Value at Risk: Meaning, Calculation and Holding-Period Scaling - Article

Introduction

Treasury positions are exposed to changes in:

• interest rates;

• bond prices;

• foreign-exchange rates;

• equity prices; and

• other market variables.

A treasury manager therefore needs a common measure that can answer a question such as:

How much could this portfolio lose over a specified period under normal market conditions at a stated confidence level?

Value at Risk, commonly called VaR, provides such a measure.

The current IIBF CAIIB BFM syllabus specifically includes “Risk Measures: VaR and Duration” under Module C – Treasury Risk Management.

RBI describes VaR as a single monetary amount estimating potential portfolio loss over a specified holding period and confidence level.

The three essential elements are:

Amount of Loss + Confidence Level + Holding Period

Without all three, a VaR number is incomplete.

Learning Objectives

After studying this topic, you should be able to:

• explain the meaning of VaR;

• interpret confidence level correctly;

• distinguish VaR from maximum possible loss;

• calculate basic parametric VaR;

• convert return volatility into monetary risk;

• understand the effect of confidence level;

• scale VaR across holding periods;

• understand portfolio diversification and correlation;

• distinguish one-day VaR from multi-day VaR; and

• identify important assumptions and CAIIB numerical traps.

What Does VaR Mean?

Suppose a treasury portfolio has a:

1-day 99% VaR of ₹X

The correct interpretation is:

Based on the model and assumptions used, there is approximately a 99% confidence that the portfolio loss over one day will not exceed ₹X under the modelled conditions.

Equivalently:

There is approximately a 1% probability that the loss will exceed ₹X.

This second interpretation is extremely important.

VaR does not say:

₹X is the maximum possible loss.

Loss beyond VaR can occur.

Three Components of Every VaR Statement

A proper VaR statement must specify:

1. Portfolio or Exposure

What position is being measured?

For example:

• bond portfolio;

• FX position;

• trading portfolio; or

• combined treasury portfolio.

2. Confidence Level

Examples:

• 95%;

• 99%.

Higher confidence means the model covers a larger part of the loss distribution.

Therefore:

Higher confidence level → Higher VaR

all else remaining equal.

3. Holding Period

Examples:

• one day;

• five days;

• ten days.

Longer holding periods generally create greater possible price movement.

Therefore:

Longer holding period → Higher VaR

under the usual scaling assumptions.

Basic Parametric VaR

For a simple linear position where returns are assumed to follow the distribution used by the model, a common parametric form is:

VaR = Portfolio Market Value × Return Volatility × Confidence Factor

This estimates the monetary loss threshold for the chosen confidence level and time horizon.

Refer to the Formula Section for a detailed explanation, variables, and solved examples.

This approach is often called:

• variance-covariance VaR;

• parametric VaR; or

• delta-normal VaR in suitable applications.

It is useful for CAIIB because it clearly shows how VaR depends on:

• exposure size;

• volatility; and

• confidence level.

If Monetary P&L Volatility Is Already Given

Sometimes a numerical question directly provides the standard deviation of daily profit or loss in rupees rather than percentage-return volatility.

In that case:

VaR = Monetary P&L Volatility × Confidence Factor

This directly converts the standard deviation of portfolio P&L into the selected confidence-level loss estimate.

Refer to the Formula Section for a detailed explanation, variables, and solved examples.

Do not multiply by portfolio value again if the volatility has already been expressed in monetary terms.

This is a common examination trap.

Confidence Factors

Under a standard one-tailed normal-distribution assumption, commonly used confidence factors are approximately:

Therefore, for the same portfolio and holding period:

99% VaR > 95% VaR

because the 99% calculation looks further into the adverse tail of the assumed distribution.

Changing the Confidence Level

Where VaR has been calculated parametrically using normal-distribution assumptions, changing only the confidence level can be done using the ratio of the relevant confidence factors.

For example:

99% VaR = 95% VaR × (2.326 ÷ 1.645)

This converts a normally distributed parametric VaR from 95% confidence to 99% confidence when all other assumptions remain unchanged.

Refer to the Formula Section for a detailed explanation, variables, and solved examples.

Do not use this conversion automatically for:

• historical-simulation VaR;

• Monte Carlo models with a different distribution; or

• non-linear portfolios

unless the question specifically supports the assumption.

VaR and Volatility

Volatility represents the degree of variation in market returns.

Everything else remaining unchanged:

Higher volatility → Higher VaR

Lower volatility → Lower VaR

This makes intuitive sense.

