> For the complete documentation index, see [llms.txt](https://zavalab.gitbook.io/statschemebook/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://zavalab.gitbook.io/statschemebook/chapter-7-decision-making-under-uncertainty/04-risk-measures.md).

# Risk Measures

* Risk measure is a scalar function $$\rho(Y(u))$$ that maps RV $$Y(u)$$ to a number.
* Risk measure is a *summarizing statistic*.
* Risk measure aims to extract key aspects/features of RV.
* We have mentioned some common risk measures that capture risk attitudes.
* We now present more sophisticated measures and analyze them in detail.
* Mean-Variance:

  $$
  \mathbb{MV}\_\kappa\[Y(u)]=\mathbb{E}\[Y(u)]+\kappa\cdot \mathbb{V}\[Y(u)]
  $$

  for some weight $$\kappa \in \mathbb{R}\_+$$; for $$\kappa\to 0$$ we emphasize the mean, for $$\kappa\gg 0$$ we emphasize the variance.
* Probability of Loss:

  $$
  \mathbb{P}(Y(u)>\bar{y})
  $$

  for some threshold $$\bar{y}\in \mathbb{R}$$. This is also often expressed as $$\mathbb{P}(Y(u)\leq \bar{y})$$.
* Event probability:

  $$
  \mathbb{P}(Y(u)\in \mathcal{A})
  $$

  where the event $$\mathcal{A}$$ captures conditions/constraints that $$Y(u)$$ should satisfy.
* Value-at-Risk:

  $$
  \mathbb{Q}\_{Y(u)}(\alpha)
  $$

  for probability $$\alpha\in \[0,1]$$. This is the $$\alpha$$-quantile of $$Y(u)$$.
* Conditional Value-at-Risk (CVaR):

  $$
  \mathbb{E}\[Y(u)|Y(u)\geq \mathbb{Q}\_{Y(u)}(\alpha)]
  $$

  for some probability $$\alpha\in \[0,1]$$. This is also known as the “expected shortfall”.
* Mean Absolute Deviation:

  $$
  \mathbb{E}\left\[|Y(u)-\mathbb{E}\[Y(u)]|\right]
  $$
* Worst-Case and Best-Case Value:

  $$
  \max Y(u)\quad \textrm{and} \quad  \min Y(u)
  $$
* Probability of loss resembles our colloquial definition of risk. But this fails to capture important aspects (magnitude of losses).
* There is a broad selection of risk measures that one can select.
* In fact, any statistic can be used as risk measure (e.g., mode, moments, median).
* It is also important to highlight that the selection of a risk measure should be based on nature of RV to be compared.
* e.g., comparing Gaussian RVs (symmetric RVs with left and right tails) is not the same as comparing exponential RVs (asymmetric RVs with only right tails).

## Example: Measuring Risk for Investment Optimization

{% hint style="danger" icon="flask" %}
We compute risk measures for investment example.

* Expected Value:

  $$\mathbb{E}\[Y(u)]=-70,891$$ USD and $$\mathbb{E}\[Y(u')]=-109,048$$ USD
* Standard Deviation:

  $$\mathbb{SD}\[Y(u)]=19,284$$ USD and $$\mathbb{SD}\[Y(u')]=79,288$$ USD
* Probability of Loss:

  $$\mathbb{P}(Y(u)>0)=0%$$ and $$\mathbb{P}(Y(u')>0)=9%$$
* Probability of Gain:

  $$\mathbb{P}(Y(u)\leq \bar{y})=6.4%$$ and $$\mathbb{P}(Y(u')\leq\bar{y})=55.90%$$ for $$\bar{y}=-100,000$$ USD.
* Value-at-Risk:

  $$\mathbb{Q}*{Y(u)}(\alpha)=-46,699$$ USD and $$\mathbb{Q}*{Y(u')}(\alpha)=-4,935$$ USD for $$\alpha=0.9$$.
* Conditional Value-at-Risk:

  $$\textrm{CVaR}*\alpha\[Y(u)]=-37,530$$ USD and $$\textrm{CVaR}*\alpha\[Y(u')]=34,810$$ USD for $$\alpha=0.9$$
* Worst-Case:

  $$\max Y(u)=-9,129$$ USD and $$\max Y(u')=181,413$$ USD.
* Best-Case:

  $$\min Y(u)=-140,609$$ USD and $$\min Y(u')=-363,708$$ USD.
  {% endhint %}

#### Python implementation: Risk measures for the two options

$$\mathbb{E}$$, $$\mathbb{SD}$$, probability of loss $$\mathbb{P}(Y>0)$$, probability of gain $$\mathbb{P}(Y\leq\bar{y})$$ with $$\bar{y}=-100$$ kUSD, $$\mathbb{Q}*Y(0.9)$$, $$\mathrm{CVaR}*{0.9}$$, worst case $$\max Y$$ and best case $$\min Y$$.

```python
def risk_measures(Y, alpha=0.9, y_bar=-100):
    Q = np.quantile(Y, alpha)
    return {'E': Y.mean(), 'SD': Y.std(ddof=1), 'P(loss)': np.mean(Y > 0), 'P(gain)': np.mean(Y <= y_bar),
            'VaR': Q, 'CVaR': Y[Y >= Q].mean(), 'max': Y.max(), 'min': Y.min()}

rm_I, rm_II = risk_measures(NPV_I), risk_measures(NPV_II)
print(f"{'measure':10s}{'Option I':>12s}{'Option II':>12s}")
for k in rm_I:
    fmt = '{:12.1%}' if k.startswith('P') else '{:12.1f}'
    print(f"{k:10s}" + fmt.format(rm_I[k]) + fmt.format(rm_II[k]))
```

```
measure       Option I   Option II
E                -70.6      -105.5
SD                19.1        74.8
P(loss)           0.0%        8.7%
P(gain)           6.7%       53.8%
VaR              -46.4        -8.5
CVaR             -37.4        24.3
max              -12.7       126.8
min             -124.8      -349.3
```

**Note:** As in the book, a risk-averse decision-maker prefers the first investment (no chance of losing money), while a risk-taker may accept a 9% chance of loss for the chance of large gains.