Common Errors In Market Risk Measurement

Why Small Conventions Cause Large Errors

Market-risk calculations often use simple formulas, but the result can be materially wrong if a rate change is entered in the wrong units, a short position is treated as long, or a liability change is added instead of subtracted. A sound calculation therefore makes its conventions explicit: what is changing, in what units, over what horizon, and from whose perspective is the change being measured?

This tutorial focuses on error detection in sensitivity and VaR calculations. The earlier material on duration, equity beta, foreign-exchange positions, and market VaR provides useful context; the emphasis here is on applying those quantities consistently and checking the result. The methods are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: A market-value change is written as a signed amount: positive for an increase in value and negative for a decrease. A loss is commonly reported as a positive amount when value falls. For an insurer with asset value $A$ and liability value $L$, both measured as positive market values, surplus is $A-L$, so its change is $\Delta A-\Delta L$.

Units: Basis Points, Decimal Changes, and Dollars

A basis point is one hundredth of a percentage point. Thus, $1$ bp is $0.01\%$, or $0.0001$ when a rate change is entered as a decimal. For example, a rise of $15$ bp is $15(0.0001)=0.0015$, equivalent to $0.15\%$. The decimal form, not the number 15 or the percentage number 0.15, belongs in a duration formula.

For a bond or other fixed-income asset with market value $V$ and modified duration $D$, the first-order duration approximation for a small yield change $\Delta y$ is

\Delta V \approx -D V\Delta y.

This is a dollar change. The corresponding approximate percentage change is $\Delta V/V\approx-D\Delta y$. Keeping those two forms separate prevents the common error of reporting a percentage as though it were dollars, or multiplying a dollar exposure by a percentage twice.

Derivation: Modified duration is defined locally by $D=-\frac{1}{V}\frac{dV}{dy}$. For a small change $\Delta y$, a first-order Taylor approximation gives $\Delta V\approx\frac{dV}{dy}\Delta y$. Substituting $\frac{dV}{dy}\approx-DV$ yields $\Delta V\approx-DV\Delta y$. The approximation is local: it ignores second-order curvature and may be inaccurate for large yield changes, nonparallel curve moves, or instruments whose cash flows change with rates.

Worked Example: Converting Basis Points Before Applying Duration

A hypothetical bond portfolio has market value $ \$2{,}000{,}000$ and modified duration $5.2$. A parallel yield increase of $15$ bp occurs. Estimate the change in value using duration and ignoring convexity.

Method. Convert the rate change to decimal form, then apply the duration approximation. The portfolio value and duration are given; no distributional assumption is needed because this is a scenario sensitivity calculation.

First convert the shock:

\Delta y=15(0.0001)=0.0015.

Then calculate the percentage change:

\frac{\Delta V}{V}\approx-5.2(0.0015)=-0.0078=-0.78\%.

Convert that percentage change to dollars:

\Delta V\approx \$2{,}000{,}000(-0.0078)=-\$15{,}600.00.

A yield increase produces an estimated value decline, as indicated by the negative sign. The result is a $0.78\%$ decline, or $ \$15{,}600.00$; these are two representations of the same estimated change. The calculation is checked by confirming that the positive yield shock and positive duration imply a negative value change. The common mistake avoided is entering $15$ or $0.15$ as $\Delta y$, rather than $0.0015$.

Signs: Exposure, Value Change, and Loss

An exposure is not automatically a loss. A long position generally gains when its underlying market price rises and loses when it falls; a short position has the opposite response. The sign should follow the position and the market move, rather than being imposed because a result is labelled “risk.” A useful convention is to calculate signed value changes first, then, if needed, express a loss as the negative of the value change.

For an insurer, liabilities require an additional sign check. Let $A$ and $L$ be positive market values, and define surplus as $A-L$. Then:

\Delta \text{Surplus}=\Delta A-\Delta L.

If rates rise, a fixed-income asset may lose value, so $\Delta A<0$. A liability valued as a fixed-income obligation may also fall in value, so $\Delta L<0$. Subtracting that negative liability change increases surplus relative to the asset change alone. It is incorrect to add $\Delta L$ to $\Delta A$ when calculating surplus.

Worked Example: Combining Asset, Liability, and Equity Changes

A hypothetical insurer holds bonds worth $ \$4{,}000{,}000$ with modified duration $2.0$. Its liabilities have market value $ \$3{,}000{,}000$ and modified duration $1.5$. It also has a long equity exposure of $ \$500{,}000$, whose value changes linearly with the equity index. Yields rise $20$ bp and the index falls $4\%$. Estimate the total surplus change, assuming a parallel yield move, duration approximations with convexity ignored, and no interaction between the rate and equity effects.

