Diversification of Risk

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Diversification of Risk

Naturally, any investor would like to receive the highest possible return from a given risk level that he/she is wants to take. Now the question is how an investor can receive the highest possible return from his comfortable risk level? Let us understand this technique of maximising return with the help of a portfolio that consists of two types of assets. These assets are risk-free assets (having no risk and a significantly low return) and risky assets (having high risk and high expected return).

We can measure investment risk from the standard deviation of investment returns. Naturally, higher risk is associated with higher standard deviation. Now, an investor has a number of options to build his/her portfolio. He/she can earn a low return without any risk by putting his/her entire funds into the risk-free asset.

In addition, the investor can also earn the maximum return by allocating his/her entire funds to the risky asset. Moreover, the investor can also settle for a risk-return trade off that would lie somewhere between these two ends, by choosing a combination of these two assets.

Capital Market Line (CML)

In a portfolio consisting of two portfolios, the return depends on the proportion of the risky and risk-free asset in the portfolio. Let us assume that that the proportion of risky assets in the portfolio is y. Naturally, the proportion of risk-free assets must be in 1-y. Therefore, the portfolio return would be given by:

Portfolio Return = yrisky asset return + (1-y)risk-free return

The total risk involved in the portfolio can be varied by changing the proportion of the two types of assets in the portfolio. The different possible combinations of the risky and risk-free assets constitute a set called investment opportunity set. The investment opportunity set can be graphed as a line when we plot return against risk in the portfolio. We can measure risk with the help of standard deviation. This line is called Capital Allocation Line (CAL).

The starting point of the capital allocation line is where the combination of minimum return and no risk given by the risk-free asset and the maximum return and risk of the risky asset intercepts. Therefore, we can say that the CAL depicts the different possible combinations of the risky and the risk-free asset. In case the risky asset provides a market return instead of a single-asset return, the CAL formed is called Capital Market Line (CML). In other words, the CML is the tangent that can be drawn from the point of the risk-free asset connecting to the feasible region of the risky asset.

Combining a Risky and Risk-Free Asset

Now, let us study how an investor can construct a portfolio in case he/ she has the choice of combining a risky asset with a risk free asset or if funds could be borrowed at the risk-free rate of interest, how the investor would shape the portfolio opportunity. We know that a risk-free asset has zero variance or standard deviation.

So, what would happen to return or risk in case a portfolio combines risky and risk-free assets? Suppose an investor invests in a risk-free security ‘f’ providing 10% return and a risky security ‘r’ having 30% returns. The risky asset has a standard deviation of 6%. Let us calculate the portfolio return and risk in case the investor invests in these two securities in equal proportion.

The portfolio return would be:

E(Rp)= wE(Rr ) + (1-w)Rf

= 0.5*0.30 + (1-0.5) 0.10

= 0.15+ 0.05

= 0.20 or 20%

We have already mentioned that the risk-free security has zero standard deviation. Therefore, the value of the covariance between the risky and risk-free security would also be zero. Therefore, the portfolio risk would be the equal to the standard deviation of the risky security and its weight, as follows:

Therefore, σp = wσr

σp = 0.5*0.06 = 0.03 = 3%.

Sharpe Portfolio Optimisation

After selecting the stocks for investment, the next step is to allocate them. There are a number of asset allocations strategies. Sharpe portfolio optimisation is a strategy that uses the Sharpe ratio for allocating assets. The ratio measures the risk-adjusted returns of a fund. Higher Sharpe ratio indicates higher returns from a fund relative to the risks involved.

In addition, we can use the ratio to compare risk-adjusted return of various funds with the help of the Sharpe ratio. It was named after Nobel Laureate William Sharpe. It should be noted that the Sharpe ratio does not gurantee the highest return or lowest risk. Simpley, it is a startegy that helps in selecting effective ways of managing risk.

We can calculate the Sharpe ratio with the help of expected return and standard deviation, and the risk-free rate (for example, the rate of US Treasury Bill), as follows:

Sharpe Ratio = ( rx – Rf ) / StdDev(x)


rx refers to the average yearly interest rate given by the asset x

Rf refers to the risk-free rate

StdDev(x) refers to the standard deviation of return of asset x

Now that we have learned to calculate the Sharpe ratio, the next step is to allocate assets for portfolio optimisation. Following steps should be followed to optimise the portfolio:

  • Determining the Sharpe ratio of all the component stocks of the portfolio.

  • Calculating the sum of the Sharpe ratio values.

  • Dividing the Sharpe ratio of the individual stocks by the sum of the Sharpe ratios. The resultant would be the percentage allocation for each stock.

Now, let us consider an example of portfolio optimisation with the help of the Sharpe ratio. Let’s assume that a portfolio consists of three stocks: P1 , P2 , and P3 . The Sharpe ratios of these three stocks are shown in Table 4.1 as follows:

StockSharpe Ratio
Total of Sharpe Ratios6.0
Sharpe Ratio of the Stocks

Now let us divide the Sharpe ratio of the individual stocks by the total of the Sharpe ratios:

  • For P1 : 1.5/6 = 0.25
  • For P2 : 2/6 = 0.33
  • For P3 : 2.5/6 = 0.42

Therefore, we can see that to obtain a optimum portfolio (minimum risk and maximum retrun), we need to invest 25% of funds in asset P1 , 33% in assetP2 , and 42% in asset P3.

Significance of Beta

In simple words, beta is a measure of the systematic risk of an asset or a portfolio compared to the systematic risk in the entire market. Therefore, we can say that beta informs an investor about the risks involved in a stock and the risk involved in the entire market. In addition, beta helps in comparing the market risk of a security with the market risk of other securities. Investment analysts use the Greek letter β to denote beta. Beta is extensively applied in the Capital Asset Pricing Model (CAPM).

