Scaling and Shifting Value-at-Risk

Why Scaling and Shifting Matter

Risk amounts are often reported in a different form from the original loss model. A per-policy loss may be scaled to represent a whole portfolio; an inflation factor may rescale all payments; a fixed expense may be added; or a result may be converted from one currency to another. For value-at-risk, these operations can be handled exactly when they are deterministic and the loss is transformed by a positive scale factor.

The key is to distinguish a genuine deterministic transformation from a change that alters the shape or ordering of outcomes. Multiplying every possible loss by the same positive number preserves which outcomes are larger. Adding the same fixed amount to every outcome also preserves their ordering. These facts are what make the value-at-risk rules work, including when the loss distribution has atoms.

Definition: For a loss $X$ and confidence level $0<\alpha<1$, define value-at-risk by the lower quantile convention $$ \operatorname{VaR}_{\alpha}(X)=\inf\{x:F_X(x)\geq\alpha\}, $$ where $F_X(x)=P(X\leq x)$. Thus, the value is the smallest loss level at which the cumulative probability is at least $\alpha$. This convention applies to both continuous and discrete losses.

The Transformation Rule

Let $a>0$ and let $b$ be a deterministic constant. Define a transformed loss $Y=aX+b$. The coefficient $a$ scales all outcomes, and $b$ shifts every outcome by the same amount. The transformation rule is

\operatorname{VaR}_{\alpha}(aX+b) =a\operatorname{VaR}_{\alpha}(X)+b,\qquad a>0.

The same confidence level $\alpha$ is used before and after transformation. A positive scale factor changes the units and size of the loss, not the percentile being measured. The shift may be positive or negative.

Why the rule holds: For any real $y$, when $a>0$, $$ F_{aX+b}(y)=P(aX+b\leq y) =P\left(X\leq\frac{y-b}{a}\right) =F_X\left(\frac{y-b}{a}\right). $$ Consequently, $F_{aX+b}(y)\geq\alpha$ exactly when $F_X((y-b)/a)\geq\alpha$. The smallest such $y$ is obtained by taking the smallest qualifying loss quantile of $X$, multiplying it by $a$, and adding $b$. Hence $\operatorname{VaR}_{\alpha}(aX+b)=a\operatorname{VaR}_{\alpha}(X)+b$. This argument uses the lower-quantile definition, so it remains valid when $X$ has a jump in its distribution function.

The rule has several immediate applications. Converting dollars to thousands of dollars uses $a=0.001$ and $b=0$. Applying a fixed inflation factor uses that factor as $a$. Adding a fixed expense uses $b$. If a transformed amount has both a scale and a shift, apply both terms to the original value-at-risk.

Transformation of loss	Value-at-risk	Condition
$aX$	$a\operatorname{VaR}_{\alpha}(X)$	$a>0$
$X+b$	$\operatorname{VaR}_{\alpha}(X)+b$	$b$ is deterministic
$aX+b$	$a\operatorname{VaR}_{\alpha}(X)+b$	$a>0$, $b$ deterministic
$0X+b$	$b$	The transformed loss is constant

Discrete Losses and Atoms

For a discrete loss, the quantile is found by accumulating probability until the chosen confidence level is reached. A common point of confusion is to use an outcome whose cumulative probability is merely close to $\alpha$, or to interpolate between outcomes. Under the stated convention, the first outcome at which the cumulative probability reaches or exceeds $\alpha$ is the value-at-risk. Once that quantile is identified, a positive scale and fixed shift transform it directly.

Worked Example: Transforming a Discrete Loss Quantile

A hypothetical equipment policy has loss $X$, in dollars, with the following distribution: $P(X=0)=0.70$, $P(X=1{,}000)=0.20$, and $P(X=4{,}000)=0.10$. The adjusted payment is $Y=1.25X+300$, where the factor represents a uniform adjustment and the fixed amount is a deterministic expense. Find $\operatorname{VaR}_{0.90}(X)$ and $\operatorname{VaR}_{0.90}(Y)$.

Method. First identify the lower 90th percentile from the cumulative probabilities; then apply the positive scaling and shift rule.

