Foundational Theories and Techniques for Risk Management, A Guide for Professional Risk Managers in Financial Services - Part I - Finance Theory

By Professional Risk Managers' International Ass

The definitive guide for any professional risk management in the financial services industry, Foundational Theories and Techniques for Risk Management provides a complete reference for financial theory and application, financial instruments, and the financial markets.

Part I provides details for financial risk managers managing

• Language: English
• Publisher: Professional Risk Managers' International Ass
• Release date: Sep 13, 2022
• ISBN: 9798986164656

Book preview

Part I - Finance Theory

This part of the textbook contains material aligned with the first major domain of the PRM Designation syllabus. The syllabus section is listed here for your reference.

Visit the PRM Resources and Case Studies page at www.prmia.org/PRM for a current list of reading materials aligned to the PRM Desigation program.

I. FINANCE THEORY

A. Arbitrage Pricing Theory

1. Describe how arbitrage pricing theory can be used for decision-making.

B. The CAPM and Multifactor Models

1. Outline the components of the Capital Asset Pricing Model (CAPM), including the risk premium, systematic and idiosyncratic risk, beta, security market line, and its underlying assumptions.

2. Compare and contrast single-factor (e.g., Capital Asset Pricing Model (CAPM)) and multifactor models.

3. Compare and contrast the capital asset pricing model and the single-index model.

C. Capital Structures

1. Outline the factors to be considered to determine the capital structure of a firm, in particular agency costs, taxes, and leverage.

D. Mean-Variance Portfolio Theory

1. Relate mean-variance portfolio theory to asset allocation decisions, with risky assets and a risk-free asset (e.g., asset correlation, efficient frontier, market portfolio, capital allocation line, capital markets line, dominated portfolio, and separation principal).

2. Describe the axioms and assumptions of utility theory with respect to expected return and risk, and describe its application to the mean-variance portfolio theory.

E. Performance Measures

1. Calculate the Sharpe ratio and Jensen’s Alpha, and interpret the results.

2. Identify and describe risk adjusted performance measures, in particular RAROC, RORAC, RARORAC, RoVaR, and the Treynor, Information, and Omega ratios, and the Sortino and Kappa indices.

3. Calculate value-at-risk (VaR) for a portfolio.

F. The Term Structure of Interest Rates

1. Define the term structure of interest rates and demonstrate how to construct a yield curve from observable bond pricing.

2. Describe the standard theories used to explain the observed shape of the yield curve: a) pure expectations theory, b) liquidity preference theory, c) preferred habitat theory, and 4) market segmentation theory.

3. Understand the concepts of duration and convexity, and describe the impact of an embedded call or put on the duration convexity and the price of a bond.

4. Compare and contrast the Ho-Lee, Hull-White, and Black-Derman- Toy models.

G. Regulatory Frameworks

1. Compare and contrast capital management and regulatory capital and relate capital management to solvency and Basel.

2. Describe the application of an internal capital adequacy process to achieve efficient capital allocation.

H. Trade Terminology

1. Illustrate the best choice of available options to execute trading for optimization of return.

2. Describe the lifecycle of a trade and distinguish between dealing and settlement.

Chapter 1: The CAPM and Multifactor Models

Authors: Keith Cuthbertson and Dirk Nitzsche

The capital asset pricing model (CAPM) is central to determining the average return required by investors for holding a stock (or portfolio of stocks). It determines the cost of equity capital for a firm raising funds when issuing more shares (i.e., a rights issue or seasoned issue). The cost of raising equity capital given by the CAPM equation is used, along with the cost of issuing debt, to give the weighted average cost of capital (WACC). The WACC is then used to discount future cash flows expected from risky physical investment projects to determine the net present value of the project—this aids in the decision as to whether the project is viable. The CAPM is useful for issues in corporate finance, but what use is the CAPM to an investor? In the CAPM, the riskiness of a stock when held as part of a well-diversified portfolio is determined by how much the stock adds to incremental portfolio risk, measured by the stock’s beta. When the market as a whole goes up or down by 10% and the return on a stock then moves by plus or minus 20%, the beta of the stock equals 2. A stock with a beta of 0.5 is, therefore, less risky (i.e., volatile) than one with a beta of 2. The CAPM implies that the required return on a stock depends directly on its beta risk—the higher the beta, the higher the return required by the investor.

