Investment Portfolio Theory PDF

Stuvia - Koop en Verkoop de Beste Samenvattingen Now, if we fix the utility score U to a particular value and solve the equation for the expected return: E[r] = U + 0,5.c.σ² We then get all the risk-return profiles that are judged as being equally...

Stuvia - Koop en Verkoop de Beste Samenvattingen Now, if we fix the utility score U to a particular value and solve the equation for the expected return: E[r] = U + 0,5.c.σ² We then get all the risk-return profiles that are judged as being equally useful by an investor with the respective risk tolerance parameter c. This set of risk-return profiles form the indifference curve. The higher the risk aversion is, the more compensation one requires for taking on additional risk if the utility of an investment remains the same. Chapter 5: The capital allocation line We extend the situation from the previous chapter by allowing investments in two assets. We can invest in a risky asset portfolio (stocks, bonds, currencies, etc.) but also in a risk-free asset (T-bills). Assume that the two investment opportunities are characterized as followed: Asset Expected return Standard Deviation Risk-free Rf = 0,05 σf = 0 Risky E[rf] = 0,10 σf = 0,08 Of course the risk-free asset has a standard deviation of 0, there is no risk. Hence it is risk-free. And not surprisingly the risky asset provides a higher expected return. An investor will now have to decide which fraction if his wealth he invests in the risky and which fraction in the risk-free asset. Let q be the fraction he invests in the risk-free asset, then is 1-q the fraction invested in the risky asset. Return from the combined portfolio = weighted average of the returns of the two individual assets: rc = q.rf +(1-q).rp = E[rc] = q.rf +(1-q).E[rp] The standard deviation of the combined portfolio is: σc = (1 - q). σp  Because the standard deviation of the risk-free asset and the correlation between the risk-free and risky asset both equal 0. Gedownload door: hannahducatteeuw | [email protected] Wil18 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen There is no correlation between the risk-free asset and the risky asset, they are uncorrelated. If the equity market has a particular risk and you invest 10% of our portfolio in equity, then your portfolio has 10% of the volatility of the equity market. Q 0,0 0,2 0,4 0,6 0,8 1,0 E[rc] 0,10 0,09 0,08 0,07 0,06 0,05 σc 0,08 0,064 0,048 0,032 0,016 0,0 This table shows how the expected return and the risk of the combined portfolio evolve with the share of the riskless asset. We can therefore create an unlimited number of risk-return profiles for two given assets. The set of potential risk-return profiles is displayed in the following figure: capital allocation line (CAL) Starting at the left end of the CAL we are able to achieve a return of 5% and a risk of 0 by purely investing in the risk-free asset. Moving to the right means that we increase the share in of the risky asset. As a consequence the expected return increases, but so does the risk. Finally we reach a portfolio that provides an expected return of 10% and a standard deviation of 8. If we are not allowed to invest borrowed money, this is the maximum in terms of expected return and risk. Capital allocation line Represents all potential ways to allocate available funds using a risky and a riskless asset. The graph displays to investors the returns they can make by taking on a certain level of risk. CAL = Capital Allocation Line Any point on the line between the two extremes can be reached, depending on how you choose the weights in your portfolio. You can have a 100% risky portfolio (right) and a 100% riskless portfolio (left). E [ r p ]−r f The slope of the capital allocation line is α: α= σp This is nothing more than the Sharpe ratio of the risky portfolio! So the slope of the CAL solely depends on the available risky portfolio and the risk-free rate. But the point on the CAL that is realized by the investor is exclusively determined by the investor’s preferences. Normally this is done by finding an indifference curve that tangents the CAL, this is the one as far as possible to the north-west and therefore represents the highest possible utility level. The investor might be able to reach a higher utility level if he could find new assets to be included in the risky portfolio. This would mean to relax the assumption that he can only invest in the risky portfolio and the risk-free asset. Gedownload door: hannahducatteeuw | [email protected] Wil19 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen Another possibility to increase risk and return is to use leverage by borrowing money to invest. Let us assume the investor borrows money from a bank and invests this amount additional to his own wealth in the risky portfolio, so he invested more than 100% of his wealth. This mean that the CAL is simply prolonged behind the risky portfolio. Assume that the investors borrows enough to buy 1,5 times the risky portfolio, which means he has 50% leverage. His investment position in the risk-free asset will be q = -0,5  the negative sign indicates the debt position rc = (-0.5) (0.05) + (1.5) (0.10) = 0.125 σc = (1.5) (0.08) = 0.12 Obviously both risk and return from the leveraged portfolio exceed the figures of the original portfolio. He might earn a very big profit now, but at the same time he also risks to have a very big loss. However, the assumption that the investor can lend at the risk-free rate is unrealistic. So imagine the risk-free rate is 5% and you have to pay 7% if you want to borrow money. The “leveraged part” of your CAL will flatten… You would have a lower expected return. rc = (-0.5) (0.07) + (1.5) (0.10) = 0.115 σc = (1.5) (0.08) = 0.12 This lowers the investor’s excess return from the investment of the borrowed money. While the expected return of the combined portfolio is now 0,115 instead of 0,125, the risk of the investment is still the same. Of course this only applies to the leveraged part of the CAL! The part between the risk- free rate and the risky portfolio P remains the same. As a result the CAL is kinked! The slope coefficients are: E [ r p ]−r f For the leveraged part: α1 = σp E [ r p ] −r b For the part without leverage: α2 = σp rb is the interest rate that you need to pay for your credit. The origin between the two different slopes are the interest rates or cost of capital that is used as a reference point: for the left part the reference point is the risk-free rate because the costs of investing in the risky assets are the opportunity costs of not being able to invest in the risk-free asset. For the leveraged part of the CAL the credit costs are the relevant reference point. Gedownload door: hannahducatteeuw | [email protected] Wil20 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen If the correlation between the assets is low, you always benefit from a mix. Because that increases diversification. U TILITY AS A FUNCTION OF ALLOCATION TO THE RISK - FREE ASSET This figure displays the utility as a function of the continuously increasing share of the risk-free asset. For the risk neutral (c = 0) and the least risk averse investor (c = 5) the maximization problem leads to the corner solution with the weight of the riskless asset q = 0. For the investor with c = 10 there is only a slight increase in utility by investing in the riskless asset. While the more risk-averse investors with c = 15 and c = 20 find their maxima more to the right. Also note that all utility functions share the point with q = 1 and utility 0,05. Investing the whole wealth in the risk-free asset means zero volatility of the outcome and therefore the coefficient for risk-averseness, c, does not matter. The pink one doesn’t care about risk. You see that spending all his wealth on the riskless asset (right) has the lowest utility for him and the more you invest in risky assets, the higher his utility becomes (left). The purple one wants to be on the safe side and likes to invest as much as possible in the riskless asset. From a certain level of risk aversion (orange line), people will start to mix between risky and less risky assets because for them that will increase utility. If the lowest two start mixing they get a higher utility and they find already quite some impact. C APITAL ALLOCATION LINE AND PORTFOLIO CHOICE We see the CAL and some representative indifference curves for 2 kinds of investors: - A more risk-tolerating (c = 10) - A more risk-averse (c = 20) Separation principle: both investors use the same risky portfolio C = 10: opts for a more risky portfolio with a higher expected risk and return. The share of the risky asset is higher in his portfolio. Gedownload door: hannahducatteeuw | [email protected] Wil21 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?