Definition of Portfolio Revision

The care taken in the construction of the portfolio should be extended to the review and revision of the portfolio. Fluctuations that occur in the equity prices cause substantial gain or loss to the investors. The investor should have competence and skill in the revision of the portfolio. Normally the average investor dislikes to sell in the bull market with the anticipation of further rise. Likewise, he is reluctant to buy in the bear market with the anticipation of further fall.

The portfolio management process needs frequent changes in the composition of stocks and bonds. In securities, the type of securities to be held should be revised according to the portfolio policy. If the policy of investor shifts from earnings to capital appreciation, the stocks should be revised accordingly. An investor can sell his shares if the price of shares reaches the historic high prices. Likewise, if the security does not fulfill the investor’s expectation regarding return and growth, it is better to get rid of it. The investor should also consider the factors like risk, quality and tax concessions. If another stock offers a competitive edge over the present stock, investment should be shifted to the other stock. Many investors find themselves inadequate in their ability to trade and earn profit. Mechanical methods are adopted to earn better profit through proper timing. Such types of mechanical methods are Formula Plans and Swaps.

Passive Management

Passive management is a process of holding a well diversified portfolio for a long term with the buy and hold approach. Passive management refers to the investor’s attempt to construct a portfolio that resembles the overall market returns. The simplest form of passive management is holding the Index fund that is designed to replicate a good and well defined index of the common stock such as BSE-Sensex or NSE-Nifty. The fund manager buys every stock in the index in exact proportion of the stock in that index. If Reliance Industry’s stock constitutes 5% of the index, the fund also invests 5% of its money in Reliance Industry stock.

The problem in the index fund is the transaction cost. If it is NSE-Nifty, the manager has to buy all the 50 stocks in market proportion and cannot leave the stocks with smallest weights to save the transaction costs. Further, the reinvestment of the dividends also poses a problem. Here, the alternative is to keep the cash in hand or to invest the money in stocks incurring transaction cost. Keeping away the stock of smallest weights and the money in hand fail to replicate the index fund in the proper manner. The commonly used approaches in constructing an index fund are as follows:

Keeping each stock in proportion to its representation in the index

Holding a specified number of stocks for example 20, which historically track the index in the best manner.

Holding a smaller set of stocks to match the index in a pre-specified set of characteristics. This may be in terms of sector, industry and the market capitalisation.

Active Management

Active Management is holding securities based on the forecast about the future. The portfolio managers who pursue active strategy with respect to market components are called ‘market timers’. The portfolio managers vary their cash position or beta of the equity portion of the portfolio based on the market forecast. The managers may indulge in ‘group rotation’s. Here, the group rotation means changing the investment in different industries’ stocks depending on the assessed expectations regarding their future performance.

Stocks that seem to be best bets or attractive are given more weights in the portfolio than their weights in the index. For example, Information Technology or Fast Moving Consumer Goods industry stocks may be given more weights than their respective weights in the NSE-50. At the same time, stocks that are considered to be less attractive are given lower weights compared to their weights in the index.

Here, the portfolio manager may either remain passive with respect to market and group components but active in the stock selection process or he may be active in the market, group and stock selection process.

The Formula Plans

The formula plans provide the basic rules and regulations for the purchase and sale of securities. The amount to be spent on the different types of securities is fixed. The amount may be fixed either in constant or variable ratio. This depends on the investor’s attitude towards risk and return. The commonly used formula plans are rupee cost averaging, constant rupee value, the constant ratio and the variable ratio plans. The formula plans help to divide the investible fund between the aggressive and conservative portfolios.

The aggressive portfolio consists more of common stocks which yield high return with high risk. The aggressive portfolio’s return is volatile because the share prices generally fluctuate. The conservative portfolio consists of more bonds that have fixed rate of returns. It is called conservative portfolio because the return is certain and the risk is less. The conservative portfolio serves as a cushion for the volatility of the aggressive portfolio. The capital appreciation in the conservative portfolio is rather slow and the fall in price of the bond or debenture is also alike.

