Moments of Scaled and Shifted Severity

Why Transforming a Severity Changes Its Moments

A severity may be recorded in dollars, thousands of dollars, or a price level different from the one needed for an analysis. These changes can be represented by a simple transformation: multiply the original loss by a constant and, if appropriate, add another constant. Getting the resulting mean right is only part of the task. Variance is measured in squared units, so it responds differently from the mean.

Let $X$ be a loss severity with finite mean and variance, and let $Z=aX+b$, where $a$ and $b$ are fixed constants. For example, $a$ can convert dollars to thousands of dollars or apply an inflation factor. A fixed $b$ can represent a constant addition in the same monetary units as $Z$. These are mathematical transformations; any product or adjustment described below is hypothetical.

Definition: For a severity $X$ with $E[X]$ and $\operatorname{Var}(X)$ finite, an affine transformation is $Z=aX+b$, where $a$ and $b$ are constants. Its mean and variance are $E[Z]=aE[X]+b$ and $\operatorname{Var}(Z)=a^2\operatorname{Var}(X)$. The standard deviation is $\sqrt{\operatorname{Var}(Z)}=|a|\sqrt{\operatorname{Var}(X)}$.

The mean is shifted by $b$, because every possible value increases by that amount. The variance does not change when a constant is added: variance measures spread around the mean, and both each value and the mean move together. Multiplying by $a$, however, multiplies each deviation from the mean by $a$; squaring deviations therefore introduces $a^2$.

Deriving the Mean and Variance Formulas

The derivation follows directly from the definitions of expectation and variance. Assume the moments used below exist, and take $a$ and $b$ to be nonrandom constants.

Find the transformed mean.
Linearity of expectation gives $E[Z]=E[aX+b]=aE[X]+b$.

Write a deviation from the transformed mean.
Since $E[Z]=aE[X]+b$, $Z-E[Z]=aX+b-(aE[X]+b)=a(X-E[X])$.

Square and take the expectation.
By the definition of variance, $\operatorname{Var}(Z)=E[(Z-E[Z])^2]=E[a^2(X-E[X])^2]=a^2\operatorname{Var}(X)$.

The derivation explains two frequent exam errors: using $a$ instead of $a^2$ for variance, and adding $b$ to variance. Neither operation matches the definition of variance. If $a=0$, then $Z=b$ is constant and its variance is zero, also as the formula states. To recover $X$ from $Z$, division by $a$ is possible only when $a\ne 0$.

Raw Moments and Central Moments After Transformation

A raw moment of order $r$ is $E[X^r]$; the first raw moment is the mean, and the second is used to calculate variance. For a nonnegative integer $r$, if the moments on the right exist, the binomial expansion gives the transformed raw moment:

E[Z^r] = E[(aX+b)^r] = \sum_{k=0}^{r}\binom{r}{k}a^k b^{r-k}E[X^k], \qquad E[X^0]=1.

For $r=2$, this becomes $E[Z^2]=a^2E[X^2]+2abE[X]+b^2$. This is consistent with the mean-and-variance result: since $\operatorname{Var}(Z)=E[Z^2]-(E[Z])^2$, substituting $E[Z]=aE[X]+b$ leaves $a^2(E[X^2]-(E[X])^2)=a^2\operatorname{Var}(X)$.

More generally, the $r$th central moment is $E[(X-E[X])^r]$. Because $Z-E[Z]=a(X-E[X])$, its transformed central moment is $a^rE[(X-E[X])^r]$, provided that moment exists. In particular, variance is the second central moment and scales by $a^2$. For negative $a$, the mean transformation still holds, while the standard deviation uses $|a|$, not $a$.

Worked Example: Converting Dollars to Thousands of Dollars

A hypothetical portfolio has loss severity $X$, measured in dollars, with $E[X]=18{,}000$ dollars and $\operatorname{Var}(X)=3{,}240{,}000$ dollars squared. Define $Z$ as the same loss measured in thousands of dollars. Find $E[Z]$, $\operatorname{Var}(Z)$, and the standard deviation of $Z$.