A portfolio whose market value moves sharply from day to day has greater potential loss than an equally sized portfolio whose value is very stable.

Relationship

Conceptually:

Exposure ↑ → VaR ↑

Volatility ↑ → VaR ↑

Confidence level ↑ → VaR ↑

Holding period ↑ → VaR ↑

Holding Period and VaR

A one-day VaR measures risk over one trading day.

But management may want to estimate risk over:

• two days;

• five days;

• ten days; or

• another holding period.

One common approach is the square-root-of-time rule.

If daily returns are assumed to be independent and identically distributed with stable volatility:

N-Day VaR = 1-Day VaR × √N

This scales one-day VaR to an N-day holding period.

Refer to the Formula Section for a detailed explanation, variables, and solved examples.

Therefore:

10-Day VaR = 1-Day VaR × √10

This gives the ten-day VaR under the square-root-of-time assumptions.

Refer to the Formula Section for a detailed explanation, variables, and solved examples.

Since:

√10 ≈ 3.162

ten-day VaR under this method is about 3.162 times one-day VaR.

This is a scaling relationship, not a solved portfolio example.

Scaling between Any Two Holding Periods

If VaR is already available for one holding period and must be converted to another:

VaR2 = VaR1 × √(Holding Period2 ÷ Holding Period1)

This scales VaR between two different horizons under the square-root-of-time assumption.

Refer to the Formula Section for a detailed explanation, variables, and solved examples.

This is useful when, for example, a question provides:

• 5-day VaR; and

• asks for 10-day VaR.

You do not need to return to one-day VaR first if the formula is applied correctly.

Why We Use the Square Root, Not the Number of Days

A common mistake is:

“Ten days means ten times one-day VaR.”

That is normally incorrect under the standard volatility-scaling assumption.

Variance accumulates proportionately with time under the basic independent-return assumption.

Standard deviation therefore increases with the square root of time.

Because basic parametric VaR is proportional to standard deviation:

VaR also scales with √time

under those assumptions.

Scaling Volatility

The same principle applies if daily volatility is given and multi-day volatility is required.

N-Day Volatility = Daily Volatility × √N

This converts daily volatility to the corresponding multi-day volatility under the standard independent-return assumption.

Refer to the Formula Section for a detailed explanation, variables, and solved examples.

You can then use the multi-day volatility in the VaR calculation.

However, avoid scaling both volatility and VaR again for the same period.

That would double-count the time adjustment.

Two Correct Calculation Routes

Suppose daily volatility is given and an N-day VaR is required.

You can conceptually use either route:

Route 1

Calculate one-day VaR.

Then:

One-Day VaR → Multiply by √N → N-Day VaR

Route 2

Scale daily volatility first.

Then calculate VaR once using the N-day volatility.

Both should produce the same result when the same assumptions are used.

Important Trap

Do not:

1. multiply volatility by √N; and then

2. multiply the resulting VaR by √N again.

That applies holding-period scaling twice.

Percentage VaR and Monetary VaR

A question may express risk as a percentage of portfolio value.

If percentage VaR is known:

Monetary VaR = Portfolio Market Value × VaR Percentage

This converts a percentage loss estimate into a currency amount.

Refer to the Formula Section for a detailed explanation, variables, and solved examples.

Conversely:

VaR Percentage = Monetary VaR ÷ Portfolio Market Value × 100

This expresses monetary VaR as a percentage of portfolio value.

Refer to the Formula Section for a detailed explanation, variables, and solved examples.

Be careful to distinguish:

• 2% written as a percentage; and

• 0.02 used as a decimal in calculation.

Portfolio VaR and Diversification

A portfolio may contain several positions.

Simply adding individual VaRs assumes that all positions produce their worst relevant movements together.

That can overstate risk when returns are not perfectly positively correlated.

For two positions under a simple variance-covariance framework:

Portfolio VaR = √(VaR12 + VaR22 + 2ρVaR1VaR2)

This combines two VaRs while recognising their correlation.

Refer to the Formula Section for a detailed explanation, variables, and solved examples.

The important principle is:

Perfect Positive Correlation

If correlation is +1:

Portfolio VaR approaches the sum of individual VaRs.

Lower Positive Correlation

Some diversification benefit arises.

Zero Correlation

Risk remains, but movements are not systematically linked.

Negative Correlation

Diversification benefit can become stronger.

Diversification Benefit

A simple way to express diversification benefit is:

Diversification Benefit = Sum of Standalone VaRs − Portfolio VaR

This measures how much portfolio risk is reduced by less-than-perfect correlation among positions.

