VaR Under Shifts and Rescaling of Losses

Why Loss Transformations Matter

A risk measure is useful only when its units and underlying loss definition are clear. If every possible loss in a portfolio is increased by a fixed amount, its VaR changes in a predictable way. The same is true when every loss is multiplied by a positive factor, as can happen when the exposure is scaled or monetary amounts are converted using a fixed exchange rate. These rules make it possible to adjust a known VaR without rebuilding the entire loss distribution.

The key is to transform the whole loss random variable, not just one outcome or a rounded summary. The confidence level must remain the same, and a positive multiplier is essential for the simple scaling rule: it preserves the order of losses.

Definition: For a loss random variable $X$ and confidence level $p$, where $0<p<1$, define Value at Risk using the left-quantile convention $$ \operatorname{VaR}_p(X)=\inf\{x:F_X(x)\ge p\}, \qquad F_X(x)=P(X\le x). $$ In a discrete model this is the smallest loss value at which the cumulative probability is at least $p$. All transformation results below use this same definition and the same confidence level $p$.

Shifting Every Loss by a Constant

Let $b$ be a fixed constant and define $Y=X+b$. This operation adds the same amount to every possible outcome. It does not change the probabilities or the ordering of outcomes; it only moves each loss value by $b$. Therefore the VaR moves by that same amount.

Theorem: For any real constant $b$, $$ \operatorname{VaR}_p(X+b)=\operatorname{VaR}_p(X)+b. $$

The constant need not be positive. A fixed charge added to losses has $b>0$; a fixed credit or recovery applied to every outcome can be represented by $b<0$. The rule describes a deterministic adjustment to every outcome, not a random expense or recovery that varies with the loss.

Worked Example: Adding a Fixed Expense to a Discrete Loss

A hypothetical insurer models annual losses $X$, in thousands of dollars, as follows: $X=0$ with probability 0.60, $X=80$ with probability 0.25, and $X=200$ with probability 0.15. A fixed annual expense of $12,000 is allocated to the loss measure, so $Y=X+12$. Find the 85% VaR of $Y$, using the left-quantile definition.

Method. First find the original quantile by accumulating probability in increasing order of loss. Then apply the fixed shift.

At $X=0$, the cumulative probability is $0.60$, which is less than $0.85$. At $X=80$, it is $0.60+0.25=0.85$. Thus the smallest loss with cumulative probability at least $0.85$ is $\operatorname{VaR}_{0.85}(X)=80$ thousand dollars.

The transformed outcomes are $Y=12,92,212$, with probabilities $0.60,0.25,0.15$, respectively. The cumulative probability at $Y=12$ is $0.60$; at $Y=92$, it is $0.60+0.25=0.85$. Therefore $\operatorname{VaR}_{0.85}(Y)=92$ thousand dollars. The transformation rule gives the same result: $80+12=92$ thousand dollars.

Check. The 85% quantile has cumulative probability exactly 0.85, and the next loss value is larger. The result is consistent with the left-quantile definition. A common mistake is to add the expense to only the loss value at the quantile but then report it in dollars rather than thousands; the calculation here uses thousands throughout.

Multiplying Losses by a Positive Constant

Suppose $a>0$ and $Y=aX$. Multiplication by a positive number preserves the order of every pair of possible losses: if $x_1<x_2$, then $ax_1<ax_2$. It also preserves the probability attached to each outcome. Consequently, the loss at which the cumulative probability first reaches $p$ is multiplied by $a$.

Theorem: For $a>0$, $$ \operatorname{VaR}_p(aX)=a\operatorname{VaR}_p(X). $$ Combining a positive rescaling with a fixed shift gives $$ \operatorname{VaR}_p(aX+b)=a\operatorname{VaR}_p(X)+b. $$

The condition $a>0$ matters. A positive scale factor can represent a proportional change in exposure or a fixed conversion between monetary units. A zero factor collapses every outcome to zero. A negative factor reverses the ordering of outcomes, so the positive-scaling formula does not apply; the relevant quantile would depend on the opposite side of the distribution.

Worked Example: Coinsurance and a Fixed Payment

A hypothetical portfolio loss $X$, measured in thousands of dollars, has outcomes $0,50,100,200$ with probabilities $0.50,0.30,0.15,0.05$, respectively. A contract payment is $Y=0.8X+10$, also in thousands of dollars. Find $\operatorname{VaR}_{0.90}(Y)$.

Method. Find the 90% quantile of $X$ from its cumulative distribution, then apply the specified affine transformation. The 0.8 factor applies to $X$; the fixed 10 is added afterward.

The cumulative probabilities for $X$ are $0.50$ at 0, $0.50+0.30=0.80$ at 50, and $0.80+0.15=0.95$ at 100. Since $0.80<0.90\le0.95$, the smallest loss whose cumulative probability reaches 0.90 is 100. Hence $\operatorname{VaR}_{0.90}(X)=100$ thousand dollars.

Transform each possible outcome: if $X=0$, then $Y=0.8(0)+10=10$; if $X=50$, then $Y=0.8(50)+10=50$; if $X=100$, then $Y=0.8(100)+10=90$; and if $X=200$, then $Y=0.8(200)+10=170$. The probabilities remain $0.50,0.30,0.15,0.05$. At $Y=50$, cumulative probability is $0.80$; at $Y=90$, it is $0.95$. Thus the 90% VaR is 90 thousand dollars.

The combined transformation rule confirms the result: $0.8(100)+10=90$ thousand dollars. A common mistake is to interpret the contract as $0.8(X+10)$, which would give $0.8(100+10)=88$ thousand dollars. That is a different payment formula: the stated fixed payment is added after the 80% scaling, so the correct value is 90 thousand dollars.