Method. A loss in thousands of dollars is the dollar amount divided by $1{,}000$. Thus $Z=0.001X$, so $a=0.001$ and $b=0$.

Mean. Apply the mean formula: $E[Z]=0.001(18{,}000)+0=18$. The mean is $18$ thousand dollars.

Variance. Apply the squared scale factor: $\operatorname{Var}(Z)=(0.001)^2(3{,}240{,}000)=0.000001(3{,}240{,}000)=3.24$. The variance is $3.24$ thousand-dollars squared.

Standard deviation. Take the square root: $\sqrt{3.24}=1.8$ thousand dollars. As a check, $1.8$ thousand dollars equals $1{,}800$ dollars, which is also $\sqrt{3{,}240{,}000}$ dollars.

Common mistake avoided. Multiplying the variance by $0.001$ would give $3{,}240$, not $3.24$. The variance must use $(0.001)^2$, because it is based on squared deviations.

Applying a Price-Level Adjustment

Suppose an old loss amount is to be expressed at a newer price level. If the applicable price index ratio is a fixed factor $c>0$, then the adjusted amount is $cX$. Its mean is $cE[X]$, and its variance is $c^2\operatorname{Var}(X)$. The factor is applied to the variance twice in the sense that it is squared, not because two separate adjustments are being made.

This calculation assumes that the same fixed factor applies to every loss being transformed. It is not the right model if the adjustment varies randomly from claim to claim or depends on the size of the claim. It also does not by itself account for changes in the distribution of losses beyond the stated scale adjustment.

Worked Example: Adjusting a Hypothetical Loss for a Higher Price Level

A hypothetical severity $X$, measured in dollars at an earlier price level, has $E[X]=2{,}500$ dollars and $\operatorname{Var}(X)=250{,}000$ dollars squared. A fixed index factor of $1.08$ converts amounts to a later price level. A hypothetical fixed adjustment of $35$ dollars is then added to each adjusted loss. Find the mean, variance, and standard deviation of the resulting severity $Z=1.08X+35$.

Method. Identify $a=1.08$ and $b=35$. The index factor scales both the mean and the standard deviation; the fixed addition affects the mean but not the variance.

Mean. First calculate the scaled mean, $1.08(2{,}500)=2{,}700$. Then add the fixed amount: $E[Z]=2{,}700+35=2{,}735$ dollars.

Variance. Square the scale factor: $(1.08)^2=1.1664$. Therefore, $\operatorname{Var}(Z)=1.1664(250{,}000)=291{,}600$ dollars squared.

Standard deviation. $\sqrt{291{,}600}=540$ dollars, since $540^2=291{,}600$.

Check. The original standard deviation is $\sqrt{250{,}000}=500$ dollars. Scaling it by $1.08$ gives $540$ dollars, as found from the transformed variance. Adding $35$ dollars changes the mean from $2{,}700$ to $2{,}735$, but does not change that spread.

Common mistake avoided. The incorrect variance calculation $1.08(250{,}000)+35$ both fails to square the scale factor and incorrectly adds the fixed amount to the variance. The correct calculation is $(1.08)^2(250{,}000)$.

Recovering Moments From a Transformed Severity

The formulas can also be reversed. If $Z=aX+b$, $a\ne 0$, and the mean and variance of $Z$ are known, then $E[X]=(E[Z]-b)/a$ and $\operatorname{Var}(X)=\operatorname{Var}(Z)/a^2$. The nonzero condition matters: when $a=0$, $Z=b$ contains no information about the original severity.

Worked Example: Recovering Original Moments and Checking a Raw Moment

A hypothetical reported severity is $Z=1.25X+50$, with amounts measured in dollars. The reported transformed severity has mean $1{,}050$ dollars and variance $22{,}500$ dollars squared. Recover the mean and variance of $X$, then check the transformed second raw moment using the raw-moment formula.

Method. Since $a=1.25\ne 0$, invert the mean and variance formulas. To check the second raw moment, first calculate $E[X^2]$ from $\operatorname{Var}(X)=E[X^2]-(E[X])^2$.

Original mean. Subtract the shift and divide by the scale: $E[X]=(1{,}050-50)/1.25=1{,}000/1.25=800$ dollars.

