18.4 Capital Allocation
Allocate the total risk measure to each BUs for the following purposes:
Measure the amount of risk each BU contributes
Set capacity and policy limits for each BU
Calculate risk-adjusted profitability
RORAC: \(\dfrac{\text{Profit}}{\text{Allocated Capital}}\)
EVA = Profit - Cost of Capital
Definition 18.2
\(Y\): Losses of a company with CDF \(F(y)\)
\(X_j\): Losses for each BU, \(\sum_j X_j = Y\)
\(\rho(Y)\): Risk measure of \(Y\), \(\mapsto\) a real number
- \(\rho(Y) = \sum_j r(X_j)\)
\(r(X_j)\): Allocation of risk measure to the individual BUs
- Usually different from \(\rho(X_j)\), which is the risk measure applied to BU \(j\)
18.4.1 Proportional Method
Allocate risk measure in proportion to the risk measure applied to each unit
\[\begin{equation} r(X_j) = \dfrac{\rho(X_j)}{\sum_i \rho(X_i)} \rho(Y) \tag{18.2} \end{equation}\]- This works for any risk measure
18.4.2 Co-Measures
\[\begin{equation} r(X_j) = \mathrm{E}[h(X_j) \cdot L(Y) \mid g(Y)] \tag{18.3} \end{equation}\]Given that:
- \(\rho(Y)\) is expressed as a conditional expected value
\[\rho(Y) = \mathrm{E}[h(Y) \cdot L(Y) \mid g(Y)]\]
- \(h(\cdot)\) is additive
\[h(u+v) = h(u) + h(v)\]
\(\hookrightarrow\) \(\rho(Y) = \sum_j r(X_j)\)
- This is a marginal allocation that sums to the total risk measure
| Risk Measure | \(h(Y)\) | \(L(Y)\) | Condition | Co-Measure |
|---|---|---|---|---|
| \(TVaR\) | \(Y\) | \(1\) | \(F(Y) > \alpha\) | \(\mathrm{E}[X_j \mid F(Y) > \alpha]\) |
| \(VaR\) | \(Y\) | \(1\) | \(F(Y) = \alpha\) | \(\mathrm{E}[X_j \mid F(Y) = \alpha]\) |
| Standard Deviation1 | \(Y - \mathrm{E}[Y]\) | \(\dfrac{Y - \mathrm{E}[Y]}{\sigma_Y}\) | none | \(\dfrac{\mathrm{Cov}(X_j,Y)}{\sigma_Y}\) |
| \(XTVaR\)2 | \(\mathrm{E}[X_j \mid Y > b] - \mathrm{E}[X_j]\) |
- Note for s.d.:
\[\rho(X_j) = \dfrac{\mathrm{Cov}(X_j,Y)}{\sigma_Y} = \mathrm{E} \left [ (X_j - \mathrm{E}[X_j]) \cdot \dfrac{Y - \mathrm{E}[Y]}{\sigma_Y} \right ]\]
- For \(XTVaR\):
\[\rho(Y) = \mathrm{E}[Y - \mathrm{E}[Y ]\mid Y > b]\]
Note the condition is on loss amount \(Y\) not the percentile
Expected losses in unit \(j\) when the company has a loss above \(b\) less the expected losses in unit \(j\)
Remark. Under \(TVaR\) and \(XTVaR\)
At low threshold, unit with higher mean have higher allocation
- At higher threshold, unit with higher variance have higher allocation
18.4.2.1 Co-\(TVaR\) and -\(VaR\)
\(TVaR\):
\[r(X_j) = \mathrm{E}[X_j \mid F(Y) > \alpha]\]
- Average losses in unit \(j\), condition on when the firm (Just like in co-measures) has losses XS the \(\alpha^{th}\) percentile
\(VaR\):
\[r(X_j) = \mathrm{E}[X_j \mid F(Y) = \alpha]\]
Expected value of losses in unit \(j\), condition on when the firm has losses at the \(\alpha^{th}\) percentile
Caveat: Difficult to calculate because it is estimating a single point in the distn, results from sims can be quite variable
Typical you’re conditioning on the BU’s \(\alpha^{th}\) percentile
Not marginal if the condition is not on the firm’s \(\alpha\)
Marginal allocation means as unit grows, it is charged with the additional capital it requires
Not marginal means the individuals don’t sum to the total (?)