Assumptions of the Formula Plan

The first assumption is that certain percentage of the investor’s fund is allocated to fixed income securities and common stocks. The proportion of money invested in each component depends on the prevailing market condition. If the stock market is in the boom condition lesser funds are allotted to stocks. Perhaps it may be a ratio of 80 per cent to bonds and 20 per cent to stocks in the portfolio. If the market is low, the proportion may reverse. In a balanced fund, 50 per cent of the fund is invested in stocks and 50 per cent in bonds.

The second assumption is that if the market moves higher, the proportion of stocks in the portfolio may either decline or remain constant. The portfolio is more aggressive in the low market and defensive when the market is on the rise.

The third assumption is that the stocks are bought and sold whenever there is a significant change in the price. The changes in the level of market could be measured with the help of indices like BSE-Sensitive Index and NSE-Nifty.

The fourth assumption requires that the investor should strictly follow the formula plan once lie chooses it. He should not abandon the plan but continue to act on the plan.

The investors should select good stocks that move along with the market. They should reflect the risk and return features of the market. The stock price movement should be closely correlated with the market movement and the beta value should be around 1.0. The stocks of the fundamentally strong companies have to be included in the portfolio.

Advantages of the Formula Plan

Basic rules and regulations for the purchase and sale of securities are provided.

The rules and regulations are rigid and help to overcome human emotion.

The investor can earn higher profits by adopting the plans.

A course of action is formulated according to the investor’s objectives.

It controls the buying and selling of securities by the investor.

It is useful for taking decisions on the timing of investments.

Disadvantages

The formula plan does not help the selection of the security. The selection of the security has to be done either on the basis of the fundamental or technical analysis.

It is strict and not flexible with the inherent problem of adjustment.\

The formula plan should be applied for long periods, otherwise the transaction cost may be high.

Even if the investor adopts the formula plan, he needs forecasting. Market forecasting helps him to identify the best stocks.

Rupee Cost Averaging

The simplest and most effective formula plan is rupee cost averaging. First, stocks with good fundamentals and long term growth prospects should be selected. Such stocks’ prices tend to be volatile in the market and provide maximum benefit from rupee cost averaging. Secondly, the investor should make a regular commitment of buying shares at regular intervals. Once he makes a commitment, he should purchase the shares regardless of the stock’s price, the company’s short term performance and the economic factors affecting the stock market.

In the rupee cost averaging plan, the investor buys varying number of shares at various points of the stock market cycle. In a way, it can be called time diversification. Let us assume that an investor decides to buy Rs11000 worth of particular shares for four quarters in one particular year, ignoring the transaction costs. The details are given in table

In the above example, the stock price fell in the second quarter but recovered in the third quarter. The investor was able to buy more stocks in the second quarter than in the first quarter. The benefits of this policy can be viewed by comparing the last two columns. In the second quarter, the average cost per share is lower than the average market price per share. This is the benefit derived from rupee cost averaging.

The rupee cost averaging for the Hero Honda stock is given in table. The process of investment is assumed to commence in January 1996 and end in 1998, covering 12 quarters.

Advantages

The advantages of the rupee cost averaging plan are

Reduces the average cost per share and improves the possibility of gain over a long period.

Takes away the pressure of timing the stock purchase from investors

Makes the investors to plan the investment programme thoroughly on the commitment of funds that has to be done periodically

Applicable to bothfalling and rising market, although it works best if the stocks are acquired in a declini1ig market.

In a nut shell, the investor must decide in advance the sum and periodic intervals at which he has to invest. Once it is decided, the implementation is mechanical.

Limitations

Extra transaction costs are involved with small and frequent purchase of shares

The plan does not indicate when to sell. It is strictly a strategy for buying

It does not eliminate the necessity for selecting the individual stocks that are to be purchased

There is no indication of the appropriate interval between purchases

The averaging advantage does not yield profit if the stock price is in a downward trend

The plan seems to work better when stock prices have cyclical patterns.

