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Portfolio Optimization

Topic 1Risk-return trade-off


Meaning - The risk-return trade-off is the balance an investor must decide on between the amount of risk and the possible returns that he can earn.

Taking an example, consider a bet wherein you can win $10,000 on fighting and winning against a professional wrestler. The amount involved is high but the probability or risk defined is also high in correlation with the return involved. However, the risk-assuming capacity is different for every individual with respect to his/her desire for the return.

One of the best ways to describe this trade-off is using a graph, as shown underneath:


The risk-return trade-off tells that the lower risk investments were more likely to generate lower return. These include government issues securities and treasury bills which have negligible to little risk involved. On the other side of the spectrum, higher-risk investment potentially derives higher returns. These are highly risky securities in the forms of futures, forwards, and options.

The risk-return trade-off is never a guarantee. At every stage of the curve, if low risk or high risk, we can win or lose. So, in the end the risk-return trade-off is really measuring how much you are prepared to lose.


Free Lunch

The nature of this trade-off is often defined using a term called ‘free lunch’, which means that an investor can never earn a free lunch with his/her investment alternatives (except for fixed payment investment avenues). However, there exists a constant and continuous trade-off between risk and return due to which the free lunch or a cost-free return. In most cases, even with fixed payment instruments, some or the other costs in terms of fees, charges, brokerage, etc. are involved.

Factors Involved

The trade-off depends on a variety of factors including-

  1. Risk appetite 

  2. Age of the investor

  3. Years left to retirement

  4. Amount of funds

  5. Diversification capacity

  6. Level of knowledge

  7. Ability to accept losses

  8. Level of income stability


Calculation & Important Terms

  1. Alpha (α)

Alpha is defined as the additional return after returns are accounted and adjusted for market volatilities and ghost fluctuations.

For example, a positive alpha of 2 represents that the security outperformed its benchmark by 2%, and vice-versa.


  1. Beta (β)

Beta is simply the systemic risk of a security when a reference index or a point is taken into consideration. In simple terms, it defines the movement of a security with respect to the market movement.

A beta multiple of 2 means that the security is twice as volatile as the market volatility.


  1. Standard Deviation (σ)

Volatility or standard deviation is a determinant of the variation in returns of a security observed over time. A higher level of standard deviation implies a higher level of risk and is generally correlated directly with the return.


  1. Error term (ε)

Epsilon or error term, is the residual value for a parameter which may be not included in the equation and accounts for any significant leftover terms.


  1. Equation

The equation using the above terms is given by:

Y = α + βx + ε

Where x is the independent variable and y is the dependent variable


Risk Ranking of Securities

  • Government bonds

  • Municipal bonds

  • Investment-grade corporate bonds

  • High-yield corporate bonds

  • Blue-chip stocks and large-cap funds

  • Mid-cap stocks and funds

  • Small-cap stocks and funds

  • Overseas equities (only for less developed stock markets compared to one’s home country)

  • Futures & Options



Importance of Risk-Return Trade-off


  1. It is the very basic step of financial planning and analysis . Asset allocation facilitated by creating a diversified pool of securities is extremely crucial.

  2. Portfolio creation and achievement of financial goals is dependent on this trade-off. For a longer time period, it is advisable to have a higher risk, thus having a potentially higher return.

  3. Quantification of risk is very important by means of VaR and ES. A Value-at-Risk of Rs. 10,000 with a level of confidence of 95% suggests that there is a 5% probability of having losses equal to or more than Rs. 10,000.

    4.Direction of risk is important as well. Risk depends on return, but return doesn’t necessarily depend on risk. For example, by    putting all funds in a single security has higher chances of incurring huge losses, however, even by diversifying a portfolio, there isn’t a guarantee of having high returns.

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