## **Learning Outcomes**

This article explains how liquidity risk changes market risk measurement and how margining converts market moves into cash-flow needs. After reading, you will be able to:

- Explain why standard VaR can understate risk for positions that are costly or slow to unwind
- Compute simple liquidity-adjusted VaR (LVaR) using liquidation horizons and spread add-ons
- Describe margining concepts (VM, IM, haircuts, MPOR) and link them to liquidity stress

## **GARP FRM Part 1 Syllabus**

For the GARP FRM Part 1 exam, you are expected to understand VaR-based risk measurement in realistic trading conditions, with a focus on the following syllabus points:

- VaR assumptions, holding periods, and limitations (including model and implementation risk)
- Stress testing as a forward-looking tool for capital and liquidity adequacy
- Derivatives exposure concepts (changing exposure, netting, collateral) and how risk is transmitted through margining

## **Test Your Knowledge**

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision._

1. Liquidity-adjusted VaR is most directly motivated by which limitation of VaR?  
   a) VaR cannot be computed for linear portfolios  
   b) VaR ignores losses beyond the quantile and often assumes frictionless liquidation  
   c) VaR always violates monotonicity  
   d) VaR is independent of the holding period

2. A \$40 million position has daily return volatility of 1.2%. Compute the 99% 1-day delta-normal VaR (assume zero mean and $z_{0.99} = 2.33$).

3. Name two reasons initial margin can rise sharply in stressed markets.

4. True/False: Variation margin reduces counterparty credit exposure but can increase funding liquidity risk for the payer. Briefly explain.

## **What Do You Know Already?**

- If you compute a 1-day VaR, what implicit assumption are you making about how quickly you could exit the position?
- What is the economic difference between paying variation margin and posting initial margin?
- In a stress event, why might bid-ask spreads and volatility increase at the same time?

Use these prompts to gauge your starting point; then focus your revision where your answers were least confident.

## **Introduction**

Risk measures such as value at risk (VaR) are built on a simplifying idea: you can value positions using observable prices and, if needed, reduce or close positions over the selected holding period. In practice, trading is not frictionless. Costs such as bid-ask spreads, market impact, and delays in liquidation mean realized losses can be materially worse than a mark-to-market VaR calculated from historical or model-based returns.

Margining adds a second layer. For derivatives and secured financing (such as repo), adverse price moves and higher volatility can trigger immediate cash demands through variation margin, higher initial margin, and larger haircuts. These cash outflows are liquidity events, even when the firm remains solvent on a mark-to-market basis. Because VaR is typically framed as a loss quantile over a chosen horizon “under normal conditions,” institutions complement VaR with liquidity-aware adjustments and stress tests that focus on survivability, capital, and liquid asset buffers.

> **Key Term: Liquidity risk**  
> The risk that a firm cannot meet cash-flow obligations when due (funding liquidity risk) or cannot trade assets quickly at prices close to quoted levels (market liquidity risk).

## **Liquidity risk channels in traded portfolios**

Liquidity risk matters because it changes both (1) the distribution of P&L and (2) the timing of cash flows.

### Market liquidity risk vs funding liquidity risk

Market liquidity risk is about _price and execution_: can you transact quickly and in size without moving the market too much? Funding liquidity risk is about _cash and collateral_: can you obtain cash to meet outflows such as margin calls, haircuts, or withdrawals?

They interact. If funding is tight, a firm may be forced to sell into weak markets, increasing market impact. If markets are illiquid, the firm may need more time (and therefore more price risk) to exit positions, increasing losses and generating more margin calls.

> **Key Term: Market impact**  
> The adverse price movement caused by your own trading activity, typically increasing with trade size relative to market depth.

### Why liquidity dries up when it matters most

Liquidity is state-dependent. In stress events, volatility rises, correlations can shift, and dealers and market makers may reduce balance-sheet usage. Bid-ask spreads can widen and trading depth can collapse. A key risk-management implication is that “normal conditions” liquidity assumptions can fail exactly when losses are already large.