The rupee cost averaging plan yields better results when applied to no load mutual funds. The problems of high transaction costs and stock selection are eliminated. The broad based index fund experiences profit if the once is volatile, allowing the averaging effect to result in cost reduction. The investor has only to decide on the size of the fund and the length of the interval between the purchases.

Col 7 = Col 2 x Col 4

Col 8 = Col 7 x Col 6

Col 9 = Col 6 ÷ Col 4

Col 10 = I Qr Price + II Qr Price ÷ 2 and so on.

Constant Rupees Plan

Constant rupee, constant ratio and variable ratio plans are considered to be true formula timing plans. These plans force the investor to sell when the prices rise and purchase as prices fall. Forecasts are not required to guide buying and selling. The actions suggested by the formula timing plan automatically help the investor to reap the benefits of the fluctuations in the stock prices.

The essential feature of this plan is that the portfolio is divided into two parts, which consists of aggressive and defensive or conservative portfolios. The portfolio mix facilitates the automatic selling and buying of bonds and stocks.

The plan The constant rupee plan enables the shift of investment from bonds tostocks and vice-versa by maintaining a constant amount invested in the stock portion of the portfolio. The constant rupee plan starts with a fixed amount of money invested in selected stocks and bonds. When the price of the stocks increases, the investor sells sufficient amount of stocks to return to the original amount of the investment in stocks. By keeping the value of aggressive portfolio constant, remainder is invested in the conservative portfolio.

The investor must choose action points or revaluation points. The action points are the times at which the investor has to readjust the values of the stocks in the portfolio. Stocks’ values cannot be continuously the same and the investor has to be watchful of the market price movements. Stocks’ value in the portfolio can be allowed to fluctuate to a certain extent. Percentage change in price like 5%, 10% or 20% can be fixed by the investor. Allowing only small percentage change would result in a lot of transaction cost and would not be beneficial to the investor. If the action points are too large, the investor may not be able get full benefit out of the price fluctuations. The table shows the constant rupee plan. The transaction costs are not considered.

According to the Table, the investor has ` 20,000 to invest and he divides it equally between stocks and bonds 50:50 that is 10,000:10,000. He makes quarterly adjustment if the stock portion falls or rises by 20%. In the third quarter, the stock prices fell by 20% initiating the action. He shifted ` 2000 from the bonds’ portion and bought 50 shares. This lifted the value of stock portion again to ` 10,000.

In the fifth quarter, the stock price has increased from ` 40 to ` 50, a 20 per cent increase. In this action point the investor disposes off the shares and shifts the money to the bond portion. By this the stock amount in the portfolio has remained constant but the total portfolio value has increased. The investor stands to gain by the total portfolio value appreciation.

The major advantage of this plan is that purchase and sales are determined automatically. This facilitates the investor to earn capital gain by selling the stocks when the price increases and buying it at a relatively lower price. To make the plan operate effectively, at the starting point, stocks should not be purchased either at high prices or at too low prices. If the investor starts the purchase at the extreme price level, the stock fund may be either too small or too large.

Constant Ratio Plan

Constant ratio plan attempts to maintain a constant ratio between the aggressive and conservative portfolios. The ratio is fixed by the investor. The investor’s attitude towards risk and return plays a major role in fixing the ratio. The conservative investor may like to have more of bond and the aggressive investor, more of stocks. Once the r$io is fixed, it is maintained as the market moves up and down. As usual, action points may be fixed by the investor. It may vary from investor to investor. As in the previous example, when the stock price moves up or down by 10 to 20 per cent action would be taken. Here, 10 per cent is taken as action point. The table shows the constant ratio plan.

The advantage of constant ratio plan is the automatism with which it forces the manager to counter adjust his portfolio cyclically. But this approach does not eliminate the necessity of selecting individual security.

