Liquidity-adjusted VaR

Ordinary Value at Risk answers one question: how far could the portfolio's value fall? It assumes you can simply mark the holdings at the market price.

But selling is not free. Every trade goes out at the bid and comes in at the ask, and the gap between them is a real cost. On a large or thinly traded position it is a big one.

Liquidity-adjusted VaR adds that cost in. It is the regular VaR plus the cost of unwinding positions.

There is a second reason it matters. The bid-ask spread is not fixed. In a stressed market it widens, and it moves about unpredictably.

So the worst case is not just a bad price move. It is a bad price move at the moment when getting out also became expensive.

The definition (section 17.4.4). The liquidity-adjusted VaR is the regular VaR plus the cost of unwinding positions. The workbook then adds the warning that matters: unwinding in a normal market is different from unwinding in a stressed market, where even the bid-ask spread displays random movements, so the liquidity cost in stressed conditions must take those movements into account.

Where the unwinding cost comes from. Liquidity risk is reflected in the bid-ask spread, treated as the cost of a round-trip transaction. The lower the spread, the higher the liquidity. Two parameters determine liquidity risk: (i) the bid-ask spread and (ii) the time to execute the trade without impacting the price. Quantity matters too — when the transaction value is much larger, the spread widens.

Half the spread, not all of it. Because one only looks at one end of the trade — to buy or to sell at a particular time — half of the bid-ask spread is taken as the cost.

The workbook's own worked figures. A portfolio manager has to buy 1 million shares of company A, quoted bid Rs 99 and offer Rs 101:

Since a portfolio holds more than one stock, the portfolio liquidity cost is the weighted average sum of all its constituents' proportional spreads.

Putting the two halves together. The chapter's parametric illustration produces portfolio VaR figures of 2.876%, 2.013% and 1.555% of a $100 million portfolio at the 99%, 95% and 90% confidence levels. A liquidity-adjusted figure is that loss plus the unwinding cost computed above — the market move and the exit cost, added.

Bid-Ask spread          = Ask price - Bid price
Mid-Market Price        = 1/2 x (Ask price + Bid price)
Proportional Bid-Ask spread = Bid-Ask spread / Mid-Market Price
Portfolio liquidity cost = weighted average sum of proportional bid-ask spreads

Liquidity-adjusted VaR  = regular VaR + cost of unwinding positions

Following the workbook's own method with rupee figures. A manager holds Rs 100 crore in two stocks and has computed a one-day 95% VaR of 2.0%, that is Rs 2,00,00,000.

Step 1 — the unwinding cost.

Step 2 — add it to VaR.

Liquidity-adjusted VaR = Rs 2,00,00,000 + Rs 67,00,000 = Rs 2,67,00,000.

The unwinding cost adds 34% to the reported risk figure, and almost all of it comes from the mid cap. That holding is 30% of the portfolio and 90% of the liquidity cost.

Step 3 — the stressed case. Suppose in a stressed market the mid cap's quote widens to bid Rs 380 and ask Rs 420. The spread becomes Rs 40 on a mid of Rs 400, a proportional spread of 0.10 — five times wider. That holding's exit cost alone becomes Rs 3,00,00,000, and the liquidity-adjusted VaR becomes Rs 2,00,00,000 + Rs 7,00,000 + Rs 3,00,00,000 = Rs 5,07,00,000.

The market move did not change. The cost of leaving did.

Chapter 17 (Risk), section 17.4.4 (Measuring Liquidity Risk), builds the bid-ask spread measure with the 1 million shares / Rs 99 / Rs 101 / Rs 2 mn worked example, and then defines liquidity-adjusted VaR in a single line at the end.

The chapter's sample question 4 is the one to watch: it asks what is NOT correct about Value at Risk, and takes liquidity of assets into account is one of the incorrect statements — ordinary VaR does not. That is exactly why this measure exists. Expect also a computation of a proportional bid-ask spread and a liquidity cost from a given quote.

• Ordinary VaR ignores liquidity. That is the point of the adjustment, and the chapter's sample question tests it directly.
• Half the spread is the cost, not the whole spread. You transact at one end of the quote at a time.
• The proportional spread is taken over the mid-market price, not over the bid and not over the ask.
• Portfolio liquidity cost is a weighted average, so a small illiquid holding can dominate the figure — as the mid cap does in the example above.
• Spreads are not constant. In stressed markets they widen and move randomly, which the workbook insists must be built into the stressed liquidity cost. A calm-market spread understates the exit cost precisely when it matters.
• Two parameters, not one. The bid-ask spread and the time taken to execute without impacting the price. Size feeds both.