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Risk Analysis and Returns


Academic year: 2023

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Calculate the expected NPV and the standard deviation of the NPV if the risk-free rate is 10%. The probability that the project's NPV is less than or equal to 0. ii). The probability that the NPV of the project is equal to or more than Rs. iii).

What is the expected NPV of the project and what is the standard deviation of the NPV. If the risk-free interest rate is 6%, find the expected NPV of the project.

Table 12.6:  k vs NPV
Table 12.6: k vs NPV

Selection of a project

Then we compare the distribution of the profitability index (or any other relative measure) of the project with the maximum risk profile that is acceptable for the expected profitability index of the firm. Then we see that the spread of the profitability index is less than the maximum risk acceptable to the firm for the given level of expected profitability index. If the management is risk averse, the risk distribution curve will be more compact than the curve of the firm, which is willing to take more risk. ii) Risk-adjusted rate discount method.

The discount rate of the project is compared to the cost of capital to the company. If the risk of the project is equal to the risk of the company's existing investments, the discount rate for the project is the average cost of capital to the company; if the risk of the project is greater than the risk of the company's existing investments, the discount rate for the project is higher than the average cost of capital to the company; and if the risk of the project is less than the risk of the company's existing investments, the discount rate for the project is less than the average cost of capital to the company. Depending on the magnitude of the risk of the project in question compared to the existing risk of the company, dk can be positive or negative.

The adjustment for differential risk is a function of management's perception and attitude towards risk. Once rk the company's risk-adjusted discount rate is specified, the company's NPV can be calculated and the project is acceptable if the NPV is positive, that is. However, one of the main limitations of the method is the consistent estimation of dk, which can be very difficult in practice and companies use arbitrary methods for estimating dk.

The security reserve corresponding to the expected value of the offer is called the security equivalent coefficient.


The equivalent security method is conceptually a superior method compared to the adjusted risk method because in this method the risk can change over the years and this fact is evident in different equivalent security coefficients for different years while the assumption in the method of adjusted risk is that of increasing risk at a constant rate. However, specifying different risks and thus different equivalent safety coefficients can be difficult to do. Since the risks at different time periods can be easily calculated in the risk adjusted method, so it is the risk adjusted method that is preferred by the firms.

Risk Risk is defined as the degree of variability/deviation of the actual return from its expected value. A high risk value means that the actual returns are at a distance from the expected value and as such the reliability of the return is less.

Risks and returns of portfolio

Portfolio Risk Portfolio risk is the variance of returns from the portfolio's expected return. This expression shows that portfolio risk depends on (i) the variance of the individual asset; and. ii) Covariance between different assets of a portfolio. Thus, portfolio theory helps managers in deciding the proportion of each security in the portfolio.

Now let's prepare the following table which gives the portfolio return and the portfolio risk for different values ​​of the weights wL and wH and the correlation ρLH:. Two assets can be combined so that the portfolio risk is less than the individual risk of each asset. For any given pair of weights wL and wH, the standard deviation of the portfolio σp falls as ρ varies from +1.0 to –1.0. iii).

For perfect negative correlation, the portfolio risk first decreases when it reaches a minimum value and then starts to increase. v). For each correlation, there is a minimum risk portfolio that has a risk lower than the risk of the individual assets. The following table shows weights corresponding to the minimum risk for each value of the correlation coefficient.

Calculate portfolio risk on a portfolio consisting of two assets with the same variance of 25% and zero covariance when the assets are combined.

Portfolio risk and the correlation between assets

So for variance to be zero, we must have imaginary weights, which is not possible. So it is not possible to have a zero variance portfolio in case of uncorrelated assets. So for portfolio variance to be minimal, the assets must be combined in equal proportion.

Thus, the standard deviation of the portfolio is the weighted average of the standard deviations of the two assets. For ρ = -1.0, the portfolio risks corresponding to different weights exhibit a V-shaped pattern, the tip of the extension on the line of expected returns. The pattern has a clockwise movement indicating that as the weight of the higher risk asset increases, the overall risk is gradually decreasing and the overall expected return is increasing.