Why the Rules Hold

The shift and scaling results follow directly from how the cumulative distribution changes. This argument also explains why the positive-factor condition is not optional.

Write the transformed cumulative distribution.
For $Y=aX+b$ with $a>0$, the event $Y\le y$ is equivalent to $aX+b\le y$, and therefore to $X\le (y-b)/a$. Thus $F_Y(y)=F_X((y-b)/a)$.

Locate the quantile condition.
The condition $F_Y(y)\ge p$ is equivalent to $F_X((y-b)/a)\ge p$. By the left-quantile definition, this requires $(y-b)/a$ to be at or above the $p$-quantile of $X$.

Find the smallest qualifying $y$.
Because $a>0$, multiplying an inequality by $a$ preserves its direction. The smallest qualifying $y$ is therefore $a\operatorname{VaR}_p(X)+b$, which proves the affine transformation rule.

F_Y(y)=F_X\!\left(\frac{y-b}{a}\right),\quad a>0 \quad\Longrightarrow\quad \operatorname{VaR}_p(Y)=a\operatorname{VaR}_p(X)+b.

If $a<0$, the event inequality reverses when dividing by $a$. The transformed lower-tail quantile is then connected to an upper-tail quantile of $X$, rather than to $a\operatorname{VaR}_p(X)+b$. This is why a formula derived for positive rescaling should not be carried over to a negative multiplier.

Worked Example: Transforming a Continuous Uniform Loss

Assume a hypothetical loss $X$, measured in thousands of dollars, is uniformly distributed on the interval from 2,000 to 10,000. Its cumulative distribution increases linearly from 0 to 1 over that interval. A revised exposure produces $Y=1.1X+400$, where 400 is a fixed amount in thousands of dollars. Find the 97.5% VaR of $Y$.

Method. For a uniform distribution on $[L,U]$, the cumulative probability at $x$ within the interval is $(x-L)/(U-L)$. Solving $(x-L)/(U-L)=p$ gives $\operatorname{VaR}_p(X)=L+p(U-L)$.

Here $L=2{,}000$, $U=10{,}000$, and $p=0.975$. The interval width is $10{,}000-2{,}000=8{,}000$. Therefore $\operatorname{VaR}_{0.975}(X)=2{,}000+0.975(8{,}000)=2{,}000+7{,}800=9{,}800$ thousand dollars.

Apply the affine transformation: $\operatorname{VaR}_{0.975}(Y)=1.1(9{,}800)+400=10{,}780+400=11{,}180$ thousand dollars. As a check, $Y$ is uniform from $1.1(2{,}000)+400=2{,}600$ to $1.1(10{,}000)+400=11{,}400$. Its 97.5% quantile is $2{,}600+0.975(11{,}400-2{,}600)=2{,}600+8{,}580=11{,}180$ thousand dollars, matching the transformation rule.

Common mistake avoided. Applying the fixed amount before scaling would produce $1.1(X+400)$, a different model. The stated model adds 400 after scaling, so the fixed amount is not multiplied by 1.1.

Interpreting the Result and Avoiding Traps

These transformation rules are exact when the stated relationship between losses holds for every outcome. They do not require a continuous distribution: the discrete examples show that probability masses and jumps in a cumulative distribution cause no problem under the left-quantile convention. Nor do they require a particular loss distribution, provided the same random variable is transformed as stated.

Before using a transformation, check the model and the units. In particular, determine whether the fixed amount is added before or after scaling, whether the multiplier is positive, and whether the reported VaR and adjustment use the same monetary unit. A deterministic shift is not interchangeable with an independent random loss, and multiplying VaR does not by itself describe the effect of changing the loss model in some other way.

Keep the confidence level fixed. The results compare quantiles at the same $p$; changing the confidence level is a separate change.
Use the stated loss definition. If the payment is $aX+b$, use $a\operatorname{VaR}_p(X)+b$. Do not silently change it to $a(X+b)$.
Preserve units. If losses are in thousands of dollars, a fixed amount stated in dollars must be converted before combining it with the loss values.
Do not infer the rule for a negative factor. The simple scaling identity requires $a>0$, so that loss ordering is preserved.

Key takeaway: With the left-quantile definition and a positive multiplier, a deterministic affine transformation $Y=aX+b$ changes VaR by the same transformation: $\operatorname{VaR}_p(Y)=a\operatorname{VaR}_p(X)+b$. The formula is reliable when the loss model, operation order, confidence level, and units are all stated correctly.

Check Your Understanding

For each problem, use the left-quantile definition and keep the stated units.

A hypothetical loss $X$, in thousands of dollars, is 0 with probability 0.40, 30 with probability 0.35, and 90 with probability 0.25. Find $\operatorname{VaR}_{0.75}(X-4)$. (Answer: 26 thousand dollars)
A hypothetical loss has outcomes 20, 60, and 120 thousand dollars with probabilities 0.55, 0.35, and 0.10. Find the 90% VaR after every loss is multiplied by 0.65. (Answer: 39 thousand dollars)
If $\operatorname{VaR}_{0.99}(X)=500$ thousand dollars and $Y=1.2X+20$, with amounts in thousands of dollars, find $\operatorname{VaR}_{0.99}(Y)$. (Answer: 620 thousand dollars)
A hypothetical loss $X$, in millions of dollars, is uniform on $[1,7]$. Find the 90% VaR of $Y=0.75X+0.2$. (Answer: 5.0 million dollars)
A proportional adjustment and fixed addition give $Y=aX+20$, where $a>0$ and amounts are in thousands of dollars. If $\operatorname{VaR}_{0.95}(X)=500$ and $\operatorname{VaR}_{0.95}(Y)=620$, determine $a$. (Answer: 1.2)

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