The limitation of the plan is that the money is shifted from the stock portion to bond portion. Bond is also a capital market instrument and responds to market pressures. Bond and share prices may both rise and fall at the same time. In the downtrend both prices may decline and then gain.

Variable Ratio Plan

According to this plan, at varying levels of market price, the proportions of the stocks and bonds change. Whenever the price of the stock increases, the stocks are sold and new ratio is adopted by increasing the proportion of defensive or conservative portfolio. To adopt this plan, the investor is required to estimate a long term trend in the price of the stocks. Forecasting is very essential to this plan. When there is a wide fluctuation variable ratio plan is useful. The table explains the variable ratio plan.

In the above example, the portfolio is adjusted for every 20 per cent change in the stock price. This adjustment criterion may be different for different investors depending upon their attitude towards risk and return. The portfolio is divided into two equal portions as in the case of other plans, with R,10,000 in each. Let us assume that there is a fall in the price of the stock, then, the percentage of stock in the portfolio declines. As the market price for the stock reaches a 20 per cent decline, that is to Rs,80, the adjustment action takes place. The purchase of 58 shares raises the stock portion to 72.48 per cent. Once again, when there is a 20 per cent change, the adjustment action is triggered. When the prices have increased to ` 100, the investor sells 50 shares and the stock portion in the portfolio is reduced back to 50 per cent.

The figure explains the variable ratio plan.

The middle line is the trend line that represents the investor’s expectation about of future course of prices. Zone 1 and 3 represent respectively of 10 and 20 per cent deviations above the expected trend, and zones 2 and 4 represent respectively 10 and 20 per cent deviations below the expected trend. Starting at ` 50, the portfolio’s bonds and stocks ratio is 50:50.

At point A, the portfolio is adjusted to the next proportion, in this case 60 per cent bonds and 40 per cent stocks. At B, again it is 50:50. Below point C there would be more stocks than bonds. Because of the decline in stock price, more stocks are purchased. Above the point D, it is again 50:50. The line moves closer to the trend line

Advantages

Automatically, the investor tends to correct his portfolio portions according to the price changes. The investor is not emotionally affected by the price changes in the market. With accurate forecast the variable ratio plan takes greater advantage of price fluctuations than the constant ratio plan.

Limitations

The investor has to construct the appropriate zones and trend for alterations of the proportions

The selection of security has to be done by the investor by analysing the merits of the stock. The plan does not help in the selection of scrips.

If the zones are too small frequent changes have to be done and it would limit portfolio performance.

Revision and the Cost

With the passage of time the stocks which were attractive once may turn out to be less attractive in terms of return. The investor’s attitude towards risk and return also may change and the forecast regarding the market also may undergo change. In this context, the necessary revision is thought of by the portfolio manager. In revision of traded volumes the portfolio manager has to incur brokerage commission, price impact and bid-ask spread. Price impact means the effects on the price of stock. In simple terms, if the size of the trade is heavy on the buying side, the prices of the stock may increase. The bid-ask spread is the difference between the price that the market maker is willing to buy and sell the stock. These costs may be higher in small size stocks and the benefits of revision may be nullified by it. Usually revision is done with the view of either increasing the expected return of the portfolio or to reduce the risk (standard deviation) of the portfolio.

SWAPS

Swap is a contract between two parties to exchange a set of cash flows over a pre-determined period of time. The two parties are known as counter parties. In an equity swap one counter party, say ‘A’, agrees to pay cash based on the rate of return of an agreed stock market index to the second counter party ‘B’. Since the payments are based on the market index, they vary according to index movements. The second counter party B agrees to pay the fixed amount of cash payments based on the current interest rate to the first counterparty A. Thus, the payment depends upon the underlying security. This agreement means that A has sold stocks and bought bonds while B has sold bonds and bought stocks. Here, they have restricted their portfolios without the transaction costs, even though they have to pay the swap fee to the swap bank that set up the contract between the two parties.