Original variance. Since $(1.25)^2=1.5625$, $\operatorname{Var}(X)=22{,}500/1.5625=14{,}400$ dollars squared. This division checks because $14{,}400(1.5625)=14{,}400+0.5625(14{,}400)=14{,}400+8{,}100=22{,}500$.

Original second raw moment. $E[X^2]=\operatorname{Var}(X)+(E[X])^2=14{,}400+800^2=14{,}400+640{,}000=654{,}400$.

Transformed second raw moment. The raw-moment formula gives $E[Z^2]=(1.25)^2E[X^2]+2(1.25)(50)E[X]+50^2$. The three terms are $1.5625(654{,}400)=1{,}022{,}500$, $125(800)=100{,}000$, and $2{,}500$. Thus $E[Z^2]=1{,}125{,}000$ dollars squared.

Check. Directly, $E[Z^2]=\operatorname{Var}(Z)+(E[Z])^2=22{,}500+1{,}050^2=22{,}500+1{,}102{,}500=1{,}125{,}000$, matching the raw-moment calculation.

Common mistake avoided. Dividing the reported variance by $1.25$, rather than by $1.25^2$, would not reverse the variance transformation. The shift $50$ is subtracted when recovering the mean, but it is not subtracted from the variance.

Exam Traps and Practical Checks

Before using an affine transformation, identify what the constants mean and confirm that their units are consistent. The scale $a$ must be a ratio or unit-conversion factor; the shift $b$ must have the units of the transformed loss. For example, adding $35$ dollars to a loss expressed in thousands of dollars requires first converting $35$ dollars to $0.035$ thousand dollars.

Do not confuse variance with standard deviation. Variance scales by $a^2$; standard deviation scales by $|a|$.
Do not add a fixed shift to variance. The shift changes every value and the mean equally, leaving each deviation unchanged.
Keep squared units visible. If $X$ is measured in dollars, its variance is in dollars squared. If $Z$ is measured in thousands of dollars, its variance is in thousand-dollars squared.
Check whether the transformation is invertible. Recovering moments by division requires $a\ne 0$.
State what is fixed. The formulas use nonrandom constants. A random scaling factor or random addition requires additional information and different calculations.

Key takeaway: For $Z=aX+b$, the mean becomes $aE[X]+b$, while the variance becomes $a^2\operatorname{Var}(X)$. Check the units, square the scale factor for variance, and leave a fixed shift out of the variance calculation.

Check Your Understanding

For each hypothetical severity, identify the affine transformation and apply the appropriate moment formulas. Answers are given after each problem.

A loss $X$ is measured in dollars with mean $6{,}400$ and variance $160{,}000$. Let $Z=X/1{,}000$, measured in thousands of dollars. Find the mean and variance of $Z$. (Answer: mean $6.4$ thousand dollars; variance $0.16$ thousand-dollars squared.)
A hypothetical loss has mean $900$ dollars and standard deviation $120$ dollars. A new amount is $Z=1.10X+25$. Find its mean, variance, and standard deviation. (Answer: mean $\$1{,}015$; variance $\$17{,}424$ squared; standard deviation $\$132$.)
A severity $Z=0.80X$ has variance $10{,}000$ dollars squared. Find the standard deviation of $X$. (Answer: $\$125$.)
A transformed severity is $Z=2X+40$, with mean $\$840$ and variance $\$6{,}400$ squared. Recover $E[X]$, $\operatorname{Var}(X)$, and $E[X^2]$. (Answer: $E[X]=\$400$; $\operatorname{Var}(X)=\$1{,}600$ squared; $E[X^2]=\$161{,}600$ squared.)
A hypothetical severity has mean $\$3{,}000$ and variance $\$360{,}000$ squared. A fixed price adjustment gives $Z=0.95X+60$. Find $E[Z]$ and $\operatorname{Var}(Z)$, and verify the transformed standard deviation from the original standard deviation. (Answer: $E[Z]=\$2{,}910$; $\operatorname{Var}(Z)=\$324{,}900$ squared; standard deviation $\$570$.)

What You'll Learn