Refer to the Formula Section for a detailed explanation, variables, and solved examples.

Three Main VaR Approaches

CAIIB learners should understand that VaR is a concept, not only one formula.

Variance-Covariance / Parametric VaR

Uses:

• volatility;

• correlations;

• distribution assumptions; and

• confidence factors.

Advantages:

• fast;

• comparatively simple; and

• useful for linear portfolios.

Weakness:

• may perform poorly when returns are highly non-normal or positions are strongly non-linear.

Historical Simulation

Uses actual historical changes in market risk factors.

The current portfolio is revalued using historical scenarios, and losses are ordered to determine the required percentile.

Advantages:

• does not require the same normal-distribution assumption;

• can capture characteristics present in historical data.

Weakness:

• depends strongly on the historical observation window.

Monte Carlo Simulation

Generates a large number of simulated market scenarios using a specified statistical model.

Advantages:

• flexible;

• useful for complex and non-linear portfolios.

Weaknesses:

• computationally intensive;

• model assumptions remain important.

VaR Does Not Tell Us Tail Severity

Assume two portfolios both have identical 99% VaR.

Beyond the 99% threshold:

• Portfolio A may have moderately larger losses;

• Portfolio B may have catastrophic losses.

VaR alone may not distinguish this sufficiently.

This is why stress testing and other tail-risk measures remain important.

The Basel Committee's current market-risk framework uses Expected Shortfall (ES) rather than VaR as the principal internal-model tail-risk measure for regulatory market-risk capital. ES looks at the average losses beyond the selected threshold; the current Basel internal-model framework uses a 97.5% one-tailed confidence level and liquidity-horizon adjustments.

VaR and Stress Testing

VaR mainly answers:

What loss threshold does the model estimate under the selected confidence level and horizon?

Stress testing asks:

What happens under an unusually severe but plausible market shock?

Examples include:

• sharp yield increase;

• sudden currency depreciation;

• severe credit-spread widening;

• collapse in market liquidity; or

• simultaneous adverse movements across markets.

Therefore:

VaR + Stress Testing

is stronger than relying on VaR alone.

RBI's risk-management guidance also emphasises stress testing alongside market-risk models.

Backtesting VaR

Backtesting compares VaR forecasts with actual trading outcomes.

Suppose a bank uses 99% one-day VaR.

Over a sufficiently large sample, the bank examines how often actual losses exceeded the VaR estimate.

These are called:

VaR exceptions or breaches

RBI's market-risk model guidance describes backtesting as comparing actual trading outcomes with risk measures to assess whether a model calibrated to a 99% confidence level performs consistently with that intended coverage.

Too many exceptions may indicate problems with:

• volatility assumptions;

• model specification;

• data;

• correlations; or

• changing market conditions.

Calculation Workflow for CAIIB

Use this sequence.

Step 1: Identify Exposure

Is portfolio value given?

Or is P&L volatility already expressed in rupees?

Step 2: Identify Volatility Period

Is volatility:

• daily;

• weekly; or

• already matched to the required horizon?

Step 3: Identify Confidence Level

Check whether the question uses:

• 95%;

• 99%; or

• another level.

Step 4: Select the Correct Confidence Factor

Use the factor specified in the question or the standard assumption appropriate to the model.

Step 5: Calculate Base-Horizon VaR

Use exposure, volatility, and confidence factor where applicable.

Step 6: Scale Holding Period if Required

Apply the square-root-of-time rule only once.

Step 7: Check Units

Ensure:

• percentage is converted correctly;

• ₹ crore remains ₹ crore;

• ₹ lakh remains ₹ lakh; and

• volatility and holding periods are consistent.

Step 8: Interpret the Result

State:

• amount;

• confidence level; and

• holding period.

Never provide only the monetary figure.

Reasonableness Checks

Use these relationships to detect numerical errors.

Exposure Doubles

If all other inputs remain unchanged:

VaR should double.

Volatility Doubles

Under simple parametric VaR:

VaR should double.

Confidence Rises from 95% to 99%

VaR should increase.

Holding Period Increases from 1 Day to 4 Days

Under square-root scaling:

VaR increases by:

√4 = 2 times

not four times.

Holding Period Increases from 1 Day to 9 Days

VaR increases by:

√9 = 3 times

under the same assumptions.

Diversification Improves

Portfolio VaR should generally fall relative to the simple sum of standalone VaRs when correlations are less than +1.

Square-Root-of-Time Assumption

The square-root rule is convenient but rests on important assumptions.