This can be explained with the help of an example. Consider Mr. Hope, a portfolio manager having an expectation of upward trend in the stock market for the year and Mr.

Despair, another portfolio manager who feels that there would be downward trend in the market for the next year. Mr. Hope wants to sell ` 10 lakhs worth of bonds and to invest it in the stock market, whereas Mr. Despair wants to dispose off` 10 lakhs worth of stocks to be invested in the bond market. Selling and buying of bonds or stocks involve transaction cost. Hence, they approach the Swap bank. A contract has been set up between Mr. Hope and Mr. Despair by the swap bank. The contract payments have to be made for every quarter. At the end of each quarter, Mr. Despair has to pay Mr. Hope an amount equal to the rate of return on the NSE-Nifty for every quarter in terms of the basic principal amount. At the same time, Mr. Hope has to pay an amount equal to 3% of the principal. The agreed notional principal amount is ` 10 Iakhs. The contract lasts for an year. They pay fees to the swap bank.

Let us assume that the rates of return of NSE-Nifty are 5%, - 2%, 3% and 6% for the four quarters. Mr. Hope has to pay ` 30,000 to Mr. Despair each quarter, the payments Mr. Despair has to make to the Mr. Hope are as follows:

The results can be summarized

First Quarter Mr. Despair pays ` 20,000 to Mr. Hope

Second Quarter: Mr. Hope pays ` 50,000 to Mr. Despair

Third Quarter: There is no payment

Fourth Quarter: Ir. Despair pays ` 30,000 to Mr. Hope

The amount paid by Mr. Despair shows what would have transacted if Mr. Despair had sold stocks and bought bonds. Likewise the payments made by Mr. Hope indicates what would have happened if he had sold bonds and bought stocks. The equity swaps could be modified based upon the index and the prevailing interest rates.

Summary

The CAPM model is based on specific assumptions. The investor could borrow or lend any amount of money at riskiess rate of interest.

All investors hold only the market portfolio and the riskless securities.

Market portfolio consists of the investments in all securities of the market. The proportion invested in each security is equal to the percentage of the total market capitalisation represented by the security.

The capital market line represents the relationship between the expected return and standard deviation of the portfolio.

The risk of the security is indicated by its covariance with the market portfolio.

Security market line shows the linear relationship between the expected returns and betas of the securities.

The objective of the asset pricing model is to identify the equilibrium asset price for expected return and risk. If the asset prices are not equal, there is a scope for arbitrage.

An arbitrage portfolio is constructed without any additional financial commitment.

Investors indulge in arbitrage, moving the price upwards if securities are held long and driving down the price of securities if held in short position, till the elimination of the arbitrage possibilities.

The factor sensitivity in arbitrage model indicates the responsiveness of a security’s return to a particular factor.

Portfolio evaluation is carried out to assess the risk and return of the different portfolios.

Mutual funds pool together the funds from investors by selling units and invest them in different types of securities.

Closed-end funds are open for a specific period for subscription. The open-ended funds units are available continuously.

Sharpe index is a measure of risk premium related to the total risk.

Treynor index measures the fund’s performance in relation to the market performance.

Jensen index compares the actual or realised return of the portfolio with the calculated or predicted return. Better performance of the fund depends on the predictive ability of the managerial personnel of the fund.

Passive management of funds consists of indexing of the stocks to be purchased. In active management funds are allocated to buy active stocks in the market.

Aggressive portfolio consists more of common stocks while conservative portfolio consists more of bonds or debentures.

Portfolios are revised with the help of formula plans.

In the rupee cost averaging technique, varying amount of shares are bought at regular intervals. This is time diversification of the portfolio.

In the variable ratio plan, the proportions of funds on aggressive and conservative portfolios change according to the varying levels of security market prices.

In an equity swap two parties agree to make payments to each other based on the stock market price and interest rate.