18.4.3 Having a Marginal Method
Definition 18.3 Marginal Allocation:
Allocation is marginal if the change to the company’s risk measure from a small change in a single BU’s (volume) is attributed to that business unit
Consistent with concept that price should be proportional to marginal cost
- Leads to suitable allocation (18.5)
Definition 18.4 Scalable Risk Measures:
A small proportional change in the business (e.g. 5%) change the risk measure by the same proportion (e.g. 5%)
This is Homogenous of degree 1:
\(\rho(aY) = a \cdot \rho(Y)\)
Most measures in currency units have this property (e.g. S.D., TVaR) but not variance (currency2) and probability of default (unitless)
- Scalable risk measures are both marginal and additive
Remark. For many companies and BUs, growth in exposure units can approximate homogeneous growth
Proportional change can be achieve from proportional reinsurance and normal growth
- Except for companies that write large policies compared to their volume; Transformed risk measure might still work
Definition 18.5 Suitable Allocation:
Growing a unit that have above average ratio of profit/risk will increase the overall ratio of profit/risk for the company
Given:
\[\dfrac{P_j}{r(X_j)} > \dfrac{P}{\rho(Y)}\]
We can show:
\[\dfrac{P + \epsilon P_j}{\rho(Y + \epsilon X_j)} > \dfrac{P}{\rho(Y)}\]
Where: \(P_j\) is the profit for the \(j\)th business unit and \(P\) is the profit of the company
- This is based on pluggin in the definition of \(r(X_j)\) below
18.4.4 Marginal Impact
A risk measure can have many co-measures but only 1 (the following form) will be marginal
\[\begin{equation} r(X_j) = \lim \limits_{\epsilon \rightarrow 0} \dfrac{\rho(Y + \epsilon X_j) - \rho(Y)}{\epsilon} = \dfrac{\partial \rho(Y)}{\partial X_j} \tag{18.4} \end{equation}\]This is the derivative of the firm risk measure w.r.t. growth in BU \(j\)
Even under non-homogeneous growth, the risk measure is still a decomposition (sum to the total) and is often close to marginal
\(XTVaR\)
\[\dfrac{\partial\rho(Y)}{\partial X_j} = r(X_j) = \mathrm{E}[X_j \mid F(Y) > \alpha] - \mathrm{E}[X_j]\]
- Scalable when done XS of a percentile \(\alpha\) (not when done XS a constant \(b\))
Variance
\[\rho(Y) = Var(Y)\]
\[r(X_j) = \mathrm{Cov}(X_j,Y)\]
- Not scalable (since \(Var(aY) = a^2 Var(Y)\)), so not marginal?
\(VaR\)
\[\rho(Y) = VaR_{\alpha}(Y) = \mathrm{E}[Y \mid F(Y) = \alpha]\]
\[r(X_j) = \mathrm{E}[X_j \mid F(Y) = \alpha]\]
Standard Deviation
- Spreading the s.d. \(\propto\) mean is not marginal:
\[\rho(Y) = Stdev(Y) = \mathrm{E}\left[ \dfrac{Y \cdot Stdev(Y)}{\mathrm{E}[Y]} \right]\]
\[r(X_j) = \mathrm{E}\left[ \dfrac{X_j \cdot Stdev(Y)}{\mathrm{E}[Y]} \right] = \dfrac{\mathrm{E}[X_j]}{\mathrm{E}[Y]} \cdot Stdev(Y)\]
- This is marginal: \(h(Y) = Y - \mathrm{E}[Y]\) and \(L(Y) = \dfrac{(Y - \mathrm{E}[Y])}{Stdev(Y)}\)