You can reduce the overall riskiness of a portfolio of stocks by selling high-beta stocks and using the funds to purchase low-beta stocks. However, you might incur high transaction costs (e.g., bid-ask spreads and brokerage fees, especially if you just wanted to change your risk exposure for a short period [3 to 6 months]). Once you know the beta of your stock portfolio, we see how this risk reduction can be achieved more easily and cheaply by using stock index futures. Every financial institution in developed economies must hold capital (i.e., the market value of stocks issued plus any retained profits) against the riskiness of its assets and liabilities. The beta of a stock can be used to calculate the financial intermediaries’ dollar exposure to risk—called its value-at-risk on its stock portfolio—in domestic and foreign assets. Therefore, the CAPM is a key tool to analyzing and solving a broad range of important practical problems.

In this chapter, we analyze the basic, one-period CAPM, the single-index model, and multifactor models such as the arbitrage pricing theory (APT). The CAPM is used broadly in finance literature to determine the average return required by shareholders to hold a stock. It also provides a way of measuring the abnormal performance, or alpha, of a portfolio of stocks. We present a brief account of the APT, which relates the expected return on a security to a set of variables called factors, which could include market-wide effects due to interest rates, exchange rates, etc. Throughout this chapter, we consider that the only risky assets are equities stocks, though strictly the model applies to choices among all risky assets (e.g., stocks, bonds, real estate, commodities, etc.).

Overview

CAPM provides a model of the expected or required return on any individual risky asset. It predicts that the expected return on a risky asset, ERi, consists of the risk-free rate r plus a risk premium, rpi:

DESCRIBE HERE

(1)

where the risk premium, rpi = βi(ERm – r) and βi = Cov(Ri, Rm)/Var(Rm). The risk premium on the stock depends on (ERm – r) and the stock’s beta (βi)—the higher the beta, the higher the required return by investors to willingly hold the stock. Term (ERm – r) is the market risk premium since it is the average return on the market portfolio above the risk-free rate. It is an additional payment for holding the risky market portfolio rather than the risk-free asset. (ERm – r) > 0, otherwise no risk-averse investor would hold the market portfolio of risky assets when she could earn more by investing all her wealth in the risk-free asset. If ex-post returns (i.e., historical average returns) correctly measure the ex-ante expected return ERi, we can think of the CAPM as explaining the average monthly return (over, say, a 240-month period) for security i. The definition of security i’s beta, βi, indicates that it depends positively on the covariance between the return on security i and the market portfolio, Cov(Ri, Rm).

Returns on individual stocks tend to move in the same direction as the market return. Hence, in general, Cov(Ri, Rm) ≥ 0 which implies βi ≥ 0 for most stocks. The CAPM predicts that stocks that have a large positive covariance (correlation) with the market return (βi > 0) will earn a high average return. This is because the addition of this high-correlation stock to the portfolio increases overall portfolio variance (risk)—to compensate the investor for this undesirable effect, she has to be compensated with a higher expected (average) return on this high-beta stock.

CAPM predicts that a stock whose return has a zero covariance (correlation) with the market return will be held willingly by investors even though it has an expected return that equals the risk-free rate (put βi = 0 in equation [I.A.4.1]). CAPM also allows one to assess the relative volatility of the (expected) returns on individual stocks based on their βi values. Stocks for which βi = 1 have a return that is expected to move one-for-one with the market portfolio (i.e., ERi = ERm), and are termed neutral stocks. If βi > 1, the stock is an aggressive stock since, on average, it moves more than changes in the expected market return (either up or down). Conversely, defensive stocks have βi < 1. Therefore, investors can use betas to rank the relative safety of various stocks and combine stocks with different betas to obtain a desired beta for a portfolio.

