Value at Risk

Value-at-risk (VaR) is the minimum loss that would be expected a certain percentage of the time over a certain period of time given the assumed market conditions. It can be expressed in either currency units or as a percentage of portfolio value. VaR can be viewed as a probabilistic measure of the range of values a firm’s portfolio could lose due to market volatility. This volatility includes effects from changes in interest rates, exchange rates, commodities prices, and other general market risks.

A typical reporting of VAR would be the following statement:

The 5% VaR of a portfolio is $2 million over a one-day period.

The following three points are important in understanding the concept of VaR:

– VaR can be measured in either currency units (in this example, the USD) or in percentage terms. In this example, if the portfolio value is $100 million, the VaR expressed in percentage terms would be 2% ($2 million/$100 million = 0.02).

– VaR is a minimum loss. This point cannot be emphasized enough. VaR is often mistakenly assumed to represent how much one can lose. If the question is, “how much can one lose?” there is only one answer: the entire portfolio. In a $100 million portfolio, assuming no leverage, the most one can lose is $100 million.

– A VaR statement references a time horizon: losses that would be expected to occur over a given period of time. In this example, that period of time is one day. (If VaR is measured on a daily basis, and a typical month has 20–22 business days, then 5% of the days equates to about one day per month.)

These are the three explicit elements of a VaR statement—the frequency of losses of a given minimum magnitude expressed either in currency or percentage terms. Thus, the VaR statement can be rephrased as follows: A loss of at least $2 million would be expected to occur about once every month.

A 5% VaR is often expressed as its complement—a 95% level of confidence. Commonly, the notion, the 5% VaR, is used, but we should be mindful that it does imply a 95% level of confidence.


J.P. Morgan Chairman Dennis Weatherstone demanded a simple report at the end of each day on how the firm’s position could change due to market risk. Morgan analysts came up with VAR as this measure and incorporated it into what became known as the 4:15 report, the report was given to Weatherstone daily at that time to summarize the day’s market events.

The Washington-based Group of Thirty, in a 1993 study headed by Weatherstone, Derivatives: Practices and Principles, recommended using VAR as a means of identifying a firm’s overall market risk. Since then, VAR has skyrocketed in popularity and there are few financial institutions which do not envisage making it part of their day-to-day reporting. In fact, many financial regulators, such as the European Economic and Monetary Union, require banks to report their VARs on a regular basis.

Because there are a variety of ways to calculate VAR, J.P. Morgan sought to make its method the industry standard. In 1994 it began giving away copies of RiskMetrics, a program it developed to calculate VAR. Morgan’s Internet web site includes a RiskMetrics section, where individuals or firms can download parts of the program. This initial move into the VAR market did not stop other investment banks from creating their own VAR calculation programs and peddling them to other financial organizations. Many of these programs differ from RiskMetrics, and while RiskMetrics is widely recognized in VAR measurement, the industry has yet to settle on a single calculation method.

Investment Strategy

An investment strategy is an approach to investment analysis and security selection. In the broadest sense, investment strategies are passive, active, or semiactive.

In a passive investment approach, portfolio composition does not react to changes in capital market expectations (passive means not reacting). For example, a portfolio indexed to the MSCI-Europe Index, an index representing European equity markets, might add or drop a holding in response to a change in the index composition but not in response to changes in capital market expectations concerning the security’s investment value. Indexing, a common passive approach to investing, refers to holding a portfolio of securities designed to replicate the returns on a specified index of securities. A second type of passive investing is a strict buy-and-hold strategy, such as a fixed, but non-indexed, portfolio of bonds to be held to maturity.

In contrast, with an active investment approach, a portfolio manager will respond to changing capital market expectations. Active management of a portfolio means that its holdings differ from the portfolio’s benchmark or comparison portfolio in an attempt to produce positive excess risk-adjusted returns, also known as positive alpha. Securities held in different-from-benchmark weights reflect expectations of the portfolio manager that differ from consensus expectations. If the portfolio manager’s differential expectations are also on average correct, active portfolio management may add value.