Step 1: Find the quantile of $X$. At $X=0$, the cumulative probability is $0.70$. At $X=1{,}000$, it is $0.70+0.20=0.90$. This is the first loss value at which the cumulative probability is at least $0.90$, so $$ \operatorname{VaR}_{0.90}(X)=1{,}000. $$

Step 2: Transform the quantile. Here $a=1.25>0$ and $b=300$, so $$ \operatorname{VaR}_{0.90}(Y) =1.25(1{,}000)+300 =1{,}250+300 =\$1{,}550.00. $$

Check. The transformed outcomes are $300$, $1.25(1{,}000)+300=1{,}550$, and $1.25(4{,}000)+300=5{,}300$, with cumulative probabilities $0.70$, $0.90$, and $1.00$. The 90th percentile is indeed $\$1{,}550.00$. The common mistake this method avoids is interpolating between outcomes or selecting the next outcome despite the cumulative probability already reaching $0.90$.

Per-Policy and Portfolio Amounts

A value-at-risk per policy can be multiplied by a policy count only when the portfolio loss is actually that deterministic multiple of the modeled per-policy loss. For example, if every policy’s loss is driven by the same random index and each policy incurs the same modeled amount $X$, then the total is $nX$, so the scaling rule applies. A general portfolio total $X_1+\cdots+X_n$ is not necessarily $n$ times one policy’s loss; it depends on how the individual losses vary and depend on one another. Therefore, multiplying a per-policy value-at-risk by the number of policies is not a general rule for aggregate risk.

Currency conversion is another direct scaling application when the exchange rate is fixed. If one unit of the source currency converts to $r>0$ units of the target currency, the converted loss is $rX$, and its value-at-risk is $r\operatorname{VaR}_{\alpha}(X)$. This calculation assumes that $r$ is deterministic for the measurement being made. If the exchange rate is random, the converted loss is not simply a fixed positive multiple of $X$.

Worked Example: From a Per-Policy Quantile to a Converted Portfolio Amount

In a hypothetical portfolio, a shared event index produces the same modeled loss $Z$ for each of 120 policies. The total loss is therefore $T=120Z$. The 97.5th-percentile per-policy loss is $\operatorname{VaR}_{0.975}(Z)=\$18{,}600$. A uniform adjustment multiplies the portfolio loss by $1.04$, and the adjusted result is converted at a fixed rate of $0.92$ euros per dollar. Find the 97.5th-percentile total in euros.

Method. Combine the deterministic multipliers and apply them to the per-policy quantile. The shared-event assumption is essential: it makes $T$ exactly $120Z$, not a general sum of independent policy losses.

Step 1: Scale to the portfolio. $$ \operatorname{VaR}_{0.975}(T) =120\operatorname{VaR}_{0.975}(Z) =120(18{,}600) =\$2{,}232{,}000. $$

Step 2: Apply the uniform adjustment. $$ 1.04(2{,}232{,}000)=\$2{,}321{,}280. $$

Step 3: Convert to euros. $$ 0.92(2{,}321{,}280) =2{,}135{,}577.60 =\text{€}2{,}135{,}577.60. $$

Check. The combined multiplier is $120(1.04)(0.92)=114.816$, and $114.816(18{,}600)=2{,}135{,}577.60$. The calculation preserves the 97.5% quantile throughout. The common mistake this method avoids is multiplying the per-policy quantile by 120 without checking whether the total loss is truly $120Z$.

Multiple Adjustments and Their Order

Several deterministic transformations can be combined algebraically. If $Y=aX+b$, followed by $W=cY+d$, where $a>0$ and $c>0$, then $$ W=caX+(cb+d). $$ It follows that $\operatorname{VaR}_{\alpha}(W)=ca\operatorname{VaR}_{\alpha}(X)+cb+d$. In particular, a shift applied before scaling is itself scaled; a shift applied afterward is not. Keeping the transformed loss expression visible helps prevent this order error.

Worked Example: A Shift Before Scaling

A hypothetical business loss $X$, measured in thousands of dollars, has $\operatorname{VaR}_{0.99}(X)=760$. A fixed allowance of 18 thousand dollars is added first, and the resulting amount is then multiplied by a retention factor of $0.60$. Find the 99th-percentile retained amount in dollars.

Method. Write the actual sequence as $Y=0.60(X+18)$, where all amounts inside the expression are in thousands of dollars. Expanding gives $Y=0.60X+10.8$, so the shift of 18 is also multiplied by 0.60.

Step 1: Transform the quantile in thousands. $$ \operatorname{VaR}_{0.99}(Y) =0.60(760)+10.8 =456+10.8 =466.8 $$ thousand dollars.

