Modeling of the market premium for risk (MRP): from...

Modeling of the market premium for risk (MRP): from observed values ​​to imputed estimates

The rule of all models based on the design of CAPM and multifactorial models - the market premium for risk is not fixed by the current values ​​of the difference between the market yield (more precisely, in practice, the stock index) and the risk-free rate. >

Market risk premium ( MRP) - is the premium over the rate of return on risk-free assets that investors expect from medium-risk investments in a given national capital market. Since we are talking about expectations, the fixation of the current observable value is not correct. And within the framework of the historical, and in the framework of the prognostic approach, averaged estimates are constructed. The market is looking for an adequate premium indicator (based on historical observations of a time interval similar to the forecast horizon) or the market premium is modeled taking into account market expectations.

Below are the historical values ​​of the country market risk premiums in developed markets, calculated by the professors of the London Business School E. Dimson and P. Marsh (Dimson & Marsh ) na long time interval (1900-2005), %.

Germany - 6,5;

Australia - 6.2;

Japan - 6.1;

South Africa - 5.4;

Italy - 5;

US - 5;

Britain - 4.6,

Belgium - 2.8;

Spain - 2.5;

Denmark - 2;

The global market is 4.8.

Important elements of the justification MRP: 1) should analyze the profitability of a well-diversified portfolio reflecting the systematic risk of this capital market; 2) the level of the risk-free rate should be set correctly ( MRP ) either within the historical or prognostic approach. Mixing estimates is not allowed.

An adequate measure of market yield and risk premiums within the historical approach is sought through statistical analysis on long time intervals. The expected market premium is equated to the averaged past observable values. It can be used as a method of monitoring the behavior of individual investors (for example, large portfolio funds) and the method of realized profitability.

The method of realized profitability is based on the need to analyze, based on past market data, the average difference between the profitability of portfolios of shares and the yield of government securities (bonds).

Market premium for risk = Average return on portfolio of shares - Average yield of a risk-free asset.

The algorithms used by analysts vary:

1) by the selected index (for example, NYSE - New York Stock Exchange; Standard and Poor's Composite Index; combined index of Subatstance Datastream, including CRSP, Standard and Poor's Composite Index and Morgan Stanley's Capital World ) and an indicator characterizing the yield of risk-free instruments;

2) on the technique of calculating the average values ​​for the profitability of the index of shares and risk-free rates. The principal choice concerns the use of arithmetic or geometric means. In calculating the formula of the average geometric profitability, the result is usually lower;

3) for the time period of observations. For example, in a number of methodological guidelines of investment companies, one can find a provision that the duration of the observation horizon should not exceed the so-called memory of the investor-the active period during which the investor invests money in shares. For developed markets, such a period for an investor starts at age 30 and ends at 70 years, i.е. does not exceed an average of 40 years. There are analysts and supporters of the longest time horizons of analysis. Their calculations for the US market are often built on a stretch since 1900. You can find the calculations of the Ibbotson agency with a premium reflected for more than 200 years (since 1798). Naturally, bonuses vary significantly depending on the chosen segment and the last year of analysis;

4) for the selected phases of the business cycle. Often bonuses are fixed with reference to a certain phase of the business cycle (most often - to the rise). An example of such processing of statistical information is given in Table. 17.1. The idea of ​​such calculations is to take into account the stage of the business cycle when setting the parameters CAPM.

Table 17.1

Market risk premium in the US market for selected key periods

Characteristics of the beginning of the period count

Watching period, years

Average premium,%










The average value before 2001


Business cycle peak



Business cycle peak



Average for the period 1962-2001


Average for all observation periods


Long-term market risk premium



In Fig. 17.2 shows the historical return of the United States stock market from 1994 to 2008, represented by the index MSCI Russia. The average historic premium of this index is 12.43% (as of March 23, 2009).


In Table. 17.2 presents an algorithm for calculating market risk premiums for the US market at different time horizons. The table shows the differences in the estimates obtained depending on the chosen method for calculating average values ​​and the base rate of risk-free returns (short-term (three-monthly) bills or long-term bonds).

Historical profitability of the MSCI Russia index

Fig. 17.2. Historical profitability of the MSCI Russia index

Table 17.2

Historical annual returns and risk premiums for the US market



Annual Return,%

Risk premium,%



The US government G. Bills

state bonds T. Bonds


T. Bills

Shares T. Bonds

Arithmetic mean



















Geometric mean



















Historically in the XX century. the excess of the yield of the average-risk (market) portfolio of stocks over government securities was on average 6%. The real average profitability of risk-free investment instruments was 0.75% with a return on equity in real terms on the equity investment of 6.95%, which gives a premium equal to 6.2% for the 1889-1978 period. and 6% until the end of the 2000s. From an economic point of view, the premium should be lower than this estimated historical mean. In academic literature, such excess over a justified level is called the "risk puzzle riddle" (premium puzzle). Consideration of the yield of government securities as an undervalued amount compared to the return on the portfolio of shares relative to the risk taking by investors is treated as a riddle of the risk-free rate of return ( Risk-Free Rate Puzzle).

On the US Central Bank's website, you can find data on government (risk-free) short-term and long-term bonds and calculate the yield spread as an indicator of the macroeconomic situation in the country (Figure 17.4).

Yield of US government securities in the dynamics

Fig. 17.4. Yield of US government securities in the dynamics

thematic pictures

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