Now, let us study the implications of different values of beta:

  • Beta = 1: It signifies that the values of the securities would appreciate with the market.

  • Beta < 1: It indicates that the volatility of the security is less than the volatility of the market.

  • Beta > 1: It denotes that the price of the security would be more volatile than the volatility of the market.

  • Beta < 0: It signifies that there is an inverse relation in the volatility of the stock and the market.

  • Beta = 0: There is no relationship between the volatility of the stock and the market.

The beta coefficient is computed by dividing the covariance of a market and stock return variance of market return. The beta coefficient is given by the following formulas:

β =Covariance of Market Return with Stock Return
Variance of Market Return

Also, beta coefficient equals correlation coefficient multiplied by standard deviation of stock returns divided by standard deviation of market returns. This is given as follows:

β = Correlation Coefficient between Market and Stock ×

Standard Deviation of Stock Return
Variance of Market Return

A beta coefficient of 1 implies that the stock bears the same risk as the market and would give the market return only. On the other hand, a beta coefficient < 1 implies that a risk level less than the market. A beta coefficient > 1 suggests that the stock bears a risk higher than market portfolio. For example, the correlation coefficient between market and share price of an organisation is 0.75; standard deviation of market is 15% and that of share price is 8%. Calculate the beta co-efficient.

Solution: β = 0.75 × (8/15) = 0.4

Thus, the stock of the organisation bears a lower risk than the market portfolio.

Beta Estimation

We have mentioned earlier that beta denotes the volatility of a stock with respect to market volatility. Beta is the tangent of the graph that we can obtain by plotting the time series of returns of a portfolio (Rp) against the overall return from the market (Rm). Rm is plotted along the x-axis and portfolio return is plotted along the y-axis. The point where x and y axes intersect is called the cash-equivalent return by which the returns are adjusted. The best-fit line which indicates the data point helps to quantify both active and passive risk.

Figure shows the derivation of beta:

With reference to the Figure, beta is the slope of the line obtained after plotting the time series of Rp against Rm. Therefore, in case beta is 1, 1% increment in the market return also results in 1% increase in the retuen of the portfolio. As beta signifies the trade off relationship between the minimising risk and maximising return, the portfolio risk can be decreased by decreasing the beta and portfolio return can be increased by increasing the beta.

Beta in Stock Selection

Investors take the help of beta to select stocks. Optimistic investors tend to select securities with high beta values. This is because these securities have a higher probability of giving better returns than the market. On the other hand, an investor expecting the market to go down tends to select the securities of which the beta is less than 1. This is because the value of these securities is expected to decline less in value than the market index. For example, if the beta of a portfolio is 0.5 and if Sensex declines by 10%, the stock will decline by 5%.

However, we need to be careful while selecting securities on the basis of beta. This is because beta is a historical measure of the volatility of stocks. The past figures are not always good indicators of the future performance of the market. In addition, the beta of a stock is never static. According to Gene Fama and Ken French, the past beta cannot indicate the future beta with high efficiency. In addition, beta shows the risk involved in a stock in comparison with the market risk. However, volatilities that are specific to a firm are not detected by beta. Therefore, beta is not a great measure of all the risks involved in a security.

Traditional Portfolio Selection

In the last few decades, investors have been putting efforts to find ways of building optimal portfolio that would give the best returns and minimise risks. The traditional portfolio method takes a logical approval of building portfolio by evaluating the performance of a portfolio with the help of their mean and variance.

Following are the two commonly used traditional portfolio selection methods:

Markowitz Portfolio Selection Method

This portfolio selection method is also known as Modern Portfolio Theory (MPT). Nobel Laureate Economist Harry Markowitz published an article titled portfolio selection in 1952. He described his mathematic argument favouring portfolio diversification. The basic premise of the MPT is that an investor can develop an “efficient frontier” of optimal portfolios that would yield maximum possible return for a given risk level.

The main steps involved in the method are:

  • Security valuation
  • Asset allocation
  • Portfolio optimisation
  • Performance measurement

Sharpe Single Index Portfolio Selection Method

This model was developed by William Sharpe to simplify the Markowitz portfolio selection method. This model reduces the number of inputs required in the Markowitz model. The model observes that most assets yield returns in relation to the overall yield in the market.

Mutual Funds for Risk Diversification

Mutual funds are great tools for diversifying portfolio and managing risks. We have studied earlier that diversification minimises risks. Howevet, individual investors are not always in a position to diversify portfolio as they lack in professional knowledge. On the other hand, mutual funds are managed by professionals who are well informed about the technicalities of risk, return and other aspects of investment.

Following are the major types of mutual funds that help in diversifying risks:

Equity-Oriented Mutual Fund

This fund helps in diversifying portfolio with the help of various equity funds, such as large-cap, mid cap, multi-cap, and sectoral funds.

Debt-Oriented Mutual Funds

These types of funds are made of debt products; therefore, they provide a fixed return with very low risk level. However, debt-oriented funds are not very common among investors as these are low-yielding investments. However, debt-oriented mutual funds are very efficient investment tools as they provide stable return at a minimum risk.

Gold Savings Funds or Gold ETFs

Indians have cultural and emotional connection with gold. Historically, gold is considered to be a very precious personal asset. Gold ETFs provide a way of investing in gold and diversifying portfolio. Gold investment is traditionally treated as a hedging instrument against inflation. However, according to experts, the portfolio should not consist of more than 10% gold.

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