Posted on Categories:Portfolio Theory, 利率理论, 金融代写

# 金融代写|利率理论代写Portfolio Theory代考|FINC6009 Investors’ Preferences

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## 金融代写|利率理论代写Portfolio Theory代考|Investors’ Preferences

Rational investors are risk averse. That is, they prefer a higher expected (mean) return and a lower variance or standard deviation. To rank all available risky assets, one needs to quantify an investor’s trade-off between risk and expected return. The foundations of utility theory rely on fundamental axioms of rational behavior for risky prospects with any general distribution. The investor’s utility function represents the investor’s preferences in terms of risk and return (i.e., her degree of risk aversion). Hence, investor preferences along with the risk-and-return characteristics of available portfolios, serve as the basis for selecting an optimal portfolio for a given investor, the portfolio that maximizes the investor’s expected utility.

One can show that under the assumption of normally distributed returns, the risk premium (i.e., the amount of mean return an investor is willing to give up in order to eliminate variance) is proportional to the product of a measure of relative risk aversion (RRA) by variance. The RRA could vary, possibly inversely, with the investor’s wealth. Most financial applications assume a constant RRA, which is a reasonable approximation for portfolio applications that do not involve enormous variations in wealth. This is the case for most investments over most horizons.

In summary, with a constant RRA, denoted as $\gamma$, the certainty equivalent return (CE) of an asset with mean return $\mu$ and variance $\sigma^{2}$ is written as shown in Equation 2.1:
$$\mathrm{CE}(\mu, \sigma)=\mu-0.5 \gamma \sigma^{2}$$
where the term after the minus sign is the Arrow-Pratt risk premium due to the investor’s risk aversion. The investor with risk aversion $\gamma$ is indifferent between a risk-free CE return and the portfolio with mean $\mu$ and variance $\sigma^{2}$. In the meanversus-standard deviation plot for a given CE, this is a parabola with intercept CE, known as an indifference curve. In Equation 2.1, a given value of CE generates one indifference curve, plotted in Figure 2.1. All the combinations of $\mu$ and $\sigma$ on the indifference curve are worth $\mathrm{CE}$ to the investor. The investor wants to invest in assets with the highest-possible CE lying on the highest-possible indifference curve. Various key combinations of risky and risk-free assets will now be considered.

## 金融代写|利率理论代写Portfolio Theory代考|One Risky Asset and the Risk-Free Asset

Consider a single risky asset $P$ with mean and variance $\left(\mu_{p}, \sigma^{2}\right)$ and a riskfree asset with return $R_{f}$ An asset allocation with weight $w$ in $P$ has a mean $R_{f}+w\left(\mu_{p}-R_{f}\right)$ and standard deviation $|w| \sigma_{p}$. By the constraint of full investment, the weight in $R_{f}$ is $1-w$. Both the mean and the standard deviation are linear in w. Therefore, in the typical case when $\mu$ is larger than $R_{f}$, the possible combinations with $w>0$ lie on a straight line, with an intercept $R_{f}$ and a positive slope $w\left(\mu_{p}-R_{f}\right) /\left(w \sigma_{p}\right)$; for instance, $\left(\mu_{p}-R_{f}\right) / \sigma_{p}$. This investment opportunity set is denoted the capital allocation line (CAL). The CAL is the line of possible portfolio risk-and-return combinations given the risk-free rate and the risk and return of a portfolio of risky assets. Negative weights span a mirroring line with a negative slope, which is of no interest because the CAL dominates it everywhere. Figure $2.1$ shows CALs for several portfolios, $P_{1}, P_{2}$, and others, in the meanversus-standard deviation space.

If these portfolios are mutually exclusive, investors must choose between mutually exclusive CALs. In Figure 2.1, the choice is unanimous because one of these lines, $C A L_{1}$, has a steeper slope than the others: it offers more expected return per unit of risk. For any allocation on another CAL, there are allocations on $C A L_{1}$ that dominate it unanimously; that is, with lower variance and an equal or higher mean (or with a higher mean and equal or lower variance). All investors agree, irrespective of their risk aversion, to rank mutually exclusive portfolios by the slope of their CAL, also known as the Sharpe ratio (Sharpe, 1966), denoted here as Sh:
$$\operatorname{Sh}(\mathrm{P})=\frac{\mu_{p}-\mathrm{R}{f}}{\sigma{p}} .$$

# 利率理论代写

## 金融代写|利率理论代写Portfolio Theory代考|Investors’ Preferences

$$\mathrm{CE}(\mu, \sigma)=\mu-0.5 \gamma \sigma^{2}$$

## 金融代写|利率理论代写Portfolio Theory代考|One Risky Asset and the RiskFree Asset

$$\mathrm{Sh}(\mathrm{P})=\frac{\mu_{p}-\mathrm{R} f}{\sigma p}$$

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