\[\rho(Y) = \mathrm{E}\left[ \dfrac{Y - \mathrm{E}[Y]^2}{Stdev(Y)} \right] = Stdev(Y)\]
\[r(X_j) = \mathrm{E}\left[ \dfrac{(X - \mathrm{E}[X])(Y - \mathrm{E}[Y])}{Stdev(Y)} \right] = \dfrac{\mathrm{Cov}(X_j,Y)}{Stdev(Y)}\]
Exponential Moment
\[\rho(Y) = \mathrm{E}[Y \cdot e^{cY/\mathrm{E}[Y]}]\]
\[r(X_j) = r_1(X_j) + \dfrac{\mathrm{E}[X_j]}{\mathrm{E}[Y]} \cdot c \cdot \mathrm{E} \left [ Y \cdot e^{cY/\mathrm{E}[Y]} \cdot \left \{ \dfrac{X_j}{\mathrm{E}[X_j]} - \dfrac{Y}{\mathrm{E}[Y]} \right \} \right ]\]
Without the curly bracket it is just allocating \(c \cdot \rho(Y)\) \(\propto\) the mean \(\dfrac{\mathrm{E}[X_j]}{\mathrm{E}[Y]}\)
Curly bracket (XS ratio factor) adjust for correlation, where BU that have large \(X_j\) when Y is also large will have additional allocation
BUs that don’t contribute to large losses will be negative in the curly bracket, receive reduced allocation
EPD
\[EPD_{\alpha} = (TVaR_{\alpha} - VaR_{\alpha}) \cdot (1 - \alpha)\]
\[\rho(Y) = \mathrm{E}[Y - B \mid F(Y) > \alpha] \cdot (1 - \alpha)\]
where \(B = VaR_{\alpha}(Y) = F_Y^{-1}(\alpha)\)
\[r(X_j) = (CoTVaR - CoVaR) \cdot (1-\alpha)\]
- Only scalable when XS of a percentile
\(RTVaR = TVaR + c \cdot Stdev(Y \mid F(Y) > \alpha)\)
\(h_1(Y) = Y\)
\(h_2(Y) = Y - \mathrm{E}[Y|F(Y) > \alpha]\)
\(L_1(Y) = 1\)
\(L_2(Y) = \dfrac{c[Y - \mathrm{E}[Y \mid F(Y) > \alpha]]}{Stdev(Y \mid F(Y) > \alpha)}\)
Condition:
\(F(Y) > \alpha\) for both
\(r(X_j) = \mathrm{E}[X_j \mid F(Y) > \alpha] + c \dfrac{Cov(X_j, Y \mid F(Y) > \alpha)}{Stdev(Y \mid F(Y) > \alpha)}\)
| Risk Measure | \(h(Y)\) | \(L(Y)\) | Condition | Co-Measure |
|---|---|---|---|---|
| \(TVaR\) | \(Y\) | \(1\) | \(F(Y) > \alpha\) | \(\mathrm{E}[X_j \mid F(Y) > \alpha]\) |
| \(VaR\) | \(Y\) | \(1\) | \(F(Y) = \alpha\) | \(\mathrm{E}[X_j \mid F(Y) = \alpha]\) |
| Standard Deviation | \(Y - \mathrm{E}[Y]\) | \(\dfrac{Y - \mathrm{E}[Y]}{\sigma_Y}\) | none | \(\dfrac{\mathrm{Cov}(X_j,Y)}{\sigma_Y}\) |
| \(EPD\) | \(X - \mathrm{E}[X \mid F(Y) = \alpha]\) | \(1-\alpha\) | \(F(Y) > \alpha\) | See 1. |
- \((CoTVaR - CoVaR) \cdot (1-\alpha)\)
18.4.5 Using Decomposition
If the allocated risk measure \(r(X_j)\) is a decomposition of the company risk measure \(\rho(Y)\) then we can use the measure to measure risk-adjusted profitability of a BU
\[\dfrac{P_j}{r(X_j)}\]
If the risk measure \(\propto\) the market value of risk \(\Rightarrow\) the BU with higher ratio have more profit relative to the value of risk they are taking
But since don’t know how to determine the market value of risk (i.e. don’t know which risk measure is \(\propto\) the value of risk) \(\Rightarrow\) Use several risk measures and hope one is close and that the indicated strategic directions are consistent with each other
Question on how to determine the market value of risk have not been settle yet, likely a risk measure on a transformed probability but we don’t know yet