The Bangia spread adjustment worked sleeve by sleeve, with the multiplier, the stressed case and the two mistakes that make the number useless.
Liquidity-adjusted VaR adds to the market VaR the cost of crossing the bid-ask spread to sell: for each position, half the spread at a bad-day level, which is the average spread plus a multiple of its volatility, times the market value. On an illustrative 840M bond fund with a one-day 99 per cent VaR of 7,242,456, the spread cost is 4,896,896 and the LVaR is 12,139,352, 1.676 times the VaR. With spreads stressed it is 19,283,247, more than two and a half times.
Worked in full in Financial Risk Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Standard VaR prices every position at the mid. A fund that has to sell does not get the mid: it gets the bid, and on the day it most needs to sell the bid has moved further away. The adjustment below, usually credited to Bangia, Diebold, Schuermann and Stroughair, puts that cost into the same number the board already reads.
| Sleeve | Market value | Mean spread | Spread volatility |
|---|---|---|---|
| Government bonds | 210,000,000 | 0.04% | 0.02% |
| Investment grade credit | 315,000,000 | 0.30% | 0.12% |
| High yield bonds | 189,000,000 | 0.90% | 0.45% |
| Leveraged loans | 126,000,000 | 1.40% | 0.80% |
| Fund | 840,000,000 | 0.5350% |
The one-day 99 per cent parametric VaR from the fund's model is 7,242,456, or 0.862 per cent of NAV. How that figure is built is in how to calculate parametric VaR for a portfolio.
Cost of liquidity for a position = ½ × market value × (mean spread + k × spread volatility)
LVaR = VaR + Σ cost of liquidity
In Excel, per sleeve: =0.5*MV*(Spread+2.326*SpreadVol)
Step 1: the bad-day spread. Use the same confidence as the VaR, so k = 2.326 at 99 per cent. For high yield the bad-day spread is 0.90 + 2.326 × 0.45 = 1.947 per cent.
Step 2: half of it. A position marked at mid loses half the spread when sold at the bid: 0.9733 per cent.
Step 3: times the market value. 189,000,000 × 0.9733 per cent = 1,839,632. Repeat for each sleeve and add up.
| Sleeve | Bad-day spread | Half | Cost |
|---|---|---|---|
| Government bonds | 0.087% | 0.0433% | 90,846 |
| Investment grade credit | 0.579% | 0.2896% | 912,114 |
| High yield bonds | 1.947% | 0.9733% | 1,839,632 |
| Leveraged loans | 3.261% | 1.6304% | 2,054,304 |
| Total | 4,896,896 |
Step 4: add it to VaR. 7,242,456 + 4,896,896 = 12,139,352, or 1.445 per cent of NAV. The liquidity add-on is 67.6 per cent of the market VaR.
High yield and loans are 37.5 per cent of NAV and 79.5 per cent of the liquidity cost. The ranking of risk in the fund changes once liquidity is in it: government bonds, which carry real duration VaR, add almost nothing, while the loan book, which may look quiet on daily marks, carries the largest single cost.
Why the two parts are simply added. Adding assumes the worst price move and the worst spread arrive on the same day. That is conservative in normal markets and close to the truth in a crisis, when prices fall and spreads widen together. It is the assumption to state in the model documentation.
| k | Cost of liquidity | LVaR | LVaR / VaR |
|---|---|---|---|
| 0 (constant spread) | 2,247,000 | 9,489,456 | 1.310 |
| 1.645 | 4,121,066 | 11,363,522 | 1.569 |
| 2.326 | 4,896,896 | 12,139,352 | 1.676 |
| 3.000 | 5,664,750 | 12,907,206 | 1.782 |
The constant-spread version, half the average spread times the position, is the textbook starting point: 0.2675 per cent of the fund, or 2,247,000. It assumes spreads never widen, which is the one thing they do in the scenario VaR is meant to describe. Spreads are fat-tailed, so some practitioners use a k of 3 or more, calibrated on their own spread history.
| Sleeve | Bad-day spread | Cost |
|---|---|---|
| Government bonds | 0.213% | 223,692 |
| Investment grade credit | 1.458% | 2,296,728 |
| High yield bonds | 4.793% | 4,529,763 |
| Leveraged loans | 7.922% | 4,990,608 |
| Total | 12,040,791 |
In the stressed case the cost of getting out, 12,040,791, is larger than the VaR itself, and LVaR reaches 19,283,247: 2.663 times the market number, a 166.3 per cent add-on.
The first is to charge the full spread instead of half. Positions are marked at mid, so selling costs the distance from mid to bid. Charging the full spread doubles the add-on to 9,793,791 and LVaR to 17,036,247, and the overstated number tends to get ignored, which is worse than not computing it.
The second is to treat the spread as the whole liquidity cost. The quoted spread is good for a quoted size. A fund selling a fifth of its loan book moves the price, and that market impact grows with the share of daily volume sold. For large positions, add an impact term or lengthen the horizon to the time it takes to sell, using days to liquidate. Note also that the spread cost is paid once, not every day: on a 10-day horizon VaR scales to 22,902,657 while the cost stays 4,896,896, so the add-on falls to 21.4 per cent.
For each position take half of the mean spread plus 2.326 spread volatilities, multiply by the market value, add the total to VaR, and run it again with stressed spreads. Here: 12,139,352 normally, 19,283,247 stressed, against a market VaR of 7,242,456. The fund's selling policies, the cost of meeting a redemption and who pays it are worked in the free workbook for this case.
Because positions are marked at mid, and a seller receives the bid, which is half the spread below the mid. In the illustrative fund half the bad-day spread costs 4,896,896. Charging the full spread would double that to 9,793,791 and overstate LVaR at 17,036,247 instead of 12,139,352.
Matching the VaR confidence is the usual start: 2.326 at 99 per cent, which gives a liquidity cost of 4,896,896 here. Spreads are fat-tailed, so a k of 3 is common when calibrated on history; it lifts the cost to 5,664,750. A k of zero, the constant-spread version, gives 2,247,000 and assumes spreads never widen.
Not in the spread version. The quoted spread applies to normal trade sizes; selling a large share of daily volume moves the price further. For concentrated positions add an impact term or extend the horizon to the days needed to sell. In the stressed case here the spread cost alone, 12,040,791, already exceeds the 7,242,456 VaR.
This article is one calculation from Financial Risk Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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