It broadly assumes that:

• returns across periods are independent;

• volatility is stable;

• the statistical process is sufficiently similar through time; and

• risk positions remain comparable over the holding period.

Real markets may violate these assumptions.

During stress:

• volatility can rise suddenly;

• correlations can increase;

• liquidity can disappear;

• returns can show fat tails; and

• the portfolio itself can change.

Therefore, square-root scaling should not be treated as a universal law.

For CAIIB numerical questions, apply it when the problem expects the conventional holding-period scaling approach.

Current Regulatory Context

VaR remains an important treasury risk concept and is expressly included in the current IIBF BFM syllabus.

However, regulatory market-risk capital should be distinguished from a basic educational VaR calculation.

RBI's current Basel III capital framework prescribes regulatory methodologies for banks' market-risk capital, while the Basel Committee's revised global internal-model framework uses Expected Shortfall rather than traditional VaR as the core tail-risk metric.

Therefore:

VaR calculation for treasury risk management ≠ automatically regulatory market-risk capital requirement

This distinction prevents an important exam and practical misunderstanding.

CAIIB Exam Focus

VaR must specify confidence level and holding period.

VaR is a monetary loss threshold, not the maximum possible loss.

A 99% VaR can still be exceeded.

Higher confidence level produces higher VaR, all else equal.

Higher volatility produces higher VaR.

Larger position produces higher VaR.

Longer holding period produces higher VaR.

For conventional time scaling:

N-Day VaR = 1-Day VaR × √N

Do not multiply by N.

Use holding-period scaling only once.

If volatility has already been scaled to N days, do not scale the resulting VaR again.

If P&L volatility is already given in rupees, do not multiply it again by portfolio value.

Portfolio diversification depends on correlation.

VaR does not measure how severe losses can become after the VaR threshold is breached.

Stress testing complements VaR.

Backtesting compares predicted VaR with actual outcomes.

Current Basel regulatory internal models use Expected Shortfall rather than traditional VaR for market-risk capital.

Common Exam Traps

Trap 1: VaR Is the Maximum Possible Loss

Incorrect.

Loss can exceed VaR.

Trap 2: 99% VaR Means a 99% Probability of Losing the VaR Amount

Incorrect.

It represents a loss threshold associated with the selected confidence level.

Trap 3: 99% Confidence Means VaR Can Never Be Exceeded

Incorrect.

There remains a tail probability beyond the VaR threshold.

Trap 4: Ten-Day VaR Is Ten Times One-Day VaR

Incorrect under standard square-root scaling.

Use √10.

Trap 5: Higher Confidence Produces Lower VaR

Incorrect.

Higher confidence moves further into the adverse tail and therefore increases VaR.

Trap 6: Portfolio Size Does Not Affect VaR

Incorrect.

For the same percentage volatility, a larger exposure creates a larger monetary VaR.

Trap 7: Daily Volatility Can Be Used Directly for Ten-Day VaR without Adjustment

Incorrect unless the calculation separately scales the resulting one-day VaR.

Trap 8: Scale Daily Volatility and Then Scale VaR Again

Incorrect.

That double-counts the holding period.

Trap 9: Monetary P&L Volatility Must Be Multiplied by Portfolio Value

Incorrect.

If volatility is already expressed as monetary P&L, multiplying by portfolio value again overstates risk.

Trap 10: Individual VaRs Should Always Be Added

Incorrect.

Portfolio VaR should recognise diversification and correlation where the model allows.

Trap 11: Zero Correlation Means Zero Portfolio Risk

Incorrect.

Each position still has its own volatility.

Trap 12: VaR Alone Captures Every Extreme Market Event

Incorrect.

Stress testing is needed to examine severe conditions beyond normal VaR assumptions.

Trap 13: Historical Simulation Requires the Normal Z-Factor

Incorrect.

Historical simulation derives its loss percentile from historical scenario outcomes rather than applying the standard normal confidence factor in the same way as parametric VaR.

Trap 14: Basic Treasury VaR Equals Current Regulatory Capital Requirement

Incorrect.

Regulatory market-risk capital follows the applicable regulatory framework and should not be replaced by a simple educational VaR formula.

Memory Technique

Remember VaR as A-C-T:

• A – Amount of potential loss

• C – Confidence level

• T – Time/Holding period

For the four main VaR drivers, remember:

E-V-C-T

• E – Exposure

• V – Volatility

• C – Confidence

• T – Time

For holding-period conversion:

Time grows normally → Risk grows by √Time

And remember the most important warning:

VaR tells you where the tail begins—not how bad the tail can become.