A third category, the semiactive, risk-controlled active, or enhanced index approach, seeks positive alpha while keeping tight control over risk relative to the portfolio’s benchmark. As an example, an index-tilt strategy seeks to track closely the risk of a securities index while adding a targeted amount of incremental value by tilting portfolio weightings in some direction that the manager expects to be profitable.

The Philosophical Difference

There is one huge philosophical difference between active and passive investors. Active investors believe that there are systematic mis-pricing in the market that can be found. Passive investors believe that, at least for the supermajority of people, they can not consistently identify mispricing in the market. Market prices are in equilibrium.

Modern Portfolio Theory

The modern portfolio theory (MPT) is a theoretical framework for constructing and analyzing a portfolio.

The ideas of diversification and risk–return trade-off were long known before the development of the Modern Portfolio Theory.

But these ideas leave several important questions unanswered. How should one measure the risk of an asset? What should be the quantitative trade-off between risk (properly measured) and expected return? One would think that risk would have something to do with the volatility of an asset’s returns, but this guess turns out to be only partly correct. When we mix assets into diversified portfolios, we need to consider the interplay among assets and the effect of diversification on the risk of the entire portfolio. Diversification means that many assets are held in the portfolio so that the exposure to any particular asset is limited. The effect of diversification on portfolio risk, the implications for the proper measurement of risk, and the risk–return relationship are the topics which have come to be known as modern portfolio theory.

The central idea of MPT is that financial assets should not be viewed in isolation, but each asset should be viewed as part of a portfolio and how in contributes to the portfolio risk and return.

A Brief History

Investing is about making choices of how to allocate wealth among different assets. Before the development of the MPT, investors analyzed each investment option individually. If one believes a company is going to perform well, he buys the stock. The concepts of diversification (e.g. Don’t put all your eggs into one basket) and risk–return trade-off (e.g. “You have to take more risks if you want more reward”) existed, but were not formalized.

In 1952, Harry Markowitz delineated what is now known as the modern portfolio theory. The most significant insight is that assets should not be selected based on their individual characteristics, but on their contribution to the entire portfolio (in terms of expected return and risk). This way, one can construct a portfolio with the same expected return but less risk compared to a portfolio constructed the old way.

The development of this theory brought two of its pioneers, Harry Markowitz and William Sharpe, Nobel Prizes.

How Does It Work

The MPT suggests that one can construct a portfolio of lower risk, while keeping the same return. How does it work exactly?

Under the MPT framework, investors care only about two things: expected return and risks, measured by variance. For a portfolio, the expected return of the portfolio is simply the weighted average of returns of each component. However, risk, measured by variances, is not a simple weighted average of component variances.

Because asset prices tend to move together, co-variances actually reduce the overall variances.

Asset prices have a tendency to move together. If you take into account these interactions, under certain assumptions, one can construct a portfolio of lower risk, while keeping the same return. People can do that because portfolio return is a simple weighted average of component returns. However,

It is important to keep in mind of the restrictions of this framework. Under the MPT, risk is measured by variances, extrapolated from historical data.


Under this framework, investors care about two things in a portfolio: expected return and risk. Investors like expected return and dislike risk. This is not true for all investors. For example, socially responsible investing (SRI) is a strategy that seeks social or environmental benefits in addition to financial rewards. (WSJ: BlackRock Plans to Block Walmart, Dick’s From Some Funds Over Guns) Therefore, MPT is not suitable for social investors.

Risk and expected return. MPT assumes that investors are risk averse, meaning that given two portfolios that offer the same expected return, investors will prefer the less risky one. Thus, an investor will take on increased risk only if compensated by higher expected returns.

Risk is measured about standard deviation.

Modern financial theory rests on two assumptions: (1) securities markets are very competitive and efficient (that is, relevant information about the companies is quickly and universally distributed and absorbed); (2) these markets are dominated by rational, risk-averse investors, who seek to maximize satisfaction from returns on their investments.

The first assumption presumes a financial market populated by highly sophisticated, well-informed buyers and sellers. The second assumption describes investors who care about wealth and prefer more to less. In addition, the hypothetical investors of modern financial theory demand a premium in the form of higher expected returns for the risks they assume.

