16) Define utility, certainty equivalent, risk aversion (referencing A, the risk aversion parameter), and indifference curve (each with one sentence). Explain the intuition behind equation 6.1 on page 160. With indifference curves, what is the preferred direction in expected return-standard deviation space? Why is the term mean-variance criterion used with reference to the issues of risk and return? Are the mean-variance criterion and the reward-to-volatility ratio related? If yes, how?
Answer 16
The usefulness of a commodity is known as a utility. It is the level of satisfaction an individual gets from consumption of a good or using a service. For example, what level of enjoyment one gets from watching a movie, eating a food item or wearing a shirt. However, utilities vary with the level of desire. If a person is very hungry, he would like a pizza so much in a new restaurant, however, another individual who is not very hungry at the moment does not enjoy it to the same level.
Certainly, equivalent is the term used for somebody who accepts that maximum output has been achieved from the input. In other words, the best value for money has been achieved for a good or service. Risk aversion means low return is acceptable to the person who is sending the product. The equation 6.1 shows how high expectations bring high utility returns. Indifferent curves allow the users to choose the best portfolio desired. Investor’s preferences of risk and return are represented in indifferent curves. The vertical line measures expected return while the horizontal line denotes risk is taken which is measured by standard deviation and variances. In the case of a risk-averse investor, for each additional unit of standard deviation, more return is expected where utility is assumed to be constant. The level of return for each additional unit becomes higher with an increase in the level of risk.
Criteria selection is based upon its means and variances. Either higher return for a given level of variance or expected return for a lower level of variance. The mean-variance criterion and the reward-to-volatility ratio are related as they give a mean-variance efficiency portfolio.
17) Explain figure 6.4, referencing in your answer the axis, intercepts, slopes, etc. In short, what does the capital allocation line (CAL) depict? Is the CAL theoretically limited by the endpoints of the risk-free asset and portfolio P? If not, why? How can the CAL be extended? Finally, what is the complete portfolio?
Answer 17
Figure 6.4 depicts a direct relationship between standard deviation and return expected at different levels. Risky and risk-free assets are both shown on the graph. F shows a risk-free asset while P shows a risky asset. The risky asset has a standard deviation of 22% and expected returns of 15%. As risk increases, so does the return shown by the slope of CAL (Capital allocation line). Capital allocation lines deal with an asset that is risky and risk-free asset at the same time. The portfolio P is the maximum Sharpe ratio that can be obtained. Sharpe ration describes how much extra return one is receiving for extra risk held in the portfolio. Sharpe ration greater than 1 is considered to be good. A high Sharpe ratio means a better return is being obtained by a specific investment. CAL connects risk-free assets with maximum Sharpe ratio. If an investor wishes to invest wholly in a stock related portfolio, then it is suggested that risk-free assets such as treasury bills and government bonds be purchased. This means you are lending money to the Government where no risk exists.
18) Explain what table 6.5 is depicting and what figure 6.7 suggests to the reader. Explain why the two indifference curves for U = .05 and U = .09 intercept the y axis at 0 (page 185). What does the slope of the indifference curve suggest? Combining the discussion of indifference curves, capital allocation lines, etc., what does figure 6.8 conceptually suggest? What is the significance of point C versus point P? Why are the indifference curves combined with the CAL important to conceptually understand?
Answer 18
Figure 6.5 shows indifferent curves of investors with different utility values. One’s utility value is 0.05, while the other’s utility value is 0.09. One curve is shown by A=2 and others by A=4. Both curves have been plotted in figure 6.7. One intercept is 0.05, and the other intercept on the graph is 0.09. A portfolio that exists with higher indifferent curve shows the investors desire high utility. At any given level of risk, a portfolio on higher indifferent curves offers higher returns. Despite having shape, both indifferent curves offer different returns at any volatility level. Portfolio on curve of utility 0.09 offers return that is 4% higher than the other curve with utility of 0.05.
More risk-averse investors are being reflected in figure 6.7 by showing a steeper indifferent curve. A steep curve means a high return is demanded by investors who are taking high risks. Figure 6.7 further continues to represent the plot of CAL as well. The figure identifies that the highest possible indifferent curve that exists at utility 0.06 and A4 still touches CAL. Point P denotes return at the highest level of risk. C shows an indifferent curve that is tangent to CAL and the point is optimal complete portfolio.
19) What is the difference between systematic and nonsystematic risk? Which, if either, can be diversified away? Please explain your answer in the context of figure 7.1, panel B (e.g., Why are “unique risk” sloped downward and “market risk” flat in this picture?) How is the insurance principle related, if at all, to the concepts of systematic and nonsystematic risk? Explain.
Answer 19
Risk is the variability of returns around the mean. It is taken by the variance of the stock known as total Risk. The portion of the stock’s variability that arises from specific risk factor is known as Unsystematic Risk. It is measured as the variance of error terms. Since unsystematic risk factors are randomly distributed, they tend to cancel each other out in a well-diversified portfolio. So unsystematic risk is diversifiable.
