Statistical Measures for Risk
Standard Deviation 1) A measure of the dispersion of a set of data from its mean. The more spread apart the data, the higher the deviation. Standard Standard deviation is calculated as the square root of variance . 2) In finance, standard standard deviation deviation is applied to the annual rate of return return of an investment to measure the investment's volatility. Standard deviation is also known as historical volatility and is used by investors as a gauge for the amount of expected expecte d volatility volati lity..
Eg. A volatile stock will have a high standard deviation while the deviation of a stable blue chip c hip stock will be lower. A large dispersion tells us how much the return on the fund is deviating from the expected normal returns. Standard Deviation Formula for Portfolio Returns
s = Standard Deviation rk = Specific Return rexpected = Expected Return n = Number of Returns (sample size).
Correlation
A measure that determines the degree to which two variable's movements are associated. The correlation correlation coefficient is calculated as:
The correlation coefficient will vary from -1 to +1. A -1 indicates perfect negative correlation, and +1 indicates perfect positive correlation
Covariance
A measure of the degree to which returns on two risky assets move in tandem. A positive covariance means that asset returns move together. A negative covariance means returns move inversely. One method of calculating covariance is by looking at return surprises (deviations from expected return) in each scenario. Another method is to multiply the correlation between the two variables by the standard deviation of each variable. Possessing financial assets that provide returns and have a high covariance with each other will not provide very much diversification. For example, if stock A's return is high whenever stock B's return is high and the same can be said for low returns, then these stocks are said to have a positive covariance. If an investor wants a portfolio whose assets have diversified earnings, he or she should pick financial assets that have low covariance to each other.
Required Rate of Return The rate of return needed to induce investors or companies to invest in something. For example, if you invest in a stock, your required return might be 10% per p er year ye ar.. Your Your reasoning is that if you don't receive 10% return, return, then you'd be better off paying down your outstanding mortgage, on which you are paying 10% interest. in terest.
Risk Free Rate of Return
The risk-free rate represents the interest an investor would expect from an absolutely riskfree investment over a specified period of time. The risk-free rate is the minimum return an investor expects for any investment because he or she will not accept additional risk unless the potential rate of return is greater than the riskfree rate. In practice, however, the risk-free risk-free rate does not exist because even the safest investments carry a very small amount of risk. Thus, the interest rate on a three-month U.S. Treasury bill is often used as the risk-free rate.
Risk Premium The additional return investors expect to get, or investors earned in the past, for assuming additional risk. It may be calculated between 2 classes of securities that differ in their risk level.
3 types of Risk Premium a) Equity risk r isk premium- Differ Difference ence between the return on equity stocks as a class and the risk-free rate represented by the return on treasury Bill. b) Bond Horizon Premium- Difference Difference between the return on long-term government bonds and the return return on treasury Bill. c) Bond Default Premium- Difference Difference between the return on long-term corporate bonds and the return on long-term government bonds.
Determinants of Required Returns
Three Components of Required Return:
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The time value of money during the time period The expected rate of inflation during the period The risk involved
Complications of Estimating Required Required Return ◦
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A wide range of rates is available for alternative investments at any time. The rates of return on specific assets change c hange dramatically over time. The difference between the rates available on different assets change over time.
Determinants of Required Returns
The Real Risk Free Rate (RRFR) ◦
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Assumes no inflation. Assumes no uncertainty about future cash flows. Influenced by time preference for consumption of income and investment opportunities in the economy
Nominal Risk-Free Risk-Free Rate (NRFR) ◦
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Conditions in the capital c apital market market Expected rate of inflation
NRFR=(1+RRFR) x (1+ Rate of Inflation) - 1 RRFR=[(1+NRFR) / (1+ Rate of Inflation)] - 1
Determinants of Required Returns
Business Risk ◦
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Uncertainty of income flows caused c aused by the nature of a firm’s business Sales volatility and operating leverage determine the level of business risk.
Financial Risk ◦
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Uncertainty caused by the use of debt financing. Borrowing requires requires fixed payments which must be paid ahead of payments to stockholders. The use of debt increases uncertainty of stockholder income and causes an a n increase in the stock’s risk premium. p remium.
Determinants of Required Returns
Liquidity Risk ◦
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How long will it take to convert an investment into cash? How certain is the price that will be received?
Exchange Rate Risk ◦
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Uncertainty of return is introduced by acquiring ac quiring securities denominated in a currency different from that of the investor. Changes in exchange rates affect the investors return when converting an investment back into the “home” currency.
Determinants of Required Returns
Country Risk ◦
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Political Political risk is the uncertainty of returns returns caused by the possibility of a major change in the political or economic environment in a country. Individuals who invest in countries that have unstable political-economic systems must include a country risk-premium risk-premium when determining their required rate of return.
Determinants of Required Returns
Risk Premium and Portfolio Theory ◦
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From a portfolio theory perspective, the relevant risk measure for an individual asset is its comovement with the market portfolio. Systematic risk relates the variance of the investment to the variance of the market. Beta measures this systematic risk of an asset. According to the portfolio theory, the risk premium depends on the systematic risk.
Determinants of Required Returns
Fundamental Risk versus Systematic Risk ◦
Fundamental risk comprises business risk, ris k, financial risk, liquidity risk, exchange rate risk, and country risk.
Risk Premium= f ( Business Risk, Financial Financial Risk, Liquidity Risk, Exchange Rate Risk, Country Risk) Systematic risk refers to the portion of an individual asset’s total variance attributable to the variability of the total market portfolio.
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Risk Premium=
f (Systematic
Market Risk)