Mean-Variance Analysis Variance Analysis

For individual investment options:

  • The expected return, or the mean return, is denoted by \mu_i
  • The variance, the measure of risk, is denoted by Var[R_i] = E[(R_i-u_i)^2] = \sigma_i^2
  • The standard deviation is simply the square room of the variance.

For a portfolio:

  • The expected return is the simple weighted average of individual returns \mu_i
  • The variance, the measure of risk, is denoted by Var[R_i] = E[(R_i-u_i)^2] = \sigma_i^2
  • The standard deviation is simply the square room of the variance.

Capital Asset Pricing Model

The capital asset pricing model (CAPM) is a theoretical framework predicts equilibrium expected returns on risky assets. CAPM is a one-factor model.


Here are the assumptions of the CAPM:

1. Individual behavior
a. Investors are rational, mean-variance optimizers.
b. Their common planning horizon is a single period.
c. Investors all use identical input lists, an assumption often termed homogeneous expectations. Homogeneous expectations are consistent with the assumption that all relevant information is publicly available.

2. Market structure
a. All assets are publicly held and trade on public exchanges.
b. Investors can borrow or lend at a common risk-free rate, and they can take short positions on traded securities.
c. No taxes.
d. No transaction costs.

The CAPM is based on two sets of assumptions, listed above. The first set pertains to investor behavior and allows us to assume that investors are alike in most important ways, specifically that they are all mean-variance optimizers with a common time horizon and a common set of information reflected in their use of an identical input list. The second set of assumptions pertains to the market setting, asserting that markets are well-functioning with few impediments to trading. Even a cursory consideration of these assumptions reveals that they are fairly strong, and one may justifiably wonder whether a theory derived from them will withstand empirical tests. Therefore, we will devote considerable attention later in the chapter to how the predictions of the model may change when one or more of these restrictive assumptions are relaxed.

Still, the simple version of the CAPM is a good place to start. While the appropriate quantification of risk and the prediction for the exact risk–return trade-off may differ across more sophisticated variants of the model, the central implication of the basic model, that risk premia will be proportional to exposure to systematic risk and independent of firm-specific risk, remains generally valid in its extensions. In part because of this commonality, the simple CAPM remains in wide use despite its empirical shortcomings.

Therefore, we begin by supposing that all investors optimize their portfolios á la Markowitz. That is, each investor uses an input list (expected returns and covariance matrix) to draw an efficient frontier employing all available risky assets and identifies an efficient risky portfolio, P, by drawing the tangent CAL (capital allocation line) to the frontier as in Figure 9.1, Panel A. As a result, each investor holds securities in the investable universe with weights arrived at by the Markowitz optimization process. Notice that this framework employs Assumptions 1(a) (investors are all mean-variance optimizers), 2(a) (all assets trade and therefore can be held in investors’ portfolios), and 2(b) (investors can borrow or lend at the risk-free rate and therefore can select portfolios from the capital allocation line of the tangency portfolio).

The CAPM asks what would happen if all investors shared an identical investable universe and used the same input list to draw their efficient frontiers. The use of a common input list obviously requires Assumption 1(c), but notice that it also relies on Assumption 1(b), that each investor is optimizing for a common investment horizon. It also implicitly assumes that investor choices will not be affected by differences in tax rates or trading costs that could affect net rates of return (Assumptions 2[c] and 2[d]).

Not surprisingly in light of these assumptions, investors would calculate identical efficient frontiers of risky assets. Facing the same risk-free rate (Assumption 2[b]), they would then draw an identical tangent CAL and naturally all would arrive at the same risky portfolio, P. All investors therefore would choose the same set of weights for each risky asset. What must be these weights?

A key insight of the CAPM is this: Because the market portfolio is the aggregation of all of these identical risky portfolios, it too will have the same weights. (Notice that this conclusion relies on Assumption 2[a] because it requires that all assets can be traded and included in investors’ portfolios.) Therefore, if all investors choose the same risky portfolio, it must be the market portfolio, that is, the value-weighted portfolio of all assets in the investable universe. We conclude that the capital allocation line based on each investor’s optimal risky portfolio will in fact also be the capital market line, as depicted in Figure 9.1, Panel B. This implication will allow us to say much about the risk–return trade-off.


