Utility Theory

Utility is used to measure an investor’s preferences. For deterministic outcomes, the following means that asset x is preferred over asset y:

\[ U(x) > U(y)\]

Often outcomes are uncertain, being phrased as gambles:

\[ G(A,B;\alpha) = \begin{cases}A,\text{ with prob }\alpha\\ B,\text{ with prob }(1 - \alpha) \end{cases}\]

Under uncertainty we consider expected utility:

\[ \mathbb{E}[U(w_1)]>\mathbb{E}[U(w_2)]\]

Utility Axioms

Axioms

Utility Functions

The choice of utility function \(U()\) should not be arbitrary, but rather a function with desirable properties for the problem.

Principle of Non-Satiation

Individuals prefer more wealth to less. Therefore we expect the utility function to always be increasing, no matter the type of investor.

\[ U'(w) > 0\]

Investor Types

We often consider the type of investor, grouping them according to risk aversion:
Risk Averse
- \(U(\mathbb{E}[w]) > \mathbb{E}[U(w)]\)
- \(U''(w) < 0\)
Risk Neutral
- \(U(\mathbb{E}[w]) = \mathbb{E}[U(w)]\)
- \(U''(w) = 0\)
Risk Loving
- \(U(\mathbb{E}[w]) < \mathbb{E}[U(w)]\)
- \(U''(w) > 0\)

Certainty Equivalent Wealth

Certainty equivalent wealth is the level of wealth that provides the same utility as the expected utility of a gamble.

\[ U(C(w)) = \mathbb{E}[U(W)]\quad\therefore C(W) = U^{-1}(\mathbb{E}[U(W)])\]

Risk premium is the difference between the expected wealth given a gamble, and their certainty equivalent wealth.
- The investor is willing to pay up to the risk premium to avoid the gamble.

\[ \pi(w,\text{Gamble}) = \mathbb{E}[w]-U^{-1}(\mathbb{E}[U(w)])\]

Absolute and Relative Risk Aversion

Absolute risk aversion is how an investor’s behavior changes with wealth
- Decreasing risk aversion means that with increasing wealth, the investor is more likely to take risk.

\[ A(w) = -\frac{U''(w)}{U'(w)}\]

Relative risk aversion refers to the change in the proportion of wealth invested in risky assets as wealth changes.
- Increasing Relative Risk Aversion means they will hoold a smaller proportion of wealth in risky assets as wealth increases.

\[ R(w) = wA(w)\]

Investment Risk Measures

Moments

Variance

\[ Var(X) = \mathbb{E}[(X-\mu)^2] = \mathbb{E}[X^2]-\mu^2\]

Semi-variance of return
- Accounts for variation below the mean

\[ \int^\mu_{-\infty}(x-\mu)^2f_x(x) dx\]

Skewness

\[ S(X) = \mathbb{E}\left[\left(\frac{X-\mu}{\sigma}\right)^3\right]\]

Kurtosis

\[ K(X) = \mathbb{E}\left[\left(\frac{X-\mu}{\sigma}\right)^4\right] \]

Shortfall Measures

Shortfall measures provide a measure of the expected under-performance relative to a given benchmark. The measure has the general form:

\[ \int^L_{-\infty}g(L-x)f_x(x) dx\]

Common cases for the function \(g()\) include:
\(g(L-x) = 1\), the short fall probability.

\[ \int^L_{-\infty}f_x(x) dx = P(X\leq L)\]

\(g(L-x) = (L-x)\), the expected short fall
\(g(L-x) = (L-x)^2\), the short fall variance.

Value-at-Risk Measure

Value-at-risk at a level \(\alpha\) is the value that we have \(<\alpha\%\) to incur a greater loss than. Aka we have a \(99\%\) chance that the loss will not be greater than the value-at-risk for \(1\%\)
- Often assume returns are often normal distributed, and the calculations follow this idea:

\[ P\left(Z<\frac{x-\mu}{\sigma}\right) = 1\%,\quad\therefore \frac{x-\mu}{\sigma}=N_{0.99}\quad (\text{Quantile of N dist)}\]

Relationship between Risk Measures and Utility Function

Important Moments can be linked to risk measures. For example if we expand the following, we will see the following uncertain wealth is dependent on the first and second moments of the random return. We interpret the following as the expected utility of an initial wealth \(+\) a random return X.

\[ \mathbb{E}[U(w_0 + X)]\]

Mean-Variance Portfolio Theory

Portfolio theory is one way to determine optimal asset allocation for investors.
- Mean-Variance theory is one of the most popular theories as it aligns with economical assumptions, and is validated through the Taylor expansion of expected utility of end of period wealth \(W\)
- Investors like more return and dislike more variance - assuming negative concavity of utility.