Method. Calculate signed dollar changes for the bonds and liabilities, compute the equity dollar change, and combine them as asset changes minus liability changes. The liability value is recorded as a positive amount, even though it is subtracted in surplus.

Convert the yield shock:

\Delta y=20(0.0001)=0.002.

The bond change is:

\Delta A_{\text{bonds}}\approx-2.0(\$4{,}000{,}000)(0.002)=-\$16{,}000.00.

The liability value change is:

\Delta L\approx-1.5(\$3{,}000{,}000)(0.002)=-\$9{,}000.00.

The rate-related surplus change is therefore:

\Delta A_{\text{bonds}}-\Delta L =-\$16{,}000.00-(-\$9{,}000.00) =-\$7{,}000.00.

The equity exposure is long and the index return is $-4\%$, so its value change is:

\Delta A_{\text{equity}}=\$500{,}000(-0.04)=-\$20{,}000.00.

Combining the changes gives:

\Delta \text{Surplus}=-\$7{,}000.00-\$20{,}000.00=-\$27{,}000.00.

The total estimated surplus decrease is $ \$27{,}000.00$. The check is that the bond and equity assets both lose value; the falling liability value offsets $ \$9{,}000.00$ of the bond loss, not adds to it. A common wrong calculation is $-\$16{,}000-\$9{,}000-\$20{,}000=-\$45{,}000$, which incorrectly adds the liability-value decline as another loss. The correction is to subtract the signed liability change.

Percentages Need the Correct Base

A percentage return applies to a value base. For a position with current market value $V$ and return $r$, its dollar change is $\Delta V=Vr$ under a simple linear return calculation. If a portfolio’s return is estimated using beta, first calculate the portfolio return from beta and the market return, then apply that return to the portfolio’s own market value.

Worked Example: Turning a Beta-Based Return Into Dollars

A hypothetical equity portfolio has market value $ \$12{,}000{,}000$ and estimated beta $1.25$ relative to a broad index. The index falls $2\%$. Assuming the beta relationship is linear over this scenario, estimate the portfolio’s dollar value change.

Method. Beta times the index return gives the estimated portfolio percentage return; multiply that percentage by the portfolio value. This scenario assumes the stated beta is the relevant sensitivity and ignores idiosyncratic return and nonlinear effects.

The estimated portfolio return is:

r_{\text{portfolio}}\approx1.25(-0.02)=-0.025=-2.5\%.

The dollar change is:

\Delta V\approx\$12{,}000{,}000(-0.025)=-\$300{,}000.00.

The result is a $2.5\%$ decline, equal to $ \$300{,}000.00$ on the portfolio’s $ \$12{,}000{,}000$ base. A common mistake is to apply the index’s $2\%$ decline directly to the portfolio, obtaining $-\$240{,}000$, and omit beta. Another is to multiply by $1.25$ after converting the percentage return to dollars, applying beta twice. The correct sequence is return first, dollar conversion second.

Time Scaling Must Match the Model

Scaling a one-day VaR to a longer horizon is not a unit conversion like basis points. A common square-root-of-time approximation is valid only under specific assumptions. If daily losses are independent and identically distributed with mean zero and finite variance, the variance of the sum over $n$ days is $n$ times the one-day variance, and its standard deviation is therefore $\sqrt{n}$ times the one-day standard deviation. If losses are normally distributed, the corresponding zero-mean VaR scales in the same way.

Condition: For zero-mean normally distributed daily losses that are independent and identically distributed, an $n$-day VaR at a fixed confidence level is $\sqrt{n}$ times the one-day VaR. This approximation does not automatically apply when daily losses have material drift, serial dependence, changing volatility, nonlinear or path-dependent exposures, or a distribution whose quantiles do not scale with its standard deviation.

Worked Example: Scaling a VaR Without Multiplying by the Number of Days

A hypothetical portfolio has a one-day $99\%$ VaR of $ \$100{,}000$. Estimate its ten-day $99\%$ VaR using square-root-of-time scaling. Assume daily losses are independent and identically distributed, normally distributed, and have mean zero; assume the portfolio composition and volatility remain unchanged.

Method. Apply $\sqrt{n}$ to the one-day VaR because the stated assumptions make the quantile scale with standard deviation. The confidence level is unchanged.

\operatorname{VaR}_{10}= \$100{,}000\sqrt{10} =\$100{,}000(3.16227766) =\$316{,}227.77.

Rounded to the nearest cent, the estimate is $ \$316{,}227.77$. The unit check is dollars at a ten-day horizon, and the result exceeds one-day VaR but is less than ten times that amount. Multiplying by 10 would produce $ \$1{,}000{,}000$, which incorrectly assumes losses accumulate in a perfectly proportional way rather than variances adding under the stated model. The square-root result is an assumption-based approximation, not a universal scaling rule.