This is one reason stress testing is widely used as a complement to VaR: stress tests are forward-looking, focus on severe scenarios, and are used in capital and liquidity adequacy planning.

## **Why standard VaR can understate losses for illiquid positions**

VaR is commonly described as the worst expected loss under normal conditions over a specified period at a given confidence level. For delta-normal VaR, a typical expression is:

$$VaR_{\alpha,h} \approx z_\alpha \, \sigma_P \sqrt{h}$$

where $z_\alpha$ is the normal quantile, $\sigma_P$ is the one-day standard deviation of portfolio P&L (or return times value), and $h$ is the holding period in days (under square-root-of-time assumptions).

VaR can understate risk when liquidity is limited because:

1. **Holding period mismatch**: a 1-day or 10-day horizon may not reflect the time-to-liquidate in size.
2. **Transaction costs omitted**: bid-ask spreads and fees are typically not in the return series used for VaR.
3. **Market impact omitted**: the act of liquidation can move prices.
4. **Liquidity and returns are linked**: spreads often widen when prices move against you (a “wrong-way” style effect for liquidity).
5. **Implementation risk**: using stale prices or infrequent marks for illiquid assets can bias volatility and correlation estimates downward.

> **Key Term: Liquidity horizon (time-to-liquidate)**  
> The time needed to exit or hedge a position without exceeding acceptable market impact and execution costs.

## **Liquidity-adjusted VaR frameworks**

Liquidity-adjusted VaR (LVaR) is not a single formula. It is a family of approaches that incorporate liquidation time and liquidation costs into a VaR-style risk metric.

### Conceptual structure

A useful way to frame LVaR is:

$$LVaR_\alpha = VaR_\alpha(\text{price risk over liquidation horizon}) + \text{liquidity cost add-on}_\alpha$$

The first term addresses _how long you are exposed to price moves while exiting_. The second term addresses _how expensive the exit is_ (spread, fees, impact).

### Approach 1: Extend the holding period to the liquidation horizon

If you estimate a 1-day VaR but expect liquidation to take $T$ days, a basic adjustment is:

$$VaR_{\alpha,T} \approx VaR_{\alpha,1} \sqrt{T}$$

This scaling relies on assumptions (roughly independent, identically distributed returns with stable volatility). In stress, volatility clusters and liquidity can deteriorate, so this may still be optimistic. Even so, the exam-relevant point is clear: _VaR depends on holding period, and illiquidity implies a longer effective holding period than the desk’s reporting horizon._

### Approach 2: Add a bid-ask spread cost

If you hold quantity $Q$ of an asset with mid-price $P$ and proportional bid-ask spread $s$, a simple one-way liquidation cost (selling at the bid instead of mid) is:

$$\text{Spread cost} \approx \frac{s}{2} \, P \, |Q|$$

This is not “random” in the simplest version; it is an add-on. If spreads are stochastic, you can model $s$ as a random variable and take a high-quantile spread cost.

> **Key Term: Bid-ask spread**  
> The difference between the quoted ask price and bid price; a direct measure of transaction cost and a common proxy for market liquidity.

### Approach 3: Include market impact and liquidation schedule

For large positions, you rarely transact all at once. Suppose you liquidate over $T$ days in equal pieces. You are exposed to daily price changes on the remaining inventory. With i.i.d. daily price changes, an illustrative variance calculation shows that “sell over time” can reduce risk relative to “hold full position for $T$ days,” but it does not eliminate it.

Let the remaining fraction of the position at the start of day $k$ be $w_k = (T-k+1)/T$. If daily price change per unit has standard deviation $\sigma_{\Delta P}$, then:

$$
\sigma_{\text{P\&L}} = \sigma_{\Delta P}\,|Q| \sqrt{\sum_{k=1}^{T} w_k^2}
= \sigma_{\Delta P}\,|Q| \sqrt{\frac{(T+1)(2T+1)}{6T}}
$$

Impact costs are often modeled with a convex cost function such as:

$$C(q) = a q + b q^2$$

where $q$ is trade size, $a$ captures spread/fees, and $b q^2$ captures increasing marginal impact with size.

