Hedging a stock portfolio with index futures
1. The idea
A share portfolio’s beta, \(\beta\), says how strongly it moves with the market: a portfolio with a beta of 1.2 tends to rise 1.2% when the index rises 1%, and to fall 1.2% when it falls 1%. Its exposure to the market is \(\beta V_A\), where \(V_A\) is the portfolio’s value.
Futures on a stock index move with the index. Each contract has a multiplier \(m\), the dollars per index point, so one contract at the futures price \(F\) carries market exposure worth \(F \times m\). Selling futures removes exposure, and buying them adds it. To remove all of the portfolio’s market exposure, sell
\[ N^* = \beta\,\frac{V_A}{F \times m} \]
contracts. More generally, to move the beta from \(\beta\) to \(\beta^*\), trade \(N = (\beta^* - \beta)\,V_A/(F \times m)\) contracts: buy them if the number is positive, sell them if it is negative.
An example. A portfolio worth $40 million has a beta of 1.1, and index futures trade at 5,000 with a multiplier of $50, so one contract carries \(5{,}000 \times 50 = \$250{,}000\) of exposure. Removing the market’s risk takes \(1.1 \times 40{,}000{,}000/250{,}000 = 176\) contracts sold; halving the beta, to 0.55, takes 88. A portfolio hedged completely earns about the risk-free rate, plus whatever its shares earn beyond what the market explains.
Index puts. A manager who wants to keep the gains of a rising market but not the losses of a falling one can buy put options on the index instead. One contract covers the index level \(S_0\) times its multiplier, so the portfolio needs
\[ N = \beta\,\frac{V_A}{S_0 \times m} \]
puts. They offset about the part of the portfolio’s losses that comes from the market; the shares’ own risk remains, and rounding the number of contracts makes the hedge approximate. For a portfolio that follows the index closely, puts with a strike price at today’s index level protect about today’s value, less the premium. With another beta, the strike price for a given protection level is worked out from the capital asset pricing model. Section 4 uses that model to show what a portfolio hedged with futures is expected to be worth at the end.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The portfolio’s beta on a chart
First, how much one contract changes the portfolio’s beta.
Now move the number of contracts. Each contract sold lowers the beta by the same amount, and the beta reaches 0 at \(N = -N^*\): \(N^*\) contracts sold.
4. One worked example, then your turn
In the worked example, we lower a portfolio’s beta.
Now one that raises it, with hints, one step at a time.
What the hedge leaves. By the capital asset pricing model, a portfolio is expected to earn \(r_P = r + \beta\,(r_M - r)\) over a period, where \(r\) is the return at the risk-free rate and \(r_M\) the market’s return, the index’s change plus its dividends, both over the same period. If the futures expire when the hedge ends, they are then worth the index level \(S_T\), and the \(N\) contracts sold at today’s futures price, \(F_0\), gain \(N\,m\,(F_0 - S_T)\). The hedged position’s expected value at the end, \(V\), is
\[ V = V_A\,(1 + r_P) + N\,m\,(F_0 - S_T), \]
which stays close to \(V_A\,(1 + r)\), the risk-free growth, wherever the index ends.
Then the value of a hedged portfolio at the end, with hints.
Now move the index level at the end. The portfolio alone rises and falls with the market, while the hedged position stays close to $12,150,000, what $12 million grows to at the risk-free rate.
And one on your own, with index puts.
5. Practice
Each question has new numbers. You master the skill when you get three right in a row, with at least one right answer for removing the market’s risk, for changing the beta, for insuring with puts, and for the value of a hedged portfolio at the end. You can always open the solution, but then that question does not count.
6. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: