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Traversing The Portfolio Theory

 It’s definitely diversity but not a hodge-podge that helps you sail through all the unforeseen tempests.

Diversification across various classes of stocks based on industries, size, value vs. growth etc. within a portfolio helps an investor to minimize risks through summing up different risk behaviours of different stocks while keeping contribution of their respective return unaffected. So an investor has a freedom to construct his portfolio in a manner that risk is minimized and return is maximized.

So early as in 1952 Harry Markowitz changed the whole mindset of all investors who used to measure their success by the performance of each of their stocks separately.  He made it clear that one should not concern if some of the stocks did not show up well over a period of time subject to that it is compensated by the rewards of others. In brief, it is not the individual stock but the whole bunch of different stocks called portfolio that should be under purview.

Markowitz showed how an investor can reduce the standard deviation of portfolio return by choosing stocks that do not move exactly together.    

Now, if we draw a graph with Risk (standard deviation of security/portfolio) on X-axis and Return  on Y-axis the efficient portfolio of given stocks would be that which lies upper and left as much as possible.  

See the graph.          

It is really a matter of choice which one of the efficient portfolios is best suited to you depending upon the level of your desire of return and aversion to risk. If you love to play with security market ups and down you are most likely to pick up the portfolio which is expected to go uppermost point with least regard to its dangerous shift towards right direction. Though most of the people like you and me are very conscious about uncertainties in the market and try to minimize risk and stay left considerably. In any case, there is perfectly no point choosing a portfolio given on the graph which lies lower than O as there is always a better return available on the given risk level as the graph shows. 

Expected Portfolio Return is just the weighted average of the Expected Returns of the stocks present in it but Risk (Standard Deviation) has to be calculated by a formula given below.

First we will see how variance is calculated in respect of a portfolio consisting of only two stocks (for the sake of simplicity) and then as we all know that the square root of that will be the standard deviation.

Portfolio Variance =  X12σ12  +  X22σ22  + X1X2 ρ12σ1σ2    

where x1 and, x2  are proportions of stock no.1 and 2, σ1, σ2  are  their standard deviations and  ρ12 is the co-variance between them.


It is to be noted that  The desired portfolio should be only one that lies upper than the critical point ‘O’ and necessarily on the line drawn on the graph which is in fact the maximum return possible on a given level of risk.  

The another important point is that we can achieve any level of return represented by any point on the curve and even more than that by adding lending and borrowing at government treasury bill rates that is termed as risk-free investment. And so our investment in part can both be positive or negative depending upon whether we choose to buy a quantity of treasury bill, lend money at treasury bill rate or borrow at the same rate. If we lend money or buy treasury bills then our investment in that part is positive and if we borrow money at treasury bill rate then our investment in that part is negative and the risk is zero in both of such cases.

The government treasury bills give less interest than the security market but it is totally risk-free and suppose if we expect our present portfolio will give a return of 20% and treasury bills is giving just  10% then some of us may like to take more risk that is associated with security investment. They may think sometimes to sell the treasury bills under their possession for buying more stocks from the market. Though it is always advisable to keep a good mix of treasury bills and stocks all the while.

Suppose portfolio P consisting of a different number of shares of selected stocks is predicted (based on past data) to have an expected return of 20% and a standard deviation of 18%. We have planned to mix some quantity of treasury bills that give an interest of suppose  10% p.a. on zero risk (s.d.).

Now, if we put half of our investment in market securities as in the portfolio mentioned above and half in the treasury bills then 

Return (r) = 1/2 x expected return on P + 1/2 x interest rate  = 15%

Variance = Standard Deviation2 = (σ2) =  X12σ12  +  X22σ22  + 2x1x2  ρ12 σσ2  

                                                                  σ  =  X1σ(since  σ2 = 0)

                                                                        = (.5) 18% =  9% 

Now, suppose we borrow money equal to our initial investment at the rate of treasury bill and invest that amount in our portfolio. Then,

Expected Return, r = 2 x 20% - 1 x 10% = 30%

And, Standard Deviation, σ  =  X1σ(since  σ2 = 0)

                                                          = 2 x 18% = 36%

So, if we assume that there is an option   for everyone to lend and borrow to any limit on the rate of treasury bills then the range of investment possibilities can be maximised to a further extent.

In the above graph please see the straight line. Below point M is the case of risk free lending while above M is the case of risk free borrowing. We can see that rightmost curves represent different portfolios of certain combinations of stocks. When they are combined together then we find the curve situated in middle and when we go further by introducing the treasury bills (buy/sell) or lending and borrowing, we find a straight line. The efficient portfolio at the tangent point M is better than all the others. It offers the highest ratio (Sharpe Ratio) of risk premium to standard deviation.

Sharpe Ratio = Risk Premium / Standard Deviation  =  r - r/  σ 

William Sharpe, John Lintner alnd Jack Treynor took this discussion to a more simplistic level in mid-60s. in the Capital Asset Pricing Model (CAPM) they said that 

"Expected Risk Premium varies in direct proportion to beta"        
                          r - rf   b (rm - rf)

where, Risk Premium (rm - rf)  is the difference between the investment's expected return and the risk -free rate
and beta is a measure of a stock's volatility in relation to the overall market. 

Now, before we conclude we must take a glance over the important alternative theories propounded in this regard. 

Stephen Ross's Arbitrage Pricing Theory (APT) is altogether on a different footing. It says that each stock's return depends partly on pervasive macroeconomic influences or 'factors' and partly on 'noise' events that are unique to that company. And so,

Return (r= a + b1 (r factor1) + b2 (rfactor2) b3 (rfactor3) + ......... + noise 

and, Risk Premium (r - rf)  b1 (r factor1- rf) + b2 (rfactor2- rf) b3 (rfactor3- rf) + .........

Making a more categorical statement Fama and French gave a Three-Factor Model.

r - rf  bmarket (r market factor) + bsize (rsize factor) bbook to market (rbook -to -market factor) .
In a nutshell it can be said that diversification of stocks in security investment helps tremendously in minimising risks though it is still to be found out as what are the actual base of ideology behind that. Though we must be cautious and logical while making investment in security market but there is no reason to hesitate as since 1978-79, the market risk premium (rm - rf) has averaged 9.81% a year in India.

(Author - Hemant K Das)


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