After this point, when the high-return, high-risk asset is given a higher weight, the expected return is i. The graph corresponding to ρ = 1.0 and ρ = -1.0 forms a triangle ABC, with B and C being the points corresponding to the net assets. This triangle marks the boundaries for the portfolios. ersification will help reduce the risk s for some values ​​of WL and WH, σP2 < σL2 and σP2 < σH2. ortfelj consists of two assets X and Y. The following information is available regarding the assets.

Impact of portfolio diversification- limits of diversification gain

This aspect of portfolio selection is related to determining the set of efficient portfolios from the available feasible set. ii) Personal aspect. This aspect of portfolio selection is related to determining the best (risk-return) opportunities from the efficient portfolio set, consistent with the investor's attitude to risk. The purpose of the portfolio theory is to determine the portfolios with maximum possible returns for a given risk or to determine the portfolios with minimum risk for return.

Among all these feasible set portfolios, the efficient portfolios are said to be the ones that maximize the expected return for a given standard deviation. The highest point of the curve represents the global maximum return portfolio, while the leftmost point represents the global minimum variance portfolio. The combination of M with the risk-free portfolio F is the best compromise between risk and return.

Since the investment in the risky asset is greater than the investor's capital, the weight of the risky asset will exceed one. As a result, the risk-averse investor's portfolio will be at the lower bound of the efficient portfolio frontier. Thus, 40% of the investor's capital would have to be borrowed to invest in a risky asset, i.e. market portfolio to get the desired return.

A measure of the investor's risk tendency is given by the investor's risk aversion index.

Fig 12.15 Efficient portfolios
Fig 12.15 Efficient portfolios

Capital asset pricing model (CAPM)

The three components of risk are (i) the risk-free rate of return; (ii) risk premium on the market portfolio; and (iii) the systematic risk coefficient β. i) Risk-free rate of return Risk-free rate of return is a theoretical concept and refers to an absolutely certain return. As a proxy for this return are the returns on assets such as treasury bills and bank deposits; and interest rates prevailing in the money market may be used. Maturity period of treasury bills. whether bank deposits are considered for less than one year.

Although the risk associated with these assets is almost negligible, in inflationary conditions the real return can be zero or even negative. ii) Market portfolio risk premium The market portfolio risk premium is the difference between the expected return on the market portfolio and the risk-free rate of return. Here βi is the ratio of the premium over the ith value to the risk premium in the market portfolio. If βi > 1, the risk premium on its asset is greater than that on the market portfolio.

If βi < 1, the risk premium on the i asset is less than on the market portfolio.

Using CAPM in capital budgeting

We have seen that SML is the relationship between the risk premium of an individual asset and the risk premium of the market portfolio. The difference between the predicted return (using SML) and the actual expected return is called α of the asset. The model establishes a direct proportional relationship between risk and expected return. ii) CML in the model indicates the relationship between risk and return of the assets, while SML indicates the role of non-diversifiable systematic risk to be considered in the pricing of securities and portfolios.

Therefore, the model may miss some aspects of real-life problems. Some limitations of the model are listed below. i) The model emphasizes a risk-free asset that would yield a risk-free return. Expected returns and standard deviations of the shares of two companies ABC Ltd.

Calculate the expected returns and standard deviations of the following portfolios: i) Find the equation and slope of CML. Assets Expected return (%) Standard deviation (%). i) Calculate the correlation coefficient ρXY if the standard deviation of the portfolio is 20%. 13 A company is considering a project that has the following characteristics that affect the project's NPV.

17 A firm has defined its risk profile - the maximum acceptable standard deviation for a given expected value of the profitability index.


Table 12.6:  k vs NPV
Table 12.11:  When SP decreases by 10%
Table 12.20  Year
Fig 12.15 Efficient portfolios


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