### Worked Example 1.1

A desk holds a \$50 million corporate bond position. The bond’s daily return volatility is 1.0%. Assume mean return is zero and $z_{0.99} = 2.33$.

- (1) Compute the 99% 1-day VaR.
- (2) The position is expected to take 5 days to unwind in size. Compute a 5-day VaR using square-root-of-time scaling.
- (3) The bond’s bid-ask spread is 80 bps of price. Compute a one-way spread cost add-on and the resulting simple LVaR.

> **Answer:**  
> **Step 1: 99% 1-day VaR**
>
> $$
> VaR_{0.99,1} = z \sigma V = 2.33 \times 0.01 \times 50{,}000{,}000 = 1{,}165{,}000
> $$
>
> So $VaR_{0.99,1} \approx \$1.165$ million.
>
> **Step 2: Scale to a 5-day liquidation horizon**
>
> $$
> VaR_{0.99,5} \approx VaR_{0.99,1}\sqrt{5}
> = 1{,}165{,}000 \times 2.2361 \approx 2{,}604{,}000
> $$
>
> So $VaR_{0.99,5} \approx \$2.604$ million.
>
> **Step 3: Add one-way spread cost**
>
> $$
> \text{Half-spread} = 80\text{ bps}/2 = 40\text{ bps} = 0.004
> $$
>
> $$
> \text{Spread add-on} \approx 0.004 \times 50{,}000{,}000 = 200{,}000
> $$
>
> So the spread add-on is about \$0.200 million.
>
> **Step 4: Simple liquidity-adjusted VaR**
>
> $$
> LVaR \approx 2{,}604{,}000 + 200{,}000 = 2{,}804{,}000
> $$
>
> Therefore, a simple 99% LVaR is approximately \$2.804 million.

## **Margining concepts and how they create liquidity stress**

Margining is a mechanism to control counterparty credit exposure, but it transmits market moves into cash needs.

### Variation margin and initial margin

- **Variation margin (VM)** transfers mark-to-market losses frequently (often daily, sometimes intraday). VM is essentially settlement of P&L.
- **Initial margin (IM)** is collateral posted to cover potential future exposure over the time it may take to close out a defaulted counterparty’s positions.

A standard regulatory intuition is: if a counterparty defaults, the surviving party faces a _margin period of risk_ before it can replace or unwind trades. IM is meant to cover adverse moves during that period at a high confidence level.

> **Key Term: Margin period of risk (MPOR)**  
> The period between the last successful exchange of collateral and the closeout (or replacement) of positions after default, including delays from disputes and liquidation.

### Haircuts on collateral

When collateral is posted in securities rather than cash, the receiver applies a **haircut** to protect against collateral value falling during liquidation. A simple haircut logic mirrors VaR:

$$\text{Haircut} \approx z_\alpha \sigma_C \sqrt{h}$$

where $\sigma_C$ is collateral price volatility and $h$ is the liquidation horizon for the collateral.

### Why margining can worsen liquidity conditions

Margining reduces _credit exposure_ but can raise _liquidity risk_ because:

1. **Cash is needed quickly**: VM calls are due on short timelines.
2. **IM is procyclical**: IM models often rise when volatility rises, increasing required posted collateral in stress.
3. **Haircuts can rise**: secured lenders may demand more collateral for the same borrowing.
4. **Concentrated funding sources**: reliance on short-term secured funding can lead to rapid increases in collateral demands.
5. **Feedback loops**: forced asset sales to raise cash can widen spreads and increase volatility, which then triggers larger margins.

This is closely aligned with why stress testing is used for liquidity planning: institutions need to assess whether liquid assets are sufficient to survive severe but plausible margin and funding shocks.

