VaR & CaR¶

Why do we need to measure financial risk?¶

Creditworthiness $\implies$ ability to conduct business
Customers are their creditors (e.g. bank, insurance company, investment bank)
- Such firms are highly leveraged
Credit risk cannot be diversified nor can it be hedged.
- $\implies$ Customers are sensitive to this risk
- - Their attitude to creditor risk differs from investors' in cap markets.
The more business (=loans/credit) a financial firm has for a given amount of capital (Owner's equity). The greater the leverage. Also understand that some assets and securities are marked to market. #doubt
Any adverse shock can quickly wipe off its capital. Thus

Value at Risk (VaR)¶

Definition¶

VaR asks us

"What loss level is, $\ni$ we are $X\%$ confident that it will not be exceeded [by any other loss level] in $N$ business days?"

For a $99\%$ VaR, we are looking for a loss level $L$ such that

\[ P(\text{Loss}\geq L) = 1\%\text{ (or 0.01)} \]

VaR = loss level
- will not be exceeded
- with a specified probability
- in a specified period
It captures an important aspect of risk in a single number

Expected Shortfall¶

The expected loss, given that the loss > VaR

\[ C - VaR \]

Say, it did exceed VaR then, by how much?

ES answers the question

"If things doo get bad, just how bad will they be?"

Evaluation of VaR¶

Summary

Pros
- Single measure of capital adequacy (vs bankruptcy)
- Comparison and aggregation
- Decomposition of risk (risk budgeting)
- No distributional assumptions
Cons
- Not a general measure of risk
- Tail behavior unexplained
- Might fail to respect benefits of diversification

Pros¶

VaR is the single most popular measure of market risk of portfolios, for gauging CAPITAL ADEQUACY.

1. VaR provides a summary picture of capital adequacy¶

Quantifies in probabilities manner, how safe the firm is from events leading to bankruptcy.

E.g. If equity capital is $\$10M$ and 95% one year VaR is $\$12M$ $\implies$ There is a 5\% chance of losing more money than its capital

2. Facilitates comparison & enables aggregation¶

Comparison
- Across portfolios
- Across Markets
Aggregation across business units

E.g. Calculate VaR for $X_{1}$ and $X_{2}$ to compare

3. Facilitates decomposition of risk¶

As it is an aggregate forward looking¹ measure, it enables decomposition of risk into sources

asset class
manager or risk factor

This decomposition helps in reallocation of assets, setting limits or monitoring allocations and portfolio managers. $\implies$ Risk Budgeting.

4. Simply involves the point-in-tail identification¶

beyond which a mass of distribution (say 5%) lies.
Doesn't make any distributional assumptions concerning returns (we can choose our own). Continuous/Discrete can be used.

This simplicity has led to widespread use of VaR as a measure of risk.

Limitations¶

1. Solely a measure of downside risk¶

Downside risk = losses in the left tail of returns distribution.

Doesn't pay attention to the shape of the return distribution outside this tail.
Thus, should NOT be treated as a general measure of portfolio risk.

VaR should not be treated as a general measure of portfolio risk

Both the distributions have the same 99% VaR, i.e. -10. But Distribution 1 is clearly more risky!

	Distribution 1	Distribution 2
Outcome	Prob	Prob
-10	0.02	0.02
0	0.90	0.08
+10	0.08	0.90

2. Tail-behavior of returns unexplained¶

VaR doesn't say much about how returns behavior in the left tail itself.

Two distributions can have different left tails yet same VaRs

95% VaR for both distributions is -10. But (1) has more tail risk than (2).

	Distribution 1	Distribution 2
Outcome	Probability	Probability
- 50	0.025	0.000
- 10	0.035	0.06
+10	0.940	0.940

3. Might fail to respect benefits of diversification.¶

A distributions VaR could go up even when it becomes more diversified and intuitively less risky, an undesirable feature in a risk measure.

Hence, use and interpret VaR with care. It is only indicative and not comprehensive.

Historical Simulation to Calculate One-Day VaR or ES¶

Create database of daily moments in all market variables
Perform simulations! We are simulating the prices of tomorrow. Since we collected 501 days of historical data ($\text{Day 1}\to\text{Day 500}$) we can perform 500 simulations for what the prices tomorrow are going to be. Let's see what each simulation would look like
1. First simulation trial (Using $0\to 1$ days' movements)
  - Assume all \% changes in all market variable are as on the first day.
2. Second simulation trial (Using $1\to 2$ days' movements)
  - Assume all \% changes in all market variables are as on the second day
3. and so on… so $i$-th trial assumes: Value of the market variable tomorrow. $\hat{v}_{501} = v_{500} \dfrac{v_{i}}{v_{i-1}}$. Assuming, $v_{500}$ is the price of the variable today, and we are trying to estimate $\hat{v}_{501}$.

Example¶

Wish to calculate 1-day, 99% VaR or ES for Portfolio on July 8, 2020
We have four return Indices (S&P 500, FTSE 100, CAC 40, Nikkei 225) and their prices today (July 7, 2020)

Steps

Collect historical data (Day 0 to day 500)
Perform 100 simulations with $v_{500} \times \dfrac{v_{i}}{v_{i-1}}$
Sort all the losses in various scenarios.
We want the 99% VaR $\implies$ Take the fifth worst loss (out of 500, which will be 99th percentile)
We want the 99% one day ES $\implies$ Take the average of the five worst losses.

Cash Flow at Risk (CaR)¶

Cash Flow Shortfall = $E(\text{Cash Flow}) -\text{Actual CF}$

Cash Flow at Risk (CaR) at $p\%$ is cash-flow shortfall $\ni$ there is a probability of $p\%$ that the firm will have a larger cash flow shortfall.

CaR Example

If the $E(CF) = \$80$ and forecasted volatility² = $\$50$ then.

CaR at 5% $= -1.65 \times 50 = -82.5$

NOTE that here we are only looking at the $z_{0.05}\times \sigma$ value since it is the shortfall and not the value we are expecting.
$v_{i}:$ Value of variable on day $i$
There are 500 simulation trial

Prospective, not retrospective. It is speculative. ↩
GARCH!?? ↩