Solved Problems

Security J has a beta of 0.75 while security K has a beta of 1.45. Calculate the expected return for these securities, assuming that the risk free rate is 5 per cent and the expected return of the market is 14 per cent.

Solution

The expected return can be calculated using CAPM

R_i = R_f + β_i (R_m = R_f)

For Security J

R_i = 5 + 0.75 (14 – 5)

 = 5 + 6.75 = 11.75 per cent

For Security K

R_i = 5 + 1.45 (14 – 5)

 = 5 + 13.05 = 18.05 per cent

A security pays a dividend of ` 3.85 and sells currently at ` 83. The security is expected to sell at ` 90 at the end of the year. The security has a beta of 1.15. The risk free rate is 5 per cent and the expected return on market index is 12 per cent. Assess whether the security is correctly priced.

Solution

To assess whether a security is correctly priced, we need to calculate (a) the expected return as per CAPM formula, (b) the estimated return on the security based on the dividend and increase in price over the holding period.

Expected return

R_i = R_f + β_i (R_m = R_f)

 = 5 + 1.15 (12 – 5)

 = 5 + 8.05 = 13.05 per cent

Estimated return

As the estimated return on the security is more or less equal to the expected return, the security can be assessed as fairly priced.

The following data are available to you as portfolio manager:

In terms of the security market line, which of the securities listed above are underpriced?

Assuming that a portfolio is constructed using equal proportions of the five securities listed above, calculate the expected return and risk of such a portfolio.

Solution

We can use CAPM to determine which of the securities listed are underpriced. For this we have to calculate the expected return on each security using CAPM equation:

R_i = R_f + β_i (R_m = R_f)

Given that R_f (Govt. security return rate) = 7 and R_m 15

The equation becomes

R_i = 7 + β_i(15 – 7)

Now,

Security A = 7 + 2.0 (15 - 7) = 7 + 16 = 23 per cent

Security B = 7 + 1.5 (15 - 7) = 7 + 12 = 19 per cent

Security C = 7 + 1.0 (15 - 7) 7 + 8 = 15 per cent

Security D = 7 + 0.8 (15 - 7) 7 + 6.4 = 13.4 per cent

Security E = 7+ 0.5 (15-7) = 7+4 = 11 percent

The expected return as per CAPM formula and the estimated return of each security can be tabulated.

A security whose estimated return is greater than the expected return is assumed to be underpriced because it offers a higher return than that expected from securities with the same risk.

Accordingly, securities A, B and C are underpriced.

To calculate the expected return and risk R_P and β_p we need to calculate β_pr firs

As the proportion of investment in each security is equal, ω_i = 0.20

β_p = (0.2) (2.0) + (0.2) (1.5) + (0.2) (1.0) + (0.2) (0.8) + (0.2) (0.5)

= (0.2) (2.0 + 1.5 + 1.0 + 0.8 + 0.5)

= (0.2) (5.8) = 1.16 Expected return of portfolio

R_P = R_f+β_p (R_m - R_f)

= 7 + 1.16 (15 - 7)

= 7 + 9.28 = 16.28 per cent

Systematic risk of the portfolio β_p = 1.16

An investor owns a portfolio that over the last five years has produced 16.8 per cent annual return. During that time the portfolio produced a 1.10 beta. Further, the risk free return and the market return averaged 7.4 per cent and 15.2 per cent per year respectively. How would you evaluate the performance of the portfolio?

Solution

The Treynor ratio can be used to evaluate the performance of the portfolio in this case.

The ratio for the market index can be taken as the benchmark for evaluation.

The portfolio has a reward to volatility ratio higher than that of the market index.

Hence, the performance of the portfolio can be considered superior.

You are given the following historical performance information on the capital market and a mutual fund:

Calculate the following risk adjusted return measures for the mutual fund:

Reward-to-variability ratio

Reward-to-volatility ratio

Comment on the mutual fund’s performance.

Solution

As the first step in calculation, the average values of the four variables may be calculated.