### Worked Example 2.1

A firm has a derivatives portfolio with current mark-to-market of \$0 (at trade inception). One day later, the portfolio value moves to \$-3.2 million for the firm (a loss). The CSA requires daily VM with no threshold and cash collateral. The IM requirement is \$6 million and was posted at inception.

- (1) What VM payment is due today?
- (2) After paying VM, what collateral is posted in total (IM + VM)?
- (3) Explain in one sentence why this can still be a liquidity problem even though credit exposure is reduced.

> **Answer:**  
> **Step 1: VM due**  
> The firm is \$3.2 million out-of-the-money, so it must pay \$3.2 million of VM today.
>
> **Step 2: Total collateral posted**  
> IM posted at inception: \$6.0 million  
> VM paid today: \$3.2 million  
> Total posted collateral (IM + cumulative VM): \$9.2 million
>
> **Step 3: Liquidity explanation**  
> The firm must source \$3.2 million in cash on short notice, which is a funding liquidity demand even though collateralization reduces counterparty credit exposure.

## **Connecting LVaR and margining in risk management**

Liquidity-aware market risk measurement and margin-aware liquidity management should be consistent:

1. **Liquidation horizons vs MPOR**  
   If a position cannot be closed quickly, both (a) LVaR holding periods and (b) margin models (IM based on MPOR) should reflect that.

2. **Joint stress of prices and liquidity**  
   A realistic stress event often includes both adverse price moves and wider spreads/higher margins. Treating spreads and margins as fixed can understate tail outcomes.

3. **Liquidity buffer sizing**  
   A firm needs liquid assets not only for expected outflows but also for high-quantile outflows (a “cash-flow-at-risk” idea). Margin calls are a central driver.

4. **Interaction with derivatives exposure measurement**  
   Derivatives exposures vary daily, and regulatory frameworks often define exposure measures as current exposure plus add-ons. Margining reduces current exposure, but it can create liquidity needs through frequent VM and larger IM in stress.

### **Exam Warning**

> Candidates often treat “collateral reduces risk” as universally true. For the exam, separate _credit exposure_ from _funding liquidity_. VM and IM can lower counterparty loss given default while simultaneously increasing the probability of a liquidity shortfall from margin calls, higher haircuts, and longer closeout periods.

## **Summary**

VaR is a quantile-based loss measure over a chosen holding period “under normal conditions.” Liquidity risk challenges this framing because the ability to exit positions is not guaranteed at quoted prices or within the reporting horizon. Liquidity-adjusted VaR responds by extending holding periods to realistic liquidation horizons and adding liquidation costs such as bid-ask spreads and (for large trades) market impact.

Margining reduces counterparty credit exposure through frequent VM and protective IM, but it can convert market volatility into rapid cash outflows. In stress, IM models, VM calls, and collateral haircuts can rise together, making liquidity buffers and stress testing central to survivability analysis. For FRM Part 1, focus on the conceptual links: holding period choice, transaction costs, liquidation time, MPOR, and how margining transmits market shocks into funding needs.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- VaR depends on holding period and typically assumes frictionless liquidation
- Liquidity-adjusted VaR increases horizons to time-to-liquidate and adds spread/impact costs
- Market liquidity risk and funding liquidity risk interact and can make each other worse in stress
- Variation margin settles mark-to-market losses quickly, creating immediate cash demands
- Initial margin and haircuts are designed to cover exposure during MPOR but can rise sharply in stress
- Stress testing complements VaR because it is forward-looking and supports capital and liquidity planning

## **Key Terms and Concepts**

- Liquidity risk
- Market liquidity risk
- Funding liquidity risk
- Liquidity horizon (time-to-liquidate)
- Liquidity-adjusted VaR (LVaR)
- Bid-ask spread
- Market impact
- Variation margin (VM)
- Initial margin (IM)
- Haircut
- Margin period of risk (MPOR)
- Procyclicality of margin


Liquidity risk and funding - Liquidity-adjusted VaR and margining (concepts)