to the task. identify unusual circumstances that could lead to losses Extensions and supplements to VAR in excess of those typically expected. Different scenarios will have attached probabilities of occurring that vary In the evaluation of the portfolio effect of a given risk, from the highly likely to the almost totally improbable. isolating the effect of a risk, particularly in complex portfolios with high correlation effects is important. Broad approaches in stress testing Incremental VAR (IVAR) is used to investigate the effect. 1. Scenario analysis – the process of evaluating a 1. IVAR measures the incremental effect of an portfolio under different states of the world. It asset on the VAR of a portfolio by measuring the involves designing scenarios with deliberately difference between the portfolio’s VAR while large movements in the key variables that affect including a specified asset and the portfolio’s the values of a portfolio’s assets and derivatives. VAR with that asset eliminated. IVAR is also use a. Stylized scenarios analysis – involves to assess the incremental effect of a subdivision simulating a movement in at least one on an enterprise’s overall VAR. Although IVAR interest rate, exchange rate, stock price, or gives a limited picture of the asset or portfolio’s commodity price relevant to the portfolio. contribution to risk, however it provides useful These movement might range from fairly information about how adding the asset will modest changes to quite extremes shifts. affect the portfolio’s overall risk as reflected in its VAR. b. Recommended scenarios from derivative policy group 2. Cash flow at risk (CFAR) – measures the risk to a company’s cash flow instead of its market value a. Parallel yield curve shifting by +-100 as in the case of VAR. CFAR is the minimum cash basis points (1 percentage point flow loss that we expect to be exceeded with a given probability over a specified period of time. b. Yield curve twisting by +-25 basis points It can be used when a company (or portfolio of c. Each of the four combinations of the assets) generates cash flows but cannot readily above shifts and twists valued in a publicly traded market, or when the analyst’s focus is on the risk to cash flow in a d. Implied volatilities changing by +-20 valuation. percent from current levels 3. Earnings at risk (EAR) measures the risk to a e. Equity index levels changing by +-10 company’s earning instead of its market value as percent in the case of VAR. EAR is the minimum earning  f. Major currencies moving by +-6 percent the loss, perhaps by having the debtor sell assets and and other currencies by +-20 percent pay the creditors a portion of their claim. g. Swap spread changing by +-20 basis Credit losses have two dimensions. points a. The likelihood of loss – the likelihood of loss is a c. Another approach to scenario analysis probabilistic concept. In every credit-based involves using actual extreme events that transaction, a given probability exists that the occurred in the past and Hypothetical debtor will default. events that have never happened in the b. Associated amount of loss (reflects the amount markets or market outcomes to which we of credit outstanding and the associatd recovery attached a small probability. date) 2. Stressing models In the risk management business, exposure must often Another approach might to use an existing model and be viewed from two different time perspectives. We apply shocks and perturbations to the model inputs in must assess first the risk associated with immediate some mechanical way. This approach might be credit events and second the risk associated with events considered more scientific because it emphasizes a that may happen later. range of possibilities rather than a single set of Elements of credit risk scenarios, but it will be more computationally demanding. 1. Current credit risk (jump-to-default risk) – the risk of events happening in the immediate a. Factor push – a form of stressing model future which is the idea of pushing prices and risk factors of an underlying model in the most 2. Potential credit risk – Though the counterparty disadvantageous way and to work out the can pay on time, still the risk remains that the combined effect on the portfolio’s value. entity will default at the later date. This exercise might be appropriate for a wide range of models, including option- 3. Cross-default provisions – which blends current pricing models such as Black-Scholes- and potential credit risk, is the possibility that a Merton, multifactor equity risk models, and counterparty will default on a current payment term structure factor models. to a different creditor, and in direct lending or derivative-based credit contracts stipulate that if b. Maximum loss optimization, which is to the borrower defaults on any outstanding credit optimize mathematically the risk variable obligations, the borrower is in default on them that will produce the maximum loss. all. Creditors stipulate this condition as one means of controlling credit exposure. c. Worst-case scenario analysis, which is to examine the worst case that we actually VAR is also used to measure credit risk known as