The averages are as follows:

Mutual fund beta = 0.87

Mutual fund return = 11.05 per cent

Return on market index = 8.69 per cent

Return on govt. securities = 5.95 per cent

Reward to variability ratio or Sharpe ratio

For the calculation of this ratio, σ_pr or mutual fund’s standard deviation of returns, is required.

Calculation of Standard deviation

Reward to volatility ratio or Treynor ratio

Mutual Fund performance

For evaluating the mutual fund performance we have to calculate the Sharpe and Treynor ratios for the market index to be used as the benchmark.

For calculating the Sharpe ratio for the market index, the standard deviation of returns on the market index has to be calculated.

                     Calculation of Standard deviation

Sharpe and Treynor ratios for the market index:

Ratios of the mutual fund and the market index may be tabulated as:

Mutual fund has performed better than the market.

Information regarding two mutual funds and a market index are given below:

Assuming the risk – free return as 5 per cent, calculate the differential return for the two funds.

Solution

Differential return, as per Jensen ratio, is calculated as:

α_p = R_p – E(R_p)

The expected return of the portfolio, E(R_p), can be calculated using the CAPM formula.

E(R_p) = R_f + β_p(R_m – R_f)

Gold fund: E(R_p) = 5 + 0.72 (10 – 5)

= 5 + 3.6 = 8.6 per cent

Platinum fund: E(R_p) = 5 + 1.33 (10 – 5)

= 5 + 6.65 = 11.65 per cent

Differential return

Gold fund:α_p = 7 – 8.6 = -1.6 per cent

Platinum fundα_p = 16 – 11.65 = 4.35 per cent

From the information given in example 3, calculate net selectivity measure for the platinum fund using Fama’s framework of performance components.

Solution

We have the following information:

R_p = 16 per centσ_p = 35 per cent

R_M = 10 per centσ_M = 24 per cent

R_f = 5 per centβ_p = 1.33

Fama’s decomposition may be stated as:

R_p = R_f + R₁ + R₂ + R₃

R_f = 5 per cent

R₁ = β_p(R_m – R_f)

= 1.33 (10 – 5) = 6.65 per cent

R₂ = [(σ_p/σ_m) – β_p](R_m - R_f)

= [(35/24) – 1.33](10-5)

= (1.46 – 1.33) (5)

0.65 per cent

R₃ = 16 – (5 + 6.65 + 0.65) = 16 – 12.3 = 3.70 per cent

Thus,

R_p = 5 + 6.65 + 0.65 + 3.70 = 16 per cent

Alternatively, Fama’s net selectively can be directly calculate as follows:

Fama’s net selectively

R_p = [R_f + (σ_p/σ_m) (R_m - R_f)

= 16 – [5 + (35/24) (10 – 5)]

= 16 – (5 + 7.3)

=  16 – 12.30 = 3.70 per cent.

Input Data

The values of the portfolio alpha, portfolio beta, and portfolio residual variance can be calculated as the first step.

α_p = ∑ i=1ⁿω_iα_i

= (0.2)(2) + (0.1)(3.5) + (0.4) (1.5) + (0.3) (0.75)

= 1.575

β_p = ∑ i=1ⁿω_iβ_i

= (0.2)(1.7) + (0.1)(0.5) + (0.4) (0.7) + (0.3) (1.3)

= 1.06

Portfolio residual variance = ∑ i=1ⁿω²_iσ²_ei

= (0.2)2(370) + (0.1)2(240) + (0.4)2 (410) + (0.3)2 (285)

= 108.45

These values are noted in the last row of the table. Using these values, we can calculate the expected portfolio return for any value of projected market return. For a market return of 15 per cent, the expected portfolio return would be:

R_p = α_p + β_pR_m = 1.575 + (1.06)(15) = 17.475

For calculating the portfolio variance we nee the variance of the market returns.

Assuming a market return variance of 320, the portfolio variance can be calculated as: