22Portfolio Trades and Risk Bids

A pension fund asks three brokers for a price at which they will take on a basket of five hundred stocks at tonight’s close, knowing only its size, its sectors and its liquidity. The broker who wins learns the names afterwards, and learns whether it bid too much. This chapter prices such a bid on a simulated basket: the cost of liquidating it, the risk carried while doing so, and the shading that competition forces on a bidder who does not want to win only when it is wrong; it then compares a disclosed basket with a blind one and ends with a transition between two portfolios.

22.1 Agency and principal portfolio trades

A portfolio trade (One Quant Book 2, chapter 22) moves a whole list at once: an index rebalance, a fund’s inflows, a change of manager. The client chooses between paying the costs itself (agency) and paying a broker to take them (principal); the risk bid is priced like any risk transfer (chapter 21), with two twists: the list is large and diversified, so its risk can be hedged in part, and in the blind version the bidder prices a description. Kavajecz and Keim (2005) studied such blind principal auctions and found that they gave the liquidity demander a transaction-cost saving relative to more traditional trading, and liquidity suppliers an efficient way to obtain order flow.

The simulated basket comes from Book 7’s synthetic market (firm.synthmkt): 786 names listed on its last day, with a median daily volume of USD 37.5 million and a median daily volatility of 171 basis points; half-spreads follow a size rule (an assumption: a median full spread of 5.9 basis points), and a ten-factor statistical risk model on 250 days of returns gives the covariance (Book 7’s firm.riskmodel). The pension fund’s basket holds 500 of them, USD 250 million, with positions from USD 38 000 to USD 4.8 million, a median of 1.0% of a day’s volume.

22.2 The cost of liquidating a basket

Each position is sold at no more than 20% of its daily volume, so a position of $k$ days’ volume takes $\max(1,5k)$ days; its cost is its half-spread plus the square-root law’s $Y\sigma\sqrt{\min(Q/V,0.2)}$ (chapter 11, $Y=0.7$). The basket’s liquidity profile is Figure 22.1: 98.8% of its notional is below 5% of a day’s volume, and no position needs more than a day. The winner’s cost of unwinding it is 14.9 basis points of the basket (firm_riskbid.liquidation).

22.3 Pricing a risk bid

While it sells, the winner holds what is left. With each position sold down in a straight line over its days $d_i$, the variance of the P&L is $\sum_{ij}w_iw_jC_{ij}I(d_i,d_j)$, with $C$ the daily covariance and $I(a,b)=a/2-a^2/(6b)$ for $a\le b$ the overlap of the two holdings (for one day, a third of a day’s variance): this is the multi-asset Almgren–Chriss risk term (Almgren and Chriss, 2001; chapter 14) for straight-line schedules. For the basket, one standard deviation is 40.4 basis points; sold short in index futures at once to remove its market beta, 12.5.

def _overlap(d):
    a = np.minimum.outer(d, d)
    b = np.maximum.outer(d, d)
    return a / 2 - a**2 / (6 * b)


def risk_sd(notional, cov, days, hedge=None) -> float:
    w = np.asarray(notional, float)
    c = np.asarray(cov, float)
    if hedge is not None:
        load, var = hedge
        load = np.asarray(load, float)
        c = c - var * np.outer(load, load)       # the factor sold at once with a future
    var = w @ (c * _overlap(np.asarray(days, float))) @ w
    return float(np.sqrt(max(var, 0.0)))

The bid, as a discount to the close, adds four parts: the expected cost, a charge for the risk (here one standard deviation of the hedged P&L, the desk’s choice), the shading that competition requires (the next section), and a margin (2 basis points). For the disclosed basket: 14.9 of cost, 12.5 of risk, 4.2 of shading and 2.0 of margin, a bid of 33.6 basis points below the close (Figure 22.2).

22.4 Blind bids and the winner’s curse

Every bidder estimates the same cost with its own error; the one whose estimate is lowest wins, and it wins most often when its estimate is too low. This is the winner’s curse of a common-value auction (One Quant Book 2, chapter 30). If each of four bidders (the desk and three competitors) estimates the cost with an independent error of standard deviation $s$, the winner’s estimate is on average $s\,\E[\max\text{ of 4 standard normals}]=1.03s$ too low, and a bidder who does not shade by that amount loses it on the trades it wins (firm_riskbid.curse).

def curse(sigma_est: float, competitors: int) -> float:
    return sigma_est * expected_max_normal(competitors + 1)


def auction(true_cost: float, sigma_est: float, bidders: int, shade: float,
            margin: float = 0.0, trials: int = 100_000, seed: int = 1) -> float:
    """Each bidder estimates the cost with an independent error; the lowest bid wins;
    the winner's profit is its bid minus the true cost."""
    rng = np.random.default_rng(seed)
    est = true_cost + sigma_est * rng.standard_normal((trials, bidders))
    win = est.min(axis=1) + shade + margin
    return float(np.mean(win - true_cost))

With estimation errors of 15% of the cost and risk (4.1 basis points for the disclosed basket), the shading is 4.2 basis points; in 200 000 simulated auctions, bidders who add only the 2-basis-point margin make $-2.2$ basis points on the trades they win, and bidders who also shade make the 2.0 they wanted.

A blind bidder sees only the profile. To price it, the desk draws 300 baskets with the same profile (each position keeps its sector, its notional and its bucket of days of volume, with a name drawn from those that fit) and prices each: their cost averages 17.0 basis points (standard deviation 0.4) and their hedged risk 13.0 (1.1). The profile does not reveal that the fund held more of the liquid names than a random draw would: the blind estimate is 2.1 basis points above the disclosed cost. The larger estimation error adds 0.2 to the shading. The blind bid is 36.4 basis points, 2.8 more than the disclosed one; the client pays the difference for not showing its names before the trade.

The blind bid still has a use: the client’s names do not leak to the losers, who might otherwise trade ahead of the winner’s unwind (chapter 9’s leakage). Whether the 2.8 basis points buy enough protection depends on how many dealers would see the names and what they would do with them.

22.5 Transitions

A transition moves a fund from one portfolio (a manager it is leaving) to another. The transition manager (One Quant Book 8, chapter 29) nets the names the two portfolios share, crosses what it can internally against other flow, and trades the rest. For two portfolios of USD 250 million with 150 names in common: selling one and buying the other outright trades USD 500 million and costs 50.4 basis points of the fund; netting the shared names trades USD 322 million at 32.8; crossing 20% of the remainder internally leaves USD 258 million to trade, at 24.0 (firm_riskbid.transition). Internal crossing (One Quant Book 7, chapter 27) and the central risk book (Book 9, chapter 25) are the same idea at the scale of a bank: the cheapest trade is the one that never reaches the market.

22.6 Tutorial: pricing a risk bid

22.7 Build: risk bids

22.8 Exercises

22.9 Problem: Pricing the Basket Blind

22.10 Interview questions