Step 2: Convert to dollars. $$ 466.8(1{,}000)=\$466{,}800.00. $$

Check. Directly applying the stated sequence to the original quantile gives $0.60(760+18)=0.60(778)=466.8$ thousand dollars, the same result. The common mistake this method avoids is calculating $0.60(760)+18=474$ thousand dollars, which incorrectly leaves the earlier shift unscaled.

Conditions and Common Exam Traps

The transformation rule is exact under its stated conditions, but it should not be applied merely because an expression contains a multiplier or an addition. The multiplier must be positive and deterministic, and the shift must be deterministic. A random adjustment can change both the ordering and the probabilities of outcomes. A nonlinear transformation, such as applying a deductible through $Y=\max(X-d,0)$, is not a single scale-and-shift transformation over all possible losses; its value-at-risk must be determined from the transformed distribution or by an appropriate quantile argument.

Using a negative multiplier as if it were positive. When $a<0$, larger values of $X$ become smaller values of $aX$, so the ordering reverses. The positive-scaling rule does not apply.
Scaling a shift that is applied afterward. For $aX+b$, the result is $a\operatorname{VaR}_{\alpha}(X)+b$. For $a(X+b)$, it is $a\operatorname{VaR}_{\alpha}(X)+ab$.
Confusing a unit conversion with an added amount. Converting dollars to thousands of dollars multiplies by $0.001$; it does not change the percentile level or add a fixed amount.
Assuming portfolio value-at-risk is policy count times per-policy value-at-risk. This is justified for an exact deterministic multiple, not for every aggregate of multiple policy losses.
Changing the confidence level during a transformation. Deterministic positive scaling and shifting preserve $\alpha$; they do not turn, for example, a 95th percentile into a 99th percentile.

A useful exam check is to write the full transformed loss in one line, including its units, before transforming the quantile. Then verify that the scale factor is positive, every shift is deterministic, the confidence level is unchanged, and the reported amount has the requested units. The central result is simple, but the conditions determine whether it is valid.

Key takeaway: Under the lower-quantile convention, a deterministic transformation $aX+b$ with $a>0$ has $\operatorname{VaR}_{\alpha}(aX+b)=a\operatorname{VaR}_{\alpha}(X)+b$. Use this rule for fixed scaling, fixed shifts, and fixed currency conversions; do not extend it automatically to random adjustments, nonlinear loss changes, or general portfolio sums.

Check Your Understanding

For each problem, use the lower-quantile convention and show the transformed loss before calculating its value-at-risk.

A loss $X$ has $\operatorname{VaR}_{0.95}(X)=\$8{,}400$. A revised payment is $1.10X+250$, with the shift in dollars. Find its 95th-percentile amount. (Answer: $\$9{,}490.00$)
A discrete loss is 0 with probability 0.60, 500 with probability 0.25, and 2,000 with probability 0.15. Find $\operatorname{VaR}_{0.85}(X)$, then find the value-at-risk of $2X+100$ at the same level. (Answer: $\operatorname{VaR}_{0.85}(X)=500$; transformed value-at-risk $=\$1{,}100.00$)
A loss has $\operatorname{VaR}_{0.99}(X)=320$ thousand dollars. A fixed amount of 20 thousand dollars is added, and then the result is multiplied by $0.75$. Find the 99th-percentile amount in dollars. (Answer: $\$255{,}000.00$)
A modeled per-policy loss $Z$ has $\operatorname{VaR}_{0.975}(Z)=\$12{,}500$. Under a hypothetical shared-event model, a 64-policy portfolio has total loss $64Z$. Convert its 97.5th-percentile amount to thousands of dollars. (Answer: $800$ thousand dollars)
A loss has $\operatorname{VaR}_{0.90}(X)=\$40{,}000$. A fixed exchange rate is 1.35 units of another currency per dollar. The converted loss is then increased by 2,000 units of that currency. Find the 90th-percentile amount in the target currency. (Answer: 56,000 target-currency units)

Transformation of loss	Value-at-risk	Condition
\(aX\)	\(a\operatorname{VaR}_{\alpha}(X)\)	\(a>0\)
\(X+b\)	\(\operatorname{VaR}_{\alpha}(X)+b\)	\(b\) is deterministic
\(aX+b\)	\(a\operatorname{VaR}_{\alpha}(X)+b\)	\(a>0\), \(b\) deterministic
\(0X+b\)	\(b\)	The transformed loss is constant

What You'll Learn