Pricing of financial assets: principles of modeling - Investments

Financial Asset Pricing: Modeling Principles

Two fundamentally different approaches are proposed for modeling the pricing of risky financial assets (Figure 16.1, Table 16.1):

1) absolute an approach that takes into account the interests and behavior of investors, their consumer preferences. The key parameter of the model is the function of investor consumption in the market, showing how the profitability of a financial asset is related to the value of its final consumption;

2) relative an approach in which the price of the risky asset under consideration (the interest rate on it) is derived through comparisons with a known price for other assets. A vivid example of this approach is the arbitrage model for determining the fair price of the Black-Scholes option.

Consumer models (absolute pricing) of the pricing of financial assets treat risk as the contribution of changes in the prices of financial assets to the uncertainty of the level of consumption.

The intermediate state between the first and the second approach is occupied by models (including the popular single-factor equilibrium model CPRM), which link asset prices to the level of monetary well-being. Note that the preferences of the investor are not always correctly expressed through changes in cash receipts, but the intermediate approach realizes this substitution. Market risk (changing the phases of the business cycle, shifts in industry dynamics) makes the investor's wealth or monetary wealth uncertain. It is this feature that is fixed in intermediate models (including the well-known model of CAPM).

Intermediate models of formation of prices for financial assets link the changes in the prices of these assets (yields) with the uncertainty of the investor's income received from the investment portfolio.

The characteristics of the actual consumption are not affected. In intermediate models, monetary wealth (wealth) is an intermediate link between the yield of financial assets and final consumption. In classical consumption patterns (absolute approach), this intermediate is omitted.

An example of a relative approach can be the market capitalized pricing model (MCPM), proposed by practitioners Tony Yeah and James McNulty and adjusted by American professors William Shultz and Michael Lyubatkin. MCPR focuses only on changes in the prices of financial assets, applying optional approaches (future volatility of the yield is estimated by the options market). In the formula for calculating the required yield, the risk premium is added to the company's borrowing rate.

Linking Investor Behavior to Financial Assets in the Market

Fig. 16.1. Linking investor behavior to financial assets in the marketplace

Intermediate consumer models were the most successful in practical application. An example of such a model is the widely used in practice model of portfolio construction CAPM.

In the standard version of CAPM , investors maximize the utility function, depending on the mean and variance of the expected return on their investment portfolios ( mean-variance behavior, MVB).

The value of the risk premium in CAPM is defined as the product of the risk price by the amount of risk accepted by the investor. The peculiarity of investor's consideration in such constructions is portfolio ownership of capital, i.e. the assumption of a diversification of the capital between various risky assets of the market. As a result of diversification, part of the overall risk is leveled and the investor actually assumes only non-diversifiable risk and requires compensation for it. This risk is called systematic.

Systematic risk (market, beta risk) is part of the overall investment risk, which is not eliminated by capital diversification (the formation of an asset portfolio). The part of the risk of the asset that has been eliminated by diversification has been called the unique (specific, unsystematic) risk.

Table 16.1

Determining asset price and increased risk return

Relative Definition

General assumptions about the utility function are an intermediate approach. Instead of consumption - welfare

Absolute Definition

Arbitration methods (for example, the Black-Scholes model, MCPM)


Sharp-Lintner model

Consumption patterns (eg, consumption-based CAPM)

The consumption-based pricing model for long-term financial assets ( consumption-based CAPM, SW-CAPM ), or consumer CAPM, is based on the premise that investors maximize consumption at comparison of financial assets. The expected risk compensation for a risk-free asset (risk premium) should be proportional to the covariance of the risk asset's yield and the marginal utility of consumption in the next period of time. The required yield is equal to the sum of risk-free yield and risk premium. The value of the risk premium is defined as the product of the risk price by the amount of risk accepted by the investor. This is the general position that is used in consumer models.

The risk price in the SV-CAPM model is defined as the inverse of the expected marginal utility of future consumption (the higher the expected value of future consumption, the higher the risk price), and the amount of risk - as taken with a negative sign covariance of the return on the risky asset and the marginal utility of future consumption ( consumption beta ) (ie, the higher the absolute value of covariance - the dependence of future consumption on the yield of the risky asset, the greater the risk and the higher the risk premium ).

Therefore, an asset whose yield has high covariance with consumption (high modulus of covariance with marginal utility of consumption) will require a high risk premium, since the lower the actual yield, the lower the consumption. The investor (consumer) in the framework of the CB-CAPM model when compiling an effective asset portfolio is irrelevant to the risk profile (variation of yield) of each individual asset, it is primarily interested in the covariance of the asset's yield and future consumption. An interesting consequence of these provisions for the United States market - investors should form a portfolio mainly from foreign assets.

