Decision and Risk - Part 2

Contents: Value at Risk

Value-at-Risk is probably the most widely used risk measure in financial institutions. It is a measure for assessing the risk (i.e. the amount of potential loss) associated with an investment or a portfolio of investments.

Basel Accord II and III prescribed Value-at-Risk as the market risk to ensure that a bank has adequate capital to be able to absorb losses.
• Basel III is a set of international banking regulations established by the Basel Committee on Banking Supervision in order to promote stability in the international financial system.

Suppose a company holds a certain number of financial assets such as stocks, bonds, derivatives, etc. This collection of assets is called a portfolio. Everyday, the value of the portfolio will change, since the price of the individual assets in the portfolio change frequently. Therefore, it’s intuitive to ask “what is the probability of the value of the portfolio dropping by more than 10 million dollars on a given day?”. Knowing this is essential to risk management.

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Let the random variable L denote the loss of a portfolio over the period h. Then given a confidence level $\alpha \in (0,1)$, we define the term Value-at-Risk to be:

$$\text{VaR}_\alpha =\inf\{ l \in \mathbb{R}: P(L>l) \leq 1 - \alpha \} = \inf\{ l \in \mathbb{R}: F_L(l) \geq \alpha \}$$

In words, the Value-at-Risk of a given portfolio is the threshold such that the probability of losing more than this threshold over a time horizon h is equal to 1 − α. It’s is hence a measure of risk exposure.

For the sake of simplicity, we assume that the percentage change in the portfolio on each day t is a random variable $Y_t$, and that the percentage change on each day is independent of the percentage change on all other days. In other words, If $V_t$ denotes the value of the portfolio on day t, then $V_t = Y_t × V_{t−1}$ where $Y_t$ is drawn from some distribution $F_Y (⋅)$. By the above assumption, the $Y_t$ values are i.i.d.

Frequentist Approach

The VaR will be estimated based on historical data. Suppose the company has held the portfolio for n + 1 days, they hence have n observations $\mathbf{y} = (y_1, . . . , y_n)$. Assuming we know the true distribution $F_Y (⋅)$ of $Y_t$. Then we can compute the VaR as follows:

• Use $\mathbf{y}$ to estimate parameters $\mathbf{\theta}$ of $F_Y (⋅)$
• Given a confidence level α, estimate q by solving $\int^q_{-\infty} p(y_t|\mathbf{\theta}) \ dy_t = 1 - \alpha$
  • $q = \Phi^{-1}(\alpha)\ \sigma-\mu \text{ if we have } F_Y \sim N(\mu, \sigma^2)$ 
• q is the estimated VaR

Having the estimated VaR, we can also multiply the percentage $\text{VaR}_{\alpha}$ by the price at time t when it’s measured, to express VaR in value terms:

$$\text{VaR}^*_{\alpha,t} = \text{VaR}_{\alpha} \cdot V_t$$

Returns

The simple return, $R_t$, in the asset between dates t - 1 and t, is definded as:

$$R_t \equiv \frac{V_t}{V_{t-1}}-1$$

The gross return between dates t - 1 and t, is definded as:

$$1 + R_t \equiv \frac{V_t}{V_{t-1}}$$

A altervative choice to the gross return is its natural logarithm, called the log-return $r_t$. The additive property make $r_t$ easier to derive the time-series properties of asset returns over time:

$$r_t \equiv \log(1+R_t)=\log V_t - \log V_{t-1}$$

Note: When $R_t$ is small, $r_t$ is an good approximation of it: $r_t = \log(1+R_t) \approx R_t$.

Multiperiod VaR

So far we have measured VaR over 1 day. To estimate VaR over a longer risk horizon, there is a simple formula for scaling VaR from a 1-day horizon to an h-day horizon:

$$\text{VaR}_{\alpha,h} = \Phi^{-1}(\alpha)\ \sigma * \sqrt{h}-\mu * h$$

Note: The VaR measure can be easily extended to more than one asset, which we won’t discuss here.

