# Institutional portfolio hedging

**FUT-02 · Equity Index Futures · Core**

**Lesson question:** How can broad risk change without selling the holdings?

Original fictional examples; educational writing, not live market data, orders or production approval.

## By the end

- Explain the mechanism in the lesson question.
- Reproduce the worked comparison and recognize its failure case.

**Prerequisites:** FUT-01

A manager may want to reduce market Beta while retaining individual investments. A short index future can offset part of the portfolio's common-market sensitivity. It does not remove company-specific risk or guarantee a matched return.

## Understand the mechanism

A simple estimate is N=(target Beta−current Beta)×portfolio value/(futures price×multiplier). It assumes a suitable proxy and a stable Beta estimate. Sector concentration, funding, basis and unexpected company events remain.

The hedge target and its statistical uncertainty should be stated separately. A precise contract count does not make the estimated Beta precise.

## Walk through the process

Define what risk to reduce and for how long. Choose an economically relevant index. Calculate contracts, round deliberately and record residual exposure. Then stress how cash and margin behave if the hedge moves against the account.

An index hedge cannot substitute for deciding whether individual holdings still deserve ownership.

## Operational boundary

All prices, portfolio values and account balances are fictional. Real contract calendars, broker requirements and available liquidity must be checked for the instrument actually traded. The example's mathematical hedge or payoff is not an execution guarantee.

## A worked example

A fictional portfolio is worth 10 million with estimated Beta 1.2. The future is 5,000 with a 50 multiplier, or 250,000 per contract. Targeting Beta zero implies selling 48. Targeting Beta 0.4 implies selling 32.

The chart varies target Beta under these fixed assumptions. It is a sizing illustration, not evidence that the portfolio's actual returns will follow the model exactly.

## Calculation popup: Work the example

```text
N=(target Beta−current Beta)×V/(F×multiplier)
```

Use matching units and the explicitly hypothetical inputs.

1. Contract notional=5,000×50=250,000.

2. (0.4−1.2)×10,000,000/250,000=−32.

**Result:** Sell 32 under the stated proxy assumptions.

## Graph: Institutional portfolio hedging—worked comparison

**Horizontal:** Target portfolio Beta  
**Vertical:** Target signed futures contracts

Fictional data from the written example. The chart organizes the comparison; its shape is not a forecast.

Complete numerical data are embedded in `scripts/equity-futures/FUT-02.js` and displayed as an accessible table in the HTML lesson.

## What would invalidate the reading?

An idiosyncratic earnings shock can damage holdings even while the broad hedge behaves as expected. Beta can also change during stress.

## Check your understanding

### How many contracts reduce example Beta from 1.2 to 0.4?

**Answer:** Sell 32.

**Why:** (0.4−1.2)×10,000,000/250,000=−32.

### What is the critical failure to avoid?

**Answer:** An idiosyncratic earnings shock can damage holdings even while the broad hedge behaves as expected. Beta can also change during stress.

**Why:** Explain this risk before treating the example as a decision rule. A correct calculation does not guarantee a favorable market outcome. **Critical risk check.**

## Takeaway

Return to the question: How can broad risk change without selling the holdings? State the observed facts, the assumptions and the failure condition separately.

## References and provenance

The cited references support definitions and conventions. The teaching prose, tickets and numerical scenarios are original; they are not attributed as historical market observations.

- [CME Introduction to Futures](https://www.cmegroup.com/education/courses/introduction-to-futures)

- [CME Micro E-mini overview](https://www.cmegroup.com/education/courses/micro-e-mini-futures/micro-e-mini-futures-products-overview)