Harry Markowitz laid down the foundation of modern portfolio management in 1952. The CAPM was published 12 years later in articles by William Sharpe, John Lintner, and Jan Mossin. The time for this gestation indicates that the leap from Markowitz’s portfolio selection model to the CAPM is not trivial.

Risks in CAMP

In the CAPM, total risk can be broken into two components: systematic risk and unsystematic risk. Systematic risk is the portion of risk that is related to the market and that cannot be diversified away. Unsystematic risk is non-market risk, risk that is idiosyncratic and that can be diversified away. Diversified investors can demand a risk premium for taking systematic risk, but not unsystematic risk.

In the CAPM, investors are only rewarded for taking systematic risk.

The Model

Required rate of return = risk-free rate + risk premium

The formula is:

required return = risk-free rate + beta * (expected market return – risk-free rate)

Corporate Income Tax

The final large non-operating item is the tax expense. This is often a large amount that affects profit substantially. Differences in tax rates can be an important driver of value. Generally, there are three types of tax rates:

The statutory tax rate, which is the tax rate applying to what is considered to be a company’s domestic tax base.

The effective tax rate, which is calculated as the reported tax amount on the income statement divided by the pre-tax income.

The cash tax rate, which is the tax actually paid (cash tax) divided by pre-tax income.

Differences between cash taxes and reported taxes typically result from timing differences between accounting and tax calculations and are reflected as a deferred tax asset or a deferred tax liability.

In forecasting tax expense and cash taxes, respectively, the effective tax rate and cash tax rate are key. A good understanding of their operational drivers and the financial structure of a company is useful in forecasting these tax rates.

Differences between the statutory tax rate and the effective tax rate can arise for many reasons. Tax credits, withholding tax on dividends, adjustments to previous years, and expenses not deductible for tax purposes are among the reasons for differences. Effective tax rates can differ when companies are active outside the country in which they are domiciled. The effective tax rate becomes a blend of the different tax rates of the countries in which the activities take place in relation to the profit generated in each country. If a company reports a high profit in a country with a high tax rate and a low profit in a country with a low tax rate, the effective tax rate will be the weighted average of the rates, and higher than the simple average tax rate of both countries. In some cases, companies have also been able to minimize their taxes by using special purposes entities. For example, some companies create specialized financing and holding companies to minimize the amount of taxable profit reported in high tax rate countries. Although such actions could reduce the effective tax rate substantially, they also create risks if, for example, tax laws change. In general, an effective tax rate that is consistently lower than statutory rates or the effective tax rates reported by competitors may warrant additional attention when forecasting future tax expenses. The notes on the financial statements should disclose other types of items, some of which could contribute to a temporarily high or low effective tax rate. The cash tax rate is used for forecasting cash flows and the effective tax rate is relevant for projecting earnings on the income statement. In developing an estimated tax rate for forecasts, analysts should adjust for any one-time events. If the income from equity method investees is a substantial part of pre-tax income and also a volatile component of it, the effective tax rate excluding this amount is likely to be a better estimate for the future tax costs for a company. The tax impact from income from participations is disclosed in the notes on the financial statements.

Often, a good starting point for estimating future tax expense is a tax rate based on normalized operating income, before the results from associates and special items. This normalized tax rate should be a good indication of the future tax expense, adjusted for special items, in an analyst’s earnings model.

By building a model, the effective tax amount can be found in the profit and loss projections and the cash tax amount on the cash flow statement.3 The reconciliation between the profit and loss tax amount and the cash flow tax figures should be the change in the deferred tax asset or liability. (Institute 128-129)

Source: Institute, CFA. 2018 CFA Program Level II Volume 4 Equity. CFA Institute, 07/2017. VitalBook file.

Return on Equity

Return on equity (ROE) measures the rate of return on the money invested by common stock owners and retained by the company thanks to previous profitable years. It demonstrates a company’s ability to generate profits from shareholders’ equity (also known as net assets or assets minus liabilities).

ROE shows how well a company uses investment funds to generate growth. Return on equity is useful for comparing the profitability of companies within a sector or industry.

Investors generally are interested in company’s that have high, increasing returns on equity.

Return on Equity = Net Income / Average Common Shareholder’s Equity