Testing the consumer CAPM on statistical data on national economies (time series of consumption and stock market indicators, and correspondingly the standard deviation of the risk rate of return and the growth rate of consumption) revealed a paradox: if you enter the covariance values ​​of the risk asset's yield and the growth rate of consumption, as well as assess the measure of relative disinclination for Risk Arrow-Pratt, then the magnitude of the risk premium is inexplicably high. Since the early 1990s. at the model level, options for a more correct linking of consumer behavior and risk premiums are sought (for example, modeling the behavior of a representative consumer using the approach of unexpected utility, introducing incompleteness of the financial market and heterogeneity of consumers, limiting the rationality of some investors, accounting habits in consumption).

Based on the CAPM model, first proposed by Stanford University professor William Sharpe, systematic risk affects a large number of investment assets, including financial assets, since macro actions are behind it. Non-systematic risk is inherent only in this asset (for example, the quality of management in the company - the issuer of securities). In practice, the delineation of risks by macrofactors and specific linkage to companies is not so obvious and depends on the analyst's interpretations. One and the same event can be considered as global for the market, having consequences for all participants, and can be as private (specific). Typical examples for the United States market are the Yukos case 2004-2005 or "TNK-BP case", "Mechel criticism" 2008. If you interpret tax claims against Yukos or the criticism of the "double prices" Mechel, non-interference in the dispute of TNK-BP shareholders as another step in changing the "rules of the game" in the United States market (redistribution of property, "squeezing out" of foreign capital, transition to oligarchic enforcement), then volatility in the prices of shares in the United States market (a sharp drop in the index in the winter and spring of 2004 or in the spring of 2008) should be interpreted as a manifestation of systematic risk. If you consider criticism from leading politicians, tax claims and litigations as a "dispute of economic entities", then such risks should be treated as specific, ie. specific for only one or two companies.

The required yield in portfolio designs is equal to the sum of risk-free yield and risk premium. Since the equilibrium state of the market is considered, the required yield is equal to the expected yield. The risk price is defined as exceeding the expected weighted return of the portfolio over the risk-free rate of return, this is the market risk premium (MRP) or the risk premium of the mid-risk company. The "Risk Quantity" is entered through a special indicator, the measure of systematic risk is the beta-asset coefficient, which is calculated in theoretical constructions as the ratio of the covariance of the risk asset's yield and the return of the market portfolio to the variance of the market's yield.

Portfolio models are built on general assumptions about the investor's utility function and treat the risk of welfare change in terms of statistical risk assessments of a financial asset in which money is invested. A typical example is an action and a statistical evaluation of the benefits and risks on it: "expected yield and standard deviation" (mean-variance, MVB). The expected return is calculated as the mathematical expectation of the possible outcomes and their probability, or as the arithmetic mean of the past data. The risk is estimated through the standard deviation of the yield from the expected value.

The utility function of the investor (U) is determined by the average (μ) and variance of the expected return on the investment portfolio, i.e.

The risk of an individual asset r is measured by the standard deviation of its yield from the average yield of this (σ;) asset.

This approach naturally follows from the theory of portfolio investment optimization, developed by Harry Markowitz, who received for his work in this field the Nobel Prize in Economics.

In the case where the asset r is one of many entering into a fully diversified portfolio, the degree of risk of this asset is measured by the covariance of that asset and the market portfolio of σM.

where E - is the mathematical expectation sign.

Since the covariance value is unlimited and depends on the scale of the data, its interpretation is ambiguous. Boles correct measure of risk can be obtained by dividing the covariance by the product of the standard deviation of the yield of a particular asset and the standard deviation of the yield of the market portfolio. As a result of standardization, we obtain the correlation coefficient between the asset's yield and the market's yield. If we divide the covariance into a variance of the market portfolio, we obtain an estimate of the elasticity of the asset to the portfolio, or the beta coefficient for the asset i.

Nobel laureate J. Tobin proved that if there is an opportunity to invest in risk-free instruments (and raise money at a risk-free rate), an optimal diversified risky portfolio of assets for an investor will be a market portfolio in which the weights of assets will coincide with weights emerging in the market.

Market portfolio - a portfolio that mimics the market, i.e. it presents all the assets on the market and their weights correspond to the weights of assets on the market.

In a market that allows diversifying capital, a rational investor chooses from the following opportunities:

1) risk-free investment with a return of k j,

2) the market portfolio as an optimal portfolio with yield

3) market portfolio of risky assets + risky asset (project, company), which can change the risk of final investment.

Portfolio models recommend that when considering an investment option, it is not so much an assessment of the risk of the asset itself that is important as consideration of how investing in this asset will affect the risk assessment of a diversified portfolio (increase the risk, not change or reduce it).

The impact of the method of setting a risk-free rate on the required return on equity (COE)

1. Yield on short-term government securities. When choosing a risk-free rate at the level of short-term government bonds, depending on the state's financial policy, the cost of own capital (COE) may turn out to be too high (with high inflation and tight monetary regulation) and undervalued (with low inflation and soft money regulation).

2. The yield on long-term (10-year, 30-year) government securities (Treasury bonds).

thematic pictures

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