A Practical Error-Checking Sequence

A consistent workflow catches many avoidable mistakes before a result is reported. Keep the raw market shock, the sensitivity, the signed value change, and the final risk measure distinct. In particular, do not silently switch from a change in market value to a positive loss amount halfway through a calculation.

Write the shock with units.
Convert basis points to decimal yield changes and percentage moves to decimal returns before multiplying.

State the exposure and direction.
Identify whether the position is long or short, whether the sensitivity is signed, and whether the result sought is a value change or a loss.

Use the correct dollar base.
Apply a percentage return to the market value of the position it describes; apply duration to market value and decimal yield change.

Combine assets and liabilities consistently.
For surplus defined as assets minus liabilities, calculate $\Delta A-\Delta L$, preserving each signed change.

Verify horizon and assumptions.
Use time scaling only when the assumed return process supports it, and check that the answer has the intended units and direction.

Common Mistakes and Corrections

Mistake	Why it fails	Correction
Using 10 for a 10 bp rate shock	The duration formula expects a decimal yield change, not a basis-point count.	Use $10(0.0001)=0.001$.
Calling a negative value change a positive gain	The sign is inconsistent with the position and market move.	Track signed value changes, then define loss as needed.
Adding a liability decline to an asset decline	Surplus is assets minus liabilities, not the sum of their values.	Calculate $\Delta A-\Delta L$.
Applying a return to the wrong portfolio value	The resulting dollar change has the wrong exposure base.	Use the market value of the position whose return was calculated.
Multiplying VaR by the number of days	This ignores the variance-based assumptions behind square-root scaling.	Use $\sqrt{n}$ only when its assumptions are reasonable and stated.

A final numerical check should address both magnitude and direction. A small decimal shock applied to a large portfolio can still produce a material dollar change. Conversely, a result that is ten or one hundred times larger than expected may signal a basis-point or percentage conversion error. For surplus, separately inspect the asset contribution and the liability offset before accepting the combined total.

Key takeaway: Market-risk arithmetic is only as reliable as its conventions. Convert units before calculating, preserve exposure signs, translate percentage changes using the right base, and compute surplus changes as asset changes minus liability changes. State the assumptions behind time scaling rather than treating it as an identity.

Check Your Understanding

For each problem, calculate signed market-value or surplus change as requested. Unless stated otherwise, report dollar amounts to the nearest cent.

A hypothetical bond portfolio worth $ \$6{,}000{,}000$ has modified duration $3.2$. Yields fall $25$ bp. Estimate the value change using duration and ignoring convexity. (Answer: $+\$48{,}000.00$)
An insurer has assets worth $ \$5{,}000{,}000$ with duration $4.0$ and liabilities worth $ \$4{,}000{,}000$ with duration $3.5$. Under a parallel rate increase of $12$ bp, estimate the change in surplus using duration and ignoring convexity. (Answer: $-\$7{,}200.00$)
A hypothetical equity portfolio worth $ \$2{,}400{,}000$ has beta $0.8$. Its index rises $5\%$. Under a linear beta approximation, estimate the portfolio dollar change. (Answer: $+\$96{,}000.00$)
A one-day $99\%$ VaR is $ \$80{,}000$. Assuming independent, identically distributed, normally distributed daily losses with mean zero and unchanged portfolio risk, estimate five-day $99\%$ VaR using square-root-of-time scaling. (Answer: $ \$178{,}885.44$)
A hypothetical insurer has a net long foreign-currency position consisting of an asset worth $1{,}500{,}000$ reporting-currency-equivalent units and a liability worth $400{,}000$ such units. The foreign currency falls $3\%$ against the reporting currency. Assuming proportional repricing and no other effects, calculate the change in surplus. (Answer: $-\$33{,}000.00$)

Mistake	Why it fails	Correction
Using 10 for a 10 bp rate shock	The duration formula expects a decimal yield change, not a basis-point count.	Use \(10(0.0001)=0.001\).
Calling a negative value change a positive gain	The sign is inconsistent with the position and market move.	Track signed value changes, then define loss as needed.
Adding a liability decline to an asset decline	Surplus is assets minus liabilities, not the sum of their values.	Calculate \(\Delta A-\Delta L\).
Applying a return to the wrong portfolio value	The resulting dollar change has the wrong exposure base.	Use the market value of the position whose return was calculated.
Multiplying VaR by the number of days	This ignores the variance-based assumptions behind square-root scaling.	Use \(\sqrt{n}\) only when its assumptions are reasonable and stated.

What You'll Learn