credit expect to occur. VAR or default VAR or credit at risk and it reflects the minimum loss with a given probability during a period of Measuring Credit Risk time. Credit risk is present when there is a positive probability Example: A company might quote a credit VAR of 10 that one party owing money to another wiil renege on million for one year at a probability of 0.05 (or a the obligation , that is the counterparty could default. If confidence level of 95 percent. That means the the defaulting party has insufficient resources to cover company has a 5 percent chance of incurring default- the loss or the creditor cannot impose a claim on any related losses of at least 10 million in one year. assets the debtor has that are unrelated to the line of business in which the credit extended, the creditor can Risk is the uncertainty that an investment will earn its suffer a loss. A creditor might be able to recover some of expected rate of return. An investor who is evaluating a future investment alternative expects or anticipate a certain rate of return. The investor might expect the Variance and standard deviation measure the dispersion investment will provide a rate of return of return, but of possible rates of return around the expected rate of this is actually the investor’s most likely estimate return. n referred to as point estimate. The investor would Variance = ∑ (probability of return) X (possible return – expected return)2 probably acknowledge the uncertainty of this point i=1 estimate return and under certain conditions the rate of The larger the variance for an expected rate of return, return might go as low as 10% or as high as 30%. The the greater the dispersion of expected returns and the larger range of possible rate of returns reflect the greater the uncertainty or risk. In perfect certainty, investor’s uncertainty regarding what the actual return there is no variance of return since there is no deviation will be as such, a larger range of possible returns makes from expectations so, no risk. the investment riskier. Standard Deviation is the square root of the variance. An investor determines how certain the expected rate of return on an investment is by analyzing estimates of n possible returns. As such, investors assign probability Standard deviation =√ ∑ (probability of return) X (possible return – expected return)2 values to all possible returns. These probability values range from zero (no chance of return) to 1 (complete Relative measure of risk certainty that the investment will provide the specified If conditions for two or more investment alternatives rate of return. These probabilities are typically have major differences in the expected rate of return, it subjective estimates based on the historical is necessary to use relative variability performance of the investment or similar investments modified by the investor’s expectations for the future. Coefficient of Variance = Standard deviations of return An investor may know that about 30% of the time the / expected rate of return rate of return on a particular investment was 10%. Example: Expected return is defined as Investment A has expected return of 0.07 and standard n Expected return E(R1) = ∑ (probability of return) X (possible return) deviation of 0.05. Investment B has expected return of i=1 0.12 and standard deviation of 0.07. E(R1) = [ (P1) (R1) + (P2) (R2) + (P3) (R3) + (P4) (R44) ……+ (P1) (R1)] CVa = 0.05 / 0.07 = 0.714 n CVb = 0.07 / 0.12 = 0.583 Expected return E(R1) = ∑ (Pi) (Ri) i=1 Comparing absolute measures of risk, investment B look riskier because of its standard deviation of 0.07 against Perfect certainty allows only on possible return investment’s A of 0.05. However, CV figures show that Example: investment B has less relative variability or lower risk per unit of expected return because it has a substantial E(R1) = (1.0) (0.05) = 0.05 higher expected rate of return. Alternatively, assuming investor believed an investment Risk measures for historical rates of returns is same as could provide several different rates of return for expected rate of returns except that we consider the depending on different possible economic conditions: historical holding period yields strong economy without inflation,.20 rate of return and 0.15 probability; weak economy, -.20 rate of return and Variance = [ ∑ [HPYi – E(HPY)]2] / n 0.15 probability; and, no major changes in the economy, HPYi = holding period yield during period i 0.10 rate of return and 0.70 probability. E(HPY) = the expected value of the holding period yield E(R1) = (0.15) (0.20) + (0.15) (-0.20) + (0.70) (0.10) = 0.07 that is equal to the arithmetic mean (AM) of the series Measuring the risk of expected rates of return N = the number of observations The standard deviation is the square root of the variance. Both measures indicate how much the individual HPYs over time deviated from the expected value of the series. Variance is a measure of the degree of dispersion of a series of numbers around their mean. The larger the variance the greater the spread of the series around its mean. Standard deviation is a measure of the spread of a series of values of a variable around its mean.