Comments

• VaR is easy to understand and interpret.
• However, VaR does not say anything about the size of losses once they exceed VaR.
• Furthermore, one of the main objections to using VaR is that it is not a coherent risk metric, i.e. it does not satisfy: $\text{VaR}_\alpha(X+Y) \leq \text{VaR}_\alpha(X) + \text{VaR}_\alpha(Y)$ where X and Y are two financial assets. (An alternative is to use the expected shortfall )

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Several problems with this simple frequentist approach:

• It does not take uncertainty about $\mathbf{\theta}$ into account as we used point estimates (MLE). But these estimates will not be accurate, so that we could potentially underestimate risk because we are not taking this uncertainty into account.
• It does not incorporate prior information about the future portfolio returns – using historical data is important, but we may also have beliefs about the future which aren’t reflected in previous history.
• The usual issues about the difficulty of communicating frequentist statements to non-statisticians, and how they get misinterpreted.

We will hence explore a Bayesian approach to VaR analysis instead. In
this case we need to perform Bayesian inference for the parameters $\mathbf{\theta}$ of distribution $F_{Y_t}$.

Bayesian Approach

Let’s take $F_{Y_t} \sim N(\mu, \sigma^2)$. We start with a prior distribution $p(\mu, \sigma^2)$ on the unknown parameters of Gaussian distribution governing the percentage daily change $Y_t$, this is chosen to reflect our beliefs about the future portfolio returns. Remember: $\mu$ is the average return on a given day $E[Y_t]$, and $\sigma^2$ is the variance $Var[Y_t]$.

Let’s start with the simplest case: suppose that the variance $\sigma^2$ is known, so only $\mu$ needs to be estimated.

In this case, we have a conjugate prior $p(\mu) = N(\mu_0, \sigma^2_0)$ which represent our prior beliefs about the average
change in the portfolio value.
The next step is to update this to get the posterior $p(\mu, \sigma^2|y_1, ..., y_n)$ given the historial data, which captures all our knowledge about the distribution of $Y_t$ based on both our prior knowledge, and the historical data:

$$p(\mu | \mathbf{y}, \sigma^2) \propto \exp\left ( \frac{1}{2\sigma^2_n} (\mu - \mu_n)^2 \right)$$

where $\sigma^2_n = \left ( \frac{n}{\sigma^2}+ \frac{1}{\sigma^2_0} \right )^{-1}$ and $\mu_n = \sigma^2_n \left ( \frac{\mu_0}{\sigma^2_0} + \frac{\sum^n_iy_i}{\sigma^2}\right )$

(Please refer to Page 34-40, Lecture 4.pdf for detailed derivation)

Non-informative Priors

In practice we may not have strong prior beliefs about $\mathbf{\theta}$, or we do not want our analysis to be influenced by our prior beliefs. Then we can choose a prior that has an extremely large uncertainty and doesn’t impose much prior information on the likelihood. E.g. Beta(1,1)

Note: Non-informative Priors may be an improper prior, where the prior is no longer a valid probability distribution.

Posterior predictive distribution

Let be Y a random variable with a known distribution $p(y|\mathbf{\theta})$, where $\mathbf{\theta}$ is unknown. Before the data $\mathbf{y}$ is observed, the marginal distribution of the unknown but observable y is as follows:

$$p(y) = \int p (y, \theta) \ d\theta = \int p(\theta)p(y|\theta) \ d\theta$$

Once the data $\mathbf{y}$ have been observed, we can predict an
unknown $\widetilde{y}$ from the same process:

$$\begin{align*} p(\widetilde{y}|\mathbf{y}) & = \int p(\widetilde{y}, \theta| \mathbf{y}) \ d\theta \\ & = \int p(\widetilde{y}|\theta, \mathbf{y})p(\theta|\mathbf{y}) \ d\theta \\ & = \int p(\widetilde{y}|\theta)p(\theta|\mathbf{y}) \ d\theta \end{align*}$$

This is the fundamental equation of Bayesian prediction.