The portion of the stock’s variability that arises from systematic risk factors is known as systematic risk. Since beta is positive for all stocks, systematic risk is not diversifiable. If an investor puts his entire wealth in a single stock, he should be concerned with total risk. However, most rational investors appreciate the benefit of diversification and therefore hold diversified portfolios. In such portfolios, unsystematic risk is virtually nil, and what remains is only systematic risk.
Unique risk is sloped downward as these are unsystematic and randomly distributed and cancel each other out as the number of stocks in a portfolio increases. Market risk is flat as it is an only systematic risk which cannot be nullified. To hedge the portfolio, the investor should have a diversified mix of asset classes and minimize the risk.
20)What is the minimum variance portfolio? Within the context of the portfolio opportunity set for p = 0 (using figure 7.5 on page 214), discuss the expected return and standard deviation curve. Compare and contrast the p = 0 opportunity set to the p = 1 and p = -1 opportunity sets. What is the intuitive message regarding correlation that is being conveyed to the reader? Why did Tim suggest that correlation was the silver bullet of the asset management industry?
Answer 20
It is known as portfolio opportunity set because it shows all combinations of portfolio expected return and standard deviation that can be made from the two available assets. Different lines demonstrate the portfolio opportunity set for different estimations of the correlation coefficient. The strong dark line interfacing the two finances demonstrates that there is no profit by broadening when the relationship be tween’s the two is flawlessly positive (r = 1). The line with dashes shows higher benefits by showing diversification and correlation coefficient becomes greater than zero. At last, for r = – 1, the portfolio opportunity set is direct, yet now it offers an ideal supporting chance and the most extreme preferred position from diversification. The lower the relationship, the more noteworthy the potential profit by diversification. In the outrageous instance of perfectly negative relationship, we have an ideal supporting chance and can build a zero-change portfolio.
21) Figure 7.6 depicts portfolios A, B, D, and E. Explain how (e.g., with what mathematical tools) one can indicate that CAL (B) dominates CAL (A). With respect to figure 7.7 (page 217), explain the significance of portfolio P, the CAL(P), and the D-P-E opportunity set of risky assets. What is the intuition behind the equation depicted by 7.13; what is the intuition behind the equation depicted by 7.14?
Answer 21
A comparison of CAL-A & CAL-B will have been made, which would become Sharpe ration for every line. The higher the slope, the higher the return per unit of additional risk taken. Hence investors will be better off by the decision.
In Figure 7.7, portfolio P is the optimized risky portfolio with the Sharpe ratio being the highest form portfolios that are risky; hence investors would choose this one. CAL-P plots all possible combinations of portfolio P and risk-free assets.
Where an investor would lie on CAL (P) depends on her risk strategy, where her indifference curve touches CAL (P)). The D-P-E opportunity set is the frontier of optimized risky portfolios – which will have the smallest standard deviation for a given expected return.
In figure 7.14-point, P is observed where a set of risk aversive investors would end up. Investment in P and risk-free assets of the investment denotes the point where maximum utility can be enjoyed by the investor.
Equation 7.13 describes the proportion of capital, and a person has to invest in the two risky assets comprised of portfolio P such that portfolio P is the optimized risky portfolio giving the highest Sharpe ratio.
22) What is separation property (per Tobin)? In reality, do all managers have the same optimal portfolio? If no, why?
Answer 22
James Tobin addressed the following in his separation theory and got Nobel Prize:
Preferences you may have for liquidities, Pricing of the assets and separation overall.
Balancing the portfolio and various funds flow models along with the credit Channel.
Social security viewed in the light of the model for the life cycle
Method for econometrics
The simple policy in separation theorem says that diversification of your shares investment portfolio – say during the economic boom, the jeweler industry shares rise, but during recession, they fall – hence instead of buying 100% shares in gold companies, you must buy 30% of shares in gold, 30% in Fast Moving Consumer Goods (FMCG) like food, toothpaste, toilet soaps, shower jells, clothing, and dresses, and the remaining 40% in diversified among other commodities like oil, automobiles, gas etc
In lay man’s terms, by not putting all your eggs in the same basket, you save some of your eggs in case the basket drop. Hence it is the way to dilute the risk using separation or diversification.
Optimized risk portfolio set up is not affected by the level of or magnitude of risk aversion by the investors, for the following reasons:
When an investor, for example, B, has a neutral attitude towards risk, then the Coefficient of risk aversion reduces to ground zero. The curve becomes indifferent and hence has minimal or no impact on the setting up of optimized risk portfolio.
Utility U = E(r) – ½ * A * Chi * Chi
The indifference curve further slopes downwards, with risk lovers.
Optimal risk portfolio differs between investors for the following reasons:
The reasons are that the needs and financial situations of different investors are varied – one investor needs money in the short term but may be willing for smaller growth – but another investor may not need money immediately and hence willing for locking up his investments for a longer duration but may demand higher returns as a consideration.