Capital Asset Pricing Model

CAPM is a logical consequence of mean-variance portfolio theory. It assumes:

All investors have homogeneous expectations.

Investors choose their risky asset proportions by maximizing the Sharpe ratio.

Investors can borrow or lend unlimited amounts at the risk-free rate.

The market is always in equilibrium.

It provides an equation to determine the return required by investors to hold any particular risky asset willingly, as part of a well-diversified portfolio:

Required return on asset i = Risk-free rate + Risk premium

or in symbols:

DESCRIBE HERE

(2)

where βi is the asset’s beta. CAPM implies that expected return on a stock is independent of the stock’s own variance (volatility).

Let us see how CAPM arises from portfolio theory. For the efficient frontier to be the same for all investors, they must have homogeneous expectations about the underlying market variables ERi, σi², and σij. With homogeneous expectations, all investors hold all the risky assets in the same proportions, regardless of the investor’s level of risk tolerance (or risk aversion). This portfolio is known as the market portfolio. Since all n assets are held in the market portfolio, there is also a set of equilibrium expected returns ERi (for each of the n assets). The CAPM equation represents the equilibrium return for asset i and considers that the contribution of asset i to overall portfolio variance depends crucially on the covariance between Ri and the market return Rm, that is, the asset’s beta.

To understand CAPM intuitively, investors must ask: "Would I be willing to hold only the single risky security A if the standard deviation of its return σA = 60% per annum (p.a.) and it had a rather low average return of only 5.8% per annum?" If this is the only security the investor holds, they would most likely want a higher average return than 5.8%, given its high individual risk of 60%. (Here, this single security constitutes the total portfolio risk). Now suppose the investor already holds a number of risky assets, and they are considering adding A to their existing portfolio. Also assume that A’s return has a rather low covariance (correlation) with the asset returns in your existing portfolio and, hence, has a low covariance with the market return. The low covariance implies that if the investor includes A in their portfolio, the overall increase in portfolio risk will be very small. Hence, the incremental risk of adding A to their existing portfolio is small; therefore, they might be willing to hold A even though it has a low average return of 5.8%. If one holds a diversified portfolio, the incremental contribution of A to overall portfolio risk is important to determining the average return required to add asset A willingly to the existing portfolio.

If A has a low covariance with the market return, it will have a low beta since βi = Cov(Ri , Rm)/Var(Rm). Suppose A has a beta of 0.1, the average excess return on the market is (ERm – r) = 8% p.a. and the risk-free rate r = 5% p.a. Then the required average return, according to CAPM, would be exactly 5.8% p.a., so A earns a return that just compensates for its incremental contribution to overall portfolio risk (though its stand-alone risk is very high; σA = 60%).

The reason stock A earns only 5.8% even though it has a high standard deviation of 60% is that much of the risk of this stock can be diversified away at a low cost by including this stock along with other stocks in a diversified portfolio. Since most of the variability of 60% in the return of stock A is removed by including stock A in the existing portfolio, an investor should receive no reward (i.e., average return) based on stock A’s high individual standard deviation. CAPM implies that an investor should only receive a reward (i.e., a high average return) based on how much stock A contributes to the (incremental) risk of the entire portfolio. The contribution of stock A to the overall portfolio risk depends on covariance/correlation and, hence, on stock A’s beta, not on stock A’s volatility.

Estimating Beta

An estimate of an asset’s beta can be obtained using an ordinary least-squares time series regression:

DESCRIBE HERE

(3)

where βi = Cov(Ri, Rm)/σm² is the formula for the ordinary least-squares estimate of the beta. For example, if the monthly return on asset i, in excess of the risk-free rate (Ri – r) over, say, the last 120 months, is regressed on the monthly excess return on the market (Rm – r)t, the slope gives an estimate of βi. In practice, an aggregate stock index such as the S&P 500 or the FT All Share Index is used as a measure of the return on the market portfolio Rm (for U.S. and U.K. investors, respectively). If CAPM is the correct model of equilibrium returns, we expect the estimate of the intercept αi to be close to zero.