\[ U(W) = U(\mathbb{E}[W]) + U'(\mathbb{E}[W])(W-\mathbb{E}[W]) + \frac{1}{2}U''(\mathbb{E}[W])(W-\mathbb{E}[W])^2 + ...\]

\[ \mathbb{E}[U(W)] = U(\mathbb{E}[W]) + \frac{1}{2}U''(E[W])\sigma^2_W + \mathbb{E}[...]\]

Efficient Frontier

The efficient frontier represents the efficient portfolios given a set of securities. Efficient portfolios are those with the highest expected return for a given level of risk.
- If correlation between assets is \(<1\), a combination of these assets offer better risk-return opportunities than the individual assets.
  - The lower this correlation the more potential benefit

Global Minimum Variance Portfolio

The GMVP is the portfolio with the lowest level of risk possible, regardless of the expected return.
- This portfolio also separates the efficient and inefficient portfolios.

2 Risky Assets

The global minimum portfolio is solved through the constrained optimization problem:

\[ \min_{w_A,w_B} \sigma^2_P=w^2_A\sigma^2_A+w^2_B\sigma^2_B+2w_Aw_B\sigma_{AB},\quad w_A+w_B = 1\] * This is derived and solved to get the weights:

\[ w_A^{MVP} = \frac{\sigma^2_B-\sigma_{AB}}{\sigma^2_A+\sigma^2_B-2\sigma_{AB}},\quad w_B^{MVP} = 1 - w_A^{MVP}\]

3 or More Risky Assets

Consider the notation (with 3 assets as example):
- Note that \(\mathbf{z}\) is the mean vector

\[ \mathbf{z} = \begin{pmatrix}z_1\\z_2\\z_3\end{pmatrix},\quad\mathbf{\Sigma}=\begin{pmatrix}\sigma^2_1&\sigma_{12}&\sigma_{13}\\\sigma_{21}&\sigma^2_2&\sigma_{23}\\\sigma_{31}&\sigma_{32}&\sigma_{3}^2\end{pmatrix},\quad\mathbf{w}=\begin{pmatrix}w_1\\w_2\\w_3\end{pmatrix},\quad \mathbf{1}=\begin{pmatrix}1\\1\\1\end{pmatrix} \]

Expected Return:

\[ \mu_P = \sum^3_{k=1}w_kz_k = \mathbf{w}^\top\mathbf{z} = \mathbf{z}^\top\mathbf{w}\]

Portfolio Variance:

\[ \sigma^2_P = \sum^3_{k=1}w^2_i\sigma^2_i+\sum\sum2w_iw_j\sigma_{ij} = \mathbf{w}^\top\mathbf{\Sigma}\mathbf{w}\]

Optimization Problem

We are trying to find the minimum variance for a given expected value.

\[ \min_{\mathbf{w}}\frac{1}{2}\mathbf{w}^\top\mathbf{\Sigma}\mathbf{w},\quad \mathbf{w}^\top\mathbf{1} = 1,\quad \mathbf{w}^\top\mathbf{z} = \mu\]

Setting up the Lagrangian:

\[ L = \frac{1}{2}\mathbf{w}^\top\mathbf{\Sigma}\mathbf{w} + \lambda(1-\mathbf{1^\top w}) +\gamma(\mu - \mathbf{z^\top w})\]

Using the Lagrangian we get the following equations to solve:

\[ \frac{\partial L}{\partial\mathbf{w}} = \mathbf{\Sigma w}-\lambda\mathbf{1}-\gamma\mathbf{z}= 0\] \[ \frac{\partial L}{\partial\lambda} = \mathbf{1}-\mathbf{1^\top w}= 0\] \[\frac{\partial L}{\partial\gamma} = \mu - \mathbf{z^\top w}= 0\]

Common Notation

\[A = \mathbf{1^\top\Sigma^{-1}1}\]

\[B = \mathbf{1^\top\Sigma^{-1}z} = \mathbf{z^\top\Sigma^{-1}1}\]

\[C = \mathbf{z^\top\Sigma^{-1}z}\]

\[\Delta = AC-B^2\]

Solving the Equations

Solving for weights:

\[ \mathbf{w} = \lambda\mathbf{\Sigma}^{-1}\mathbf{1} + \gamma\mathbf{\Sigma}^{-1}\mathbf{z}\]

Solving for \(\lambda\) and \(\gamma\):

\[ \lambda = \frac{C - \mu B}{\Delta},\quad\gamma = \frac{\mu A-B}{\Delta}\]

Minimum Variance Portfolio

If we sub in the values we have found to the variance formula:

\[ \begin{split}\sigma^2_P &= \mathbf{w^\top\Sigma w}\\ &= \lambda\mathbf{w^\top 1} + \gamma\mathbf{w^\top z}\\ &= \lambda +\gamma\mu_P\\ &= \frac{A\mu^2_p-2B\mu_p+C}{\Delta}\end{split}\]

