# 金融代写|利率理论代写portfolio theory代考|AN INTRODUCTION TO PREFERENCE FUNCTIONS

## 金融代写|利率理论代写portfolio theory代考|AN INTRODUCTION TO PREFERENCE FUNCTIONS

We start our formal discussion of the choice between risky assets with a simple example. Consider the two alternatives shown in Table 11.2. Investment $\mathrm{A}$ and investment $\mathrm{B}$ each have three possible outcomes, each equally likely. Investment A has less variability in its outcomes but has a lower average outcome.

One approach to choosing between them is to specify how much more valuable the large outcomes are relative to the small outcomes and then to weight the outcomes by their value and find the expected value of these weighted outcomes. The idea of adding up or averaging weighted outcomes is very common. Consider, for example, how the winning team is selected in hockey. Table 11.3 shows the hypothetical records for two hockey teams.

Current practice weights wins by two, ties by one, and losses by zero. With this weighting scheme, the Islanders would be leading the Flyers 100 to 95 . But there is nothing special about this weighting scheme. A league interested in deemphasizing the incentive for ties might weight wins by four, ties by one, and losses by zero. In this case, the Flyers would be considered the dominant team, 185 to 180 . If we denote $W$ as the result (win, tie, lose), $U(W)$ as the value of this result, and $N(W)$ as the number of times (games) that $W$ occurs, then to determine the better team, we calculate
$$\sum_W U(W) N(W)$$
The team with the higher $U$ is considered the better team. For example, utilizing current practice, $U$ (win) $=2, U$ (tie) $=1$, and $U$ (loss) $=0$. Applying the formula to the Islanders yields
$$U=2(40)+1(20)+0(10)=100$$
This is the 100 we referred to earlier. While the particular function $U(W)$ differs between situations, the principle is the same. Traditionally, instead of using the number of outcomes of a particular type, the proportion is used. There were 70 hockey games in our example. If $P(W)$ is the proportion of the total games that resulted in outcome $W$, then $P(W)=$ $N(W) / 70$. Dividing through by 70 will not affect the ordering of teams. Weighting a function by the proportion of each outcome is equivalent to calculating an average or expected value. Letting $E(U)$ designate the expected value of $U$ yields $^1$
$$E(U)=\sum_W U(W) P(W)$$
When we apply this principle to the decision problem shown in Table 11.2, we have special names for the principle. The weighting function is called a utility function and the principle is called the expected utility theorem. Consider the example shown in Table 11.2 and a set of weights as shown in Table 11.4.

## 金融代写|利率理论代写portfolio theory代考|RISK TOLERANCE FUNCTIONS

Note that the portfolio problem is expressed as a choice between mean returns and standard deviation of return. Thus any utility function can alternately be expressed the same way. This has resulted in a proposal to express expected utility maximization as maximizing
$$f=\bar{R}-\frac{\sigma^2}{T}$$
where $T$ is referred to as risk tolerance and expresses the investor’s trade-off between expected return and variance of return. The higher $T$, the “more tolerant” the investor is of risk and the higher the risk of the portfolio selected. Table 11.7 shows the choice for two investors: investor $\mathrm{A}$, with a risk tolerance of 100 , and investor $\mathrm{B}$, with a risk tolerance of 150. Their choices are applied to the investment problem shown in Table 11.1.

With these choices and risk tolerances, investor A would select investment 2 and investor B would select investment 3. One way to apply the risk tolerance idea is to simply use it to evaluate the investments being considered. When we assume riskless lending and borrowing, the optimum proportion to invest in the Tangency Portfolio $\left(X_T\right)$ and the

amount to lend or borrow $\left(1-X_T\right.$ ) can be determined directly. Using the preceding equation and substituting in the formula for the expected return and variance of the portfolio of debt and stock, finding the value of $X_T$ that maximizes the function yields ${ }^2$
$$x_T=\frac{T}{2}\left(\frac{\bar{R}_T-R_F}{\sigma_T^2}\right)$$
For the example discussed in Table 11.1, the Tangency Portfolio had a mean return of 10 and a standard deviation of 20 , and the riskless rate was $4 \%$. Thus, for investor $\mathrm{A}$, with a risk tolerance of 100 , we have
$$x_T=\frac{100}{2}\left[\frac{10-4}{400}\right]=\frac{6}{8}=\frac{3}{4}$$
And for investor $\mathrm{B}$, with a risk tolerance of 150 , we have
$$x_T=\frac{150}{2}\left[\frac{10-4}{400}\right]=1 \frac{1}{8}$$
Once again, to implement this, one needs to estimate an investor’s risk tolerance. Risk tolerance is easier to obtain from an investor because it is a single number. In implementing utility functions, one has to determine both the functional form of the investor’s utility function and the parameters. Although we can specify some general characteristics of utility functions.

# 利率理论代考

## 金融代写|利率理论代写portfolio theory代考|AN INTRODUCTION TO PREFERENCE FUNCTIONS

$$\sum_W U(W) N(W)$$

$$U=2(40)+1(20)+0(10)=100$$

$$U=2(40)+1(20)+0(10)=100$$

$$E(U)=\sum_W U(W) P(W)$$

## 金融代写|利率理论代写portfolio theory代考|RISK TOLERANCE FUNCTIONS

$$f=\bar{R}-\frac{\sigma^2}{T}$$

$$x_T=\frac{T}{2}\left(\frac{\bar{R}_T-R_F}{\sigma_T^2}\right)$$

$$x_T=\frac{100}{2}\left[\frac{10-4}{400}\right]=\frac{6}{8}=\frac{3}{4}$$

$$x_T=\frac{150}{2}\left[\frac{10-4}{400}\right]=1 \frac{1}{8}$$

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