Further reasons are illustrated in the graphs drawn.
23) Discuss the meaning of table 7.4 with a particular focus on the reduction in standard deviations for the p = 0 and p = .40 correlation cases. How should one interpret what happens to standard deviation at n = 20, 100, and 101? Mathematically, explain the derivation and implications of equation 7.21 in the context of diversification.
Answer 23
The table says that the reduction in standard deviation or risk is more when assets are less correlated that is the reason the drop in standard deviation decreases when co-relation=0 to 0.4
Reduction in standard deviation increases as you increase more stocks in a portfolio
Universal size means how many numbers of stocks is there in the portfolio. All have equal weight, so each one of it will weight 1/n
If n increases the reduction in standard deviation with adding one more stock decrease.
For co-relation of “zero”:
There would be a drop of 0.27%-point reduction in the standard deviation at n=20, if there is an addition of one more stock in the portfolio.
There would be a drop of 0.27%-point reduction in the standard deviation at n=20, if there is an addition of one more stock in the portfolio which shows that there is not much benefit if you keep adding stocks for diversification of you already have 100 stocks
At n=101, there will be 0.02 percentage point reduction in standard deviation if you increase 1 more stock in the portfolio. Which shows that there is not much benefit if you keep adding stocks for diversification of you already have 101 stocks?
For co-relation of “0.4”:
There would be a drop of 0.06%-point reduction in the standard deviation at n=20, if there is an addition of one more stock in a portfolio
There would be a drop of 0%-point reduction in the standard deviation at n=100, if there is an addition of one more stock in the portfolio. Which shows that there is not much benefit if you keep adding stocks for diversification of you already have 100 stocks?
There would be a drop of 0%-point reduction in the standard deviation at n=101, if there is an addition of one more stock in the portfolio. Which shows that there is not much benefit if you keep adding stocks for diversification of you already have 101 stocks?
A variance of portfolio=weight of each stock^2*variance of each stock+2*Co-relation*weight of each stock*weight of each stock*standard deviation of each stock*standard deviation of each stock=
As there will be n terms of variance and n*(n-1)/2 terms of co-variance
Variance of portfolio=n/n^2*variance of a stock+2*n*(n-1)/2*1/n^2*Co-relation*variance of each stock=variance of each stock/n+ (n-1)/n*co-relation*variance of each stock
If you keep adding stocks, there will be diversification benefits but only to a limit. There is not much benefit after 100 stocks. So, adding more than 100 stocks in the portfolio for diversification purposes is not good.
24) The correlation coefficients between pairs of stocks are as follows: Correlation (A,B) = .75; Correlation (A,C) = .55; Correlation (A,D) = .35. Each stock has an expected return of 6% and a standard deviation of 15%. If the entire portfolio is in stock A and you can only add some of one other stock to the portfolio, which stock—B, C, or D—would you choose? Explain.
Answer 24
Intuitively, it is expected that all the under-consideration stocks will have the same return rate and standard deviation. We will choose the stock with lowest risk. This shows that the required addition was the one with lowest co-relation with the Stock “A”, & that stock is “D”
Moreover, it is observed that the optimal portfolio is the global minimum variance portfolio (G) when all of the stocks have the same expected return rate for any risk-averse investor. It becomes the target to find the optimal portfolio (G) for any of the pairs which also includes “A” when portfolio is limited to Stock “A” with additional Stock, and the combination with lowest variance is selected. With Stocks I and J, the G weight formula would be:
w_Min (I)=(σ_J^2-Cov(r_I,r_J))/(σ_I^2+σ_J^2-2Cov(r_I,r_J))
w_Min (J)=1-w_Min (I)
While having total standard deviations equal to 20%:
Cov(r_I,r_J)=ρσ_I σ_J=400ρ ”and” w_Min (I)=w_Min (J)=0.5
The result shows that the required addition to stock A would be the stock having the lowest co-relation with it, that is, stock D. The optimized portfolio is equal investment in Stock A and Stock D, & standard deviation would be 17.03%.
25) Explain the concepts of risk pooling and the insurance principle. What does the insurance principle tell investors? In the same context, explain the concept and message of risk sharing. Finally, what is the implication of risk pooling and risk sharing for longer-term investing? Does risk fade in the longer run? Explain.
Answer 25
The risk from different insurance companies is bought together to form a pool, which is known as risk pooling. As per insurance principles, insurance contracts exist between insurer and insured. Insurance is based upon principles of good faith, principles of indemnity, contribution, and subrogation. This way risk is distributed over a huge number of individuals while each of them pays a very minimum cost. This amount is accumulated together, and a total figure is contributed. If any of the individuals from the pool of investors face any risks or mishap, then the pooled money is given to him/her. So, using risk pooling per head amount of the contribution is reduced for the insurance company. If the risk pooling activity continues, then ultimately risk factors will be eliminated from the loop and fade away.