Beta and Systematic Risk

CAPM predicts that only the covariance of returns between asset i and the market portfolio influences the average (excess) return on asset i. No additional variables such as the dividend price ratio, the size of the firm, or the price/earnings ratio should influence average (excess) returns on a stock. All contributions to the risk of asset i, when held as part of a diversified portfolio, are encapsulated in its beta. The beta of a security represents the part of risk that cannot be diversified away. Hence, beta represents a stock’s market risk and is the only source of risk that contributes to the (average) excess return of asset i.

Of course, an individual investor might choose not to use the full mean-variance optimization procedure when determining portfolio allocation. However, beta might still be useful. For example, if an investor requires a portfolio with a specific beta (e.g., an aggressive portfolio with Βp > 1), this can be constructed easily by combining securities with different betas in different proportions:

DESCRIBE HERE

(4)

where the weights wi = ($s in asset-i)/(total $s in all assets). The wi here do not represent the optimum weights given by mean-variance portfolio theory, but simply those proportions chosen by this particular investor.

Security Market Line

CAPM can be rearranged and expressed in terms of the security market line (SML). Suppose that the historic average value of the market risk premium (Rm – r) is 8% p.a. and the risk-free rate is r = 5%. CAPM gives the average required return for investors willing to hold asset i:

DESCRIBE HERE

(5)

If CAPM is correct in anticipating that there should be a linear relationship between average returns on a set of stocks Ri and each stock’s βi, then this linear relationship is known as the security market line.

Figure I.1. Security Market Line

DESCRIBE HERE

The SML (Figure I.1.) graphs average returns Ri against their betas βi, and if CAPM is correct, then the historic (average) security returns for all assets should lie somewhere on the SML.¹ The average return given by SML is the return required by the investor to willingly hold this stock (as part of her diversified portfolio) to compensate for its beta risk.

The difference between the average, historic (excess) return (Ri – r) and the required return is known as the abnormal return, or alpha, of the stock.

DESCRIBE HERE

(6)

= Actual average return – CAPM required return

CAPM/SML implies that all assets should only earn the required return and, hence, should have a zero alpha. A positive alpha implies that historically, the asset earned a positive, abnormal return; a negative alpha implies a negative, abnormal return.

SML can be used to try to pick underpriced and overpriced stocks. To see this, consider security S (Figure I.1.) with a beta of 0.5. This has a historic return below the required return given by the SML and, hence, has a negative alpha. You could duplicate the beta of security S by creating a portfolio X with 50% (of your own funds) in the safe asset (with β = 0) and 50% in another stock M with β = 1. The beta of portfolio X is then βP = 0.5 × 0 + 0.5 × 1 = 0.5. However, this synthetic portfolio X would lie on the SML (at P)—it has a higher expected return than S but the same beta risk. Hence, S would be an inferior stock, and would be sold (or short-sold) since its actual return is less than the required return given by the SML. If S is sold by many investors, its current price would fall and one could buy back the stock at a lower price and make a speculative profit. However, a lower current price for S implies a higher expected return in the future, so S moves towards P. A stock like S that has a negative, historic abnormal return (alpha) is mispriced and its market price will fall, which will cause it to eventually be priced correctly; its average return will end up on the SML. If this happens very speedily, the market is efficient and obeys the efficient-markets hypothesis.

Alternatively, consider a security like Q also with βi = 0.5 (Figure I.1.), but which currently has a higher average return than the required return indicated by the SML; it has a positive alpha. An investor should purchase Q since it earned a higher, historic return than its required return (due to its beta risk). Securities like Q and S are currently mispriced (i.e., they are not on the SML), and a speculator might short-sell S and use the funds to purchase Q. If the mispricing is corrected, the price of Q will rise as everyone seeks to purchase it because of its current high (abnormal) average return. Conversely, everyone will also seek to short-sell S, so its price falls in the market. If a portfolio manager spotted this mispricing first and executed their trades before everyone else, they would earn a handsome profit from this mispricing. A long-short portfolio does not change in value if the market