6The Head of Desk’s Job

The budget meeting in December sets next year’s revenue target at this year’s result plus ten per cent, and leaves the desk’s risk limits where they were. This year’s result was exactly what the desk expects to make; its revenue has a Sharpe ratio of 1.3. With those limits the desk meets the new target in 45 years out of a hundred. The head of desk has been handed a plan that fails more often than it succeeds, before a single trade; to meet it three years in four, the desk would have to run more than twice its risk. This chapter is about the numbers a head of desk agrees to in December and runs by for the rest of the year.

6.1 The budget: revenue, costs and the plan

A desk’s plan has three parts: revenue by business, costs (its own direct costs and those allocated to it), and the capital and limits it will use (chapter 7). The chapter’s desk (illustrative, $ millions a year) runs three businesses: client flow, expected revenue 30 and volatility 25; inventory trading, 15 and 20; structured solutions, 10 and 12; correlated 0.3, 0.2 and 0.4. Its expected revenue is 55, its volatility 42.2 and its Sharpe ratio 1.30. Direct costs of 18 and allocated costs of 10 leave, after variable pay of 20% of a positive result, an expected profit of 20.5, and a 26% chance that revenue does not cover costs.

The counter-intuitive half of the proposition is the useful one: a steadier desk is less likely to beat a stretch target, because its results cluster closer to their mean (Figure 6.1). At $k=1.1$ and $S=1.3$ the desk meets its budget with probability 44.8% (45.0% in 20 000 simulated years); to meet it with probability 75% it must scale every position by 2.28. A budget that sits above the desk’s expected revenue does not ask for effort: it asks for risk.

6.2 Risk appetite: from a loss the firm can bear to limits

The definitions follow the Financial Stability Board’s principles of 2013, which also name the upper bound, risk capacity, the most risk a firm can take before breaching its regulatory capital and liquidity constraints. A statement such as “the firm accepts a loss of at most $20 million in a one-in-twenty year from this desk” becomes limits in three steps (Figure 6.2).

With expected revenue $\mu=55$, a loss of at most $L=20$ at 95% confidence allows a volatility of $(55+20)/1.645=45.6$: the desk, at 42.2, may scale up by 8%. At 99% one day, the corresponding value at risk (One Quant Book 6, chapter 21) is $2.326\times45.6/\sqrt{252}=6.7$. The $20 million of tolerance, shared by each business’s volatility, gives stops of 8.8, 7.0 and 4.2. The cascade is the risk hierarchy of One Quant Book 6, chapter 29, read from the top: the limits are the risk appetite statement in numbers.

def risk_multiple(plan: Plan, prob: float) -> float:
    """Scale lambda of every position (mean and volatility both scale) so that P(revenue >= budget) = prob:
    lambda (mu - z sigma) = budget, possible only if the desk's Sharpe ratio exceeds z = Phi^-1(prob)."""
    s = plan_stats(plan)
    z = _inv_phi(prob)
    edge = s["mu"] - z * s["sigma"]
    if edge <= 0:
        return math.inf
    return plan.budget / edge


def simulate(plan: Plan, n: int, rng, days: int = 252) -> dict:
    """n simulated years of daily revenue for the whole desk: annual totals and maximum drawdowns."""
    s = plan_stats(plan)
    d = rng.normal(s["mu"] / days, s["sigma"] / math.sqrt(days), (n, days))
    path = np.cumsum(d, axis=1)
    peak = np.maximum.accumulate(np.maximum(path, 0.0), axis=1)
    return {"annual": path[:, -1], "max_drawdown": (peak - path).max(1)}


def cascade(plan: Plan, loss_tolerance: float, conf: float = 0.95, var_conf: float = 0.99, days: int = 252) -> dict:
    """The firm bears a loss of at most loss_tolerance in a 1-in-(1/(1-conf)) year. With the desk's expected
    revenue mu, the volatility budget is (mu + L) / z_conf; the daily VaR limit is z_var * vol_budget / sqrt(days);
    each business's stop is its share of the volatility budget (by its own sigma) times the loss tolerance."""
    s = plan_stats(plan)
    vol = (s["mu"] + loss_tolerance) / _inv_phi(conf)
    sig = np.array([b.sigma for b in plan.businesses])
    stops = {b.name: loss_tolerance * w for b, w in zip(plan.businesses, sig / sig.sum(), strict=True)}
    return {"vol_budget": vol, "scale": vol / s["sigma"], "daily_var": _inv_phi(var_conf) * vol / math.sqrt(days),
            "stops": stops}

6.3 Product scope and new-product approval

A new product is where a desk’s risks are least understood and its controls least tested: the pricing model is new, the booking may be manual, the settlement process may not exist. US bank supervisors’ guidance expects banks to manage the risks of “new, modified, or expanded products and services” with appropriate approvals before a new activity starts and with controls to identify, measure, monitor and report its risks. For the head of desk, the approval process is the price of growth: every function that signs is one that will support the product when it goes wrong. Product scope also fixes what the desk may not do; a trader who finds a profitable trade outside it has found a request for approval, not a trade.

6.4 The rhythm: the day, the week, the year

A head of desk runs to a calendar that repeats. The contents vary by firm; the structure is common, because it follows the controls:

The monthly review compares cumulative revenue with the plan. A plan spread evenly over the year is a line; the desk’s revenue is a random path around its expectation, and the band it can wander in without meaning anything widens with the square root of time (Figure 6.3). In month three, a desk exactly on its expected path is already $1.4 million behind a budget ten per cent above it, well inside its noise; a head of desk who reacts to that gap by adding risk is reacting to noise.

6.5 Tutorial: next year’s plan

What to change next. Raise the correlation between flow and inventory to 0.8 and redo the cascade; set the budget at expectation and compare the drawdowns the desk must live with.

6.6 Build: the desk plan

6.7 Exercises

6.8 Problem: Plus Ten Per Cent

6.9 Interview questions