If we derive this with respect to \(\mu\) we can get the minimum in the mean-variance space. This is equivalent to the minimum variance portfolio.

\[ \frac{\partial\sigma^2_P}{\partial\mu_P} = \frac{2A\mu_P-2B}{\Delta}=0,\quad \mu_g = \frac{B}{A},\quad\sigma_g=\sqrt{\frac{1}{A}}\]

Subbing these values into the weight equation gets us the minimum variance portfolio weights:

\[ w_g = \lambda_g\mathbf{\Sigma^{-1}1}+\gamma_g\mathbf{\Sigma^{-1}z} = \frac{1}{A}\mathbf{\Sigma^{-1}1}\]

Spanning the MVF

We have the weight equation for any portfolio on the MVF:

\[ \mathbf{w} = \lambda\mathbf{\Sigma^{-1}1}+\gamma\mathbf{\Sigma^{-1}z}\]

If we consider two portfolios, one being the global minimum variance and one being one with specific weights:

\[ w_g = \frac{\Sigma^{-1}1}{A},\quad w_d=\frac{\mathbf{\Sigma^{-1}z}}{B}\]

Therefore we can create a portfolio with the weights:

\[ \mathbf{w}=\lambda A\mathbf{w}_g+\gamma B\mathbf{w_d},\quad \lambda A+\gamma B = 1\]

Therefore, we can combine \(w_g\) and \(w_d\) to create any other MVF portfolio
Some properties of the ‘d’ portfolio:

\[ \mu_d = \frac{C}{B},\quad\sigma^2_d=\frac{C}{B^2},\quad \mathbf{w_d^\top\Sigma w_g}=\frac{1}{A}\]

Two Fund Theorem

Any other 2 MVF portfolios can also span the frontier. We can consider a portfolio made up of these two portfolios, which can be combined to form any other minimum variance portfolio.

\[ \mathbf{w_a} = (1-a)\mathbf{w_g} + a\mathbf{w_d}\] \[ \mathbf{w_b} = (1-b)\mathbf{w_g} + b\mathbf{w_d}\]

One Fund Theorem

We can consider one risky asset and a risk free asset. The one fund theorem states that all efficient portfolios are a combination of the risk free asset and the tangency portfolio
The mean of this portfolio is:

\[ (1-w)r_f+w z\]

The SD of this portfolio is:

\[ w\sigma\]

MVP with Risk Free Asset

The minimization problem is now the following:
- Note that there is no weight constraint as this has been accounted for in this expected value constraint.

\[ \min_\mathbf{w}\frac{1}{2}\mathbf{w^\top\Sigma w},\quad (\mathbf{z}-r_f\mathbf{1})\mathbf{w}=\mu-r_f\]

Forming the Lagrangian with the following first order equations:

\[ L = \frac{1}{2}\mathbf{w^\top\Sigma w}+\gamma(\mu - r_f - (\mathbf{z}-r_f\mathbf{1})^\top\mathbf{w})\] \[ \frac{\partial L}{\partial\mathbf{w}} = \mathbf{\Sigma w}-\gamma(\mathbf{z}-r_f\mathbf{1})= 0\]

\[ \frac{\partial L}{\partial\gamma} = \mu - r_f - (\mathbf{z}-r_f\mathbf{1})^\top\mathbf{w}= 0\]

The solved weight equation is:

\[ \mathbf{w} = \gamma\mathbf{\Sigma^{-1}}(\mathbf{z}-r_f\mathbf{1})\]

From this we can solve for \(\gamma\) by subbing in the weight equation:

\[ \gamma = \frac{\mu-r_f}{(C-2r_fB+r_f^2A)}\]

Upon substitution, the minimum variance set is:

\[ \sigma^2 = \mathbf{w^\top\Sigma w}=\frac{(\mu-r_f)^2}{(C-2r_fB+r_f^2A)}\] \[ \mu = r_f\pm (C-2r_fB+r_f^2A)^{1/2}\sigma \]

Solving for the Tangency Portfolio

The tangency portfolio is when the sum of the weights in the risky assets is 1 (aka no weight in the risk free asset, \(1-w = 0\)). Therefore the weights should follow the following equation:

\[ 1=\mathbf{1^\top w}_t\]

Subbing in the weight equation we found for risky assets:

\[ 1 = \mathbf{1^\top}\gamma_t\mathbf{\Sigma^{-1}}(\mathbf{z}-r_f\mathbf{1})\] \[ \gamma_t = \frac{1}{B-Ar_f}\] \[ \mathbf{w}_t=\frac{1}{B-Ar_f}\mathbf{\Sigma^{-1}}(\mathbf{z}-r_f\mathbf{1})\]

Modern Portfolio Theory

Jake

21/10/2021