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Stuvia - Koop en Verkoop de Beste Samenvattingen EGARCH Similar to the GJR-GARCH model, the EGARCH model captures some kind of asymmetry in the volatility process. ε t −1 ε t−1 ε t −1 Log(σt²) =...

Stuvia - Koop en Verkoop de Beste Samenvattingen EGARCH Similar to the GJR-GARCH model, the EGARCH model captures some kind of asymmetry in the volatility process. ε t −1 ε t−1 ε t −1 Log(σt²) = ω + a. | σ t −1 −E [ ]| σ t−1 + b.log(σt-1²) + c. σ t −1 If b < 0 the negative shocks will have a bigger impact on future volatility than positive shocks of the same magnitude. The main advantage of this model is that it does not require any constraints on the coefficients, they may all become negative thanks to the log function! Not only the unconditional variance but also the conditional variance will now be leptokurtic. Chapter 4: From Utility to the Indifference Curve We have learned about risk and return but it is still hard to make a choice between competing investment opportunities. The Sharp ratio may give a hint about how much reward you may expect for an additional unit of risk, however you may still think “Is it worth it to take the risk?”. The answer may be very different depending on whom you ask. Obviously the investor’s personal preferences play a major role. The Markowitz portfolio selection approach will learn us how to separate individual preferences from the objective choice of an optimal portfolio. Portfolio (Expected) Excess SD of excess Sharpe return return ratio A: low risk 0,02 0,04 0,5 B: medium risk 0,04 0,08 0,5 1 C: medium risk 0,05 0,09 0,55 2 D: high risk 0,06 0,12 0,5 E: extremely 0,06 0,12 0,4 high risk If we assume that all investors base their decisions on a quadratic utility function of the following form: 1 U = E(r) -. c. σ² 2 U = utility E(r)= expected return on the asset or portfolio = excess return C = coefficient of risk aversion σ² = variance of returns The utility depends solely on the expected return (or excess return) of an investment and on its risk, measured by the standard deviation. Then we also Gedownload door: hannahducatteeuw | [email protected] Wil16 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen have parameter c, the higher c is, the higher the penalty for risk and the lower is the utility of an investment opportunity. C is the parameter for risk aversion, investors will suffer more from a high risk if their risk aversion is higher and vice versa. E [ r ] −r f Sharp ratio = σ ( r−r f ) Alternativ C=0 C = 0,5 C=6 C = 10 Sharpe e ratio A 0,0200 0,0196 0,0152 0,0120 0,50 B 0,0400 0,0384 0,0208 0,0080 0,50 C 0,0500 0,0480 0,0257 0,0095 0,55 D 0,0600 0,0564 0,0168 -0,0120 0,50 E 0,0600 0,0544 -0,0075 -0,0525 0,45 The higher the risk aversion the more the investor will tend to opt for an investment opportunity with lower return and lower risk. The choice depends on the investor’s risk-aversion, NOT on the Sharp ratio! When the alternatives are close together it is hard for the alternative with a lower sharp ratio to perform better. An alternative that is inefficient or mean-variance dominated, such as E, can never be the one exclusively yielding the highest utility. I II III IV I : they all dominate investment D, they provide a higher return at a lower risk IV: D dominates all these alternative investments because it offers a higher return at a lower risk II & III: this purely depends on the investor’s personal preferences. You get a lower return for a lower risk or a higher return of a higher risk… However this only applies to stand-alone investments! The question whether you should add an asset to a portfolio does not only depend on an asset’s individual risk but on how much it contributes to the portfolio’s total risk. An asset that offers a low expected return combined with a high risk may still be worth adding to the portfolio if it is weakly or negatively correlated with returns of the assets that are already included in the portfolio. Gedownload door: hannahducatteeuw | [email protected] Wil17 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  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?