Plot = require("@observablehq/plot")
d3 = require("d3@7")
htl = require("htl")
mulberry32 = seed => {
return function () {
let t = seed += 0x6D2B79F5
t = Math.imul(t ^ t >>> 15, t | 1)
t ^= t + Math.imul(t ^ t >>> 7, t | 61)
return ((t ^ t >>> 14) >>> 0) / 4294967296
}
}
randomNormal = rng => {
let u = 0, v = 0
while (u === 0) u = rng()
while (v === 0) v = rng()
return Math.sqrt(-2.0 * Math.log(u)) * Math.cos(2.0 * Math.PI * v)
}Adjust portfolio beta with index futures
Adjusting a portfolio’s beta is about reducing or amplifying systematic risk. Equity portfolios tend to co-move with the broad market, and that co-movement is summarized by beta. A beta above 1.0 means the portfolio swings more than the market on average, while a beta below 1.0 reduces its sensitivity to broad market movements. Using index futures, a manager can offset part of the market exposure in minutes–no need to sell the underlying holdings.
Futures as a beta dial
Suppose a portfolio currently has beta \(\beta\) with respect to an equity index. To reach a desired beta \(\beta^*\), the manager can go short index futures contracts if \(\beta > \beta^*\) (to mute exposure) or go long if \(\beta < \beta^*\). With portfolio value \(V_P\), futures price \(F_0\), and contract multiplier \(Q\), the number of contracts to trade, \(N\), is given by
\[ N = \frac{(\beta^* - \beta) V_P}{F_0 Q}. \]
Each futures contract has a notional value of \(F_0 Q\). A short position reduces beta; a long position increases it. The trade shifts the slope of the portfolio’s value line versus market moves.
Interactive beta hedging lab
The dashboard below links the market index to your portfolio. Set the starting beta and a target beta, then press Apply hedge to layer in index futures. The right-hand plot shows both the original portfolio path and the hedged path. Try a high starting beta with a low target to see how the hedge flattens the volatility.
Tip
How to experiment
- Choose a starting beta with the slider.
- Set the target beta slider to your desired level.
- Decide whether the portfolio value or the contract multiplier should change and watch how that shifts the required number of futures.
- Press Apply hedge to lock the adjustment, or Clear hedge to reset the dashboard.
Effective Beta: The actual beta achieved after rounding to the nearest whole contract. Formula:
\[\beta_{eff} = \beta + N_{rounded} \frac{F_0 Q}{V_P}\]
This formula is derived from the principle that the beta of a combined portfolio is the weighted average of the betas of its individual components (the original portfolio and the futures position).
Here’s a breakdown of its origin:
Original Portfolio’s Market Exposure: The original portfolio with value \(V_P\) and beta \(\beta\) has a market exposure equivalent to \(\beta \times V_P\).
Futures Position’s Market Exposure: Each index futures contract has a notional value of \(F_0 \times Q\). Since an index futures contract tracks the market index, its beta relative to that index is approximately 1. Therefore, \(N_{rounded}\) futures contracts contribute \(N_{rounded} \times F_0 \times Q\) to the total market exposure.
Combined Market Exposure: The total market exposure of the adjusted portfolio is the sum of the original portfolio’s exposure and the futures position’s exposure: \((\beta \times V_P) + (N_{rounded} \times F_0 \times Q)\).
Effective Beta: The effective beta (\(\beta_{eff}\)) is this total market exposure divided by the original portfolio’s value (\(V_P\)): \(\beta_{eff} = \frac{(\beta \times V_P) + (N_{rounded} \times F_0 \times Q)}{V_P} = \beta + N_{rounded} \frac{F_0 Q}{V_P}\)
The hedge alters the portfolio’s sensitivity to market movements. As you can see in the ‘Portfolio value’ chart, when the target beta \(\beta^*\) is lower than the starting beta, the hedged path (orange dashed line) is flatter than the original path (blue line). This illustrates a reduction in market risk. A target beta near zero creates a nearly market-neutral portfolio. Conversely, choosing a \(\beta^*\) above the starting beta adds leverage, making the hedged path steeper and more volatile than the original.