Black Swan Bets

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Portfolio Strategy: Convexity or Concavity?

	Convex Function	Concave Function
Example	\(f(x) = x^2\)	\(f(x) = \sqrt{x}\)
Curve shape	Bends upward	Bends downward
Chord position	Lies above the curve	Lies below the curve
Financial intuition	Increasing gains in extremes	Diminishing returns

When building a portfolio, investors often gravitate toward strategies that produce smooth, stable returns. These are typically based on concave payoffs — they perform reliably during normal market conditions but are vulnerable to extreme shocks. Think of strategies like shorting volatility or yield harvesting: small profits accumulate steadily, month after month — until a market dislocation triggers a sudden and devastating loss.

In contrast, convex strategies do the opposite. They accept small, repeated losses in exchange for the possibility of outsized gains during rare events. A classic example is buying out-of-the-money options: you lose the option premium most of the time, but if a crash (or boom) occurs, the payoff can be explosive. Taleb’s “Black Swan” approach advocates precisely this — positioning a small part of the portfolio for tail events, while keeping the rest relatively safe.

	Markowitz (Finance 101)	Taleb (Black Swan)
Optimisation objective	Max mean return, given variance	Maximise convexity of tail payoff
Assumptions	Normal returns, iid shocks	Fat tails, rare shocks dominate
Mathematical nature	Concave utility, linear returns	Convex payoff, nonlinear returns
Payoff shape	Smooth, stable	Flat then explosive (e.g. options)
Exposure to tail risk	Minimised via diversification	Exploited as a source of asymmetry
Black Swans result	Vulnerable	Resilient or super-profitable

# Concavity
monthly_pl <- c(rep(100, 11), -8500)
months <- month.abb[1:12]  # Jan to Dec

df <- data.frame(
  Month = factor(months, levels = months),
  Monthly_PL = monthly_pl) %>%
  mutate(Cumulative_PL = cumsum(Monthly_PL))

ggplot(df, aes(x = Month, y = Cumulative_PL, group = 1)) +
  geom_line(linewidth = 1.2, color = "#d62728") +
  geom_point(size = 3, color = "#d62728") +
  geom_hline(yintercept = 0, linetype = "dashed", color = "gray40") +
  labs(
    title = "Concave Payoff: Short Volatility Strategy",
    subtitle = "Consistent gains wiped out by one tail event (Dec)",
    x = "",
    y = "Cumulative Profit/Loss ($)") +
  theme_sph()

# Convexity
monthly_pl_convex <- c(rep(-100, 11), 10000)
months <- month.abb[1:12]  # Jan to Dec

df_convex <- data.frame(
  Month = factor(months, levels = months),
  Monthly_PL = monthly_pl_convex) %>%
  mutate(Cumulative_PL = cumsum(Monthly_PL))

ggplot(df_convex, aes(x = Month, y = Cumulative_PL, group = 1)) +
  geom_line(linewidth = 1.2, color = "#1f77b4") +
  geom_point(size = 3, color = "#1f77b4") +
  geom_hline(yintercept = 0, linetype = "dashed", color = "gray40") +
  labs(
    title = "Convex Payoff: Long Volatility Strategy",
    subtitle = "Steady small losses with large tail event gain (Dec)",
    x = "",
    y = "Cumulative Profit/Loss ($)") +
  theme_sph() 

Both strategies — concave and convex — have their place. But which is “better” depends on your goal: do you want to look smart most of the time (concave), or be right when it matters most (convex)? There is no right answer; you pays your money and you takes your choice.

Is Diversification Overbought?

Finance 101 typically favours playing it safe. Diversification spreads risk and reduces volatility, and is widely regarded as a sensible, professional approach. But diversification also smooths out extremes. It dilutes the impact of tail opportunities as much as it protects against tail risks. For those wanting convex payoffs — seeking to profit from rare, asymmetric outcomes — diversification is not the way. Often diversification is just a cover; avoiding rather than managing risk head on.

Behold, the fool saith, “Put not all thine eggs in the one basket” - which is but a manner of saying, “Scatter your money and your attention”; but the wise man saith, “Put all your eggs in the one basket and - WATCH THAT BASKET” Quoted from Pudd’nhead Wilson, an 1894 novel by Mark Twain

There are clear parallels with the Fantasy Premier League game — one of my favourite pastimes. As a manager, playing it safe — picking the most-owned captain, spreading the risk across consistent, high-ownership players — might keep you comfortably mid-table. But you won’t reach elite status by following the crowd. The aim isn’t to average 60 points a week. Fantasy success, like portfolio convexity, is about capturing upside — grabbing that one-week megahaul when your triple-captained differential bags a hat-trick. Hedging and diversifying might be wise if you’re managing your Nan’s pension. But in Fantasy, as in certain investment strategies, the lodestar is glory — not mere survival.