# Risk management applications of forward and futures strategies

**1. Adjusting the portfolio beta: Demonstrate the use of equity futures contracts to achieve a target beta for a stock portfolio and calculate, interpret the number of futures contracts required.**

– To modify the beta of an equity portfolio with futures on an equity index, we need to know the beta of the equity portfolio to be hedged or leveraged, as well as the beta of the futures contract.

**– To calculate beta:**

**β = Cov(i,m)/ б ^{2}**

i: An individual stock, equity portfolio, or equity index

Cov(i,m) covariance of returns on asset i with the market

б ^{2 }Variance of the market return

number of contract = (β_{T} – β_{P})/ β_{f} * Vp/ [P_{f }* (multiplier)]

**Example:**

Adjusting portfolio beta: A manager of a $5.000.000 portfolio wants to increase the beta from the current value of 0.8 to 1.1. The beta on the futures contract is 1.05 and the total futures price is $ 240.000.

Calculate the required number of future contracts to achieve a beta of 1.1

Calculate the required number of futures contracts to achieve a beta of 0.0

Target beta: 1.1

number of contracts = (1.1 – 0.8) /1.05 * 5000.000/ (240.000) = 5.95

Buy 6 contracts at $240.000

Target beta: 0

number of contracts = (0.8– 0.8)/1.05 * 5000.000/240.000 = -15.87

Sell 16 contracts at $240.000

**Basis risk** occurs whenever the item hedged (in the numerator of the hedge formula calculation) is not a perfect march for the hedging vehicle ( in the denominator of the hedge formula) and as a result, the two change in relationship to each other in unpredictably way.

**The typical reasons for basis risk:**

– The numerator and denominator are not based on the same item. (A stock portfolio hedged using a contract based on the S&P index, a bond portfolio hedged with a Treasury bond contract based on a single delieverable Treasury bond)

– The betas and durations used in the hedge calculation do not reflect the actual subsequent market value changes of the portfolio or contract.

– The hedge results are measured prior to contract expiration and the hedge is closed prior to contract expiration.

– The number of contract is rounded.

– The future and spot price are not fairly priced based on the cash and carry arbitrage model.

Effective beta of the position can be measured ex post ( after the fact) as:

**Effective beta = % change in value of the portfolio/ % change in the index.**

**2. Synthetic positions: Construct a synthetic stock index fund using cash and stock index futures ( equitizing cash)**

– Synthetic positions are based on the same formulas using beta or duration to modify portfolio risk.

– Synthetic positions more precisely replicate the same initial investment and ending results that would have occurred if the replicated position had been owned instead.

– Synthetic equity/bond positions require purchasing contracts and holding sufficient cash equivalents earning the risk-free rate to pay for the contracts at expiration.

– Synthetic cash positions involve holding the underlying and shorting contracts to hedge the position in such a way that the hedged position “earns” the risk free rate over the hedging period.

– The number of contract is computed using the previous risk modification formulas. The quantity to hedge ( in the numerator of the hedging formula) is the FV of the amount to modify.

– If the objectives is to create synthetic equity from cash and the desired β_{T} is the same as β_{f}, then the first term in the calculation becomes

**( β _{T}-0)/ β_{f }= 1**

Because it has no effect on the calculation, the betas can be ignores.

– If the objectives is to create synthetic cash from equity and the β_{p }is the same as β_{f }and the first term of the calculation becomes:

**( 0 – β _{p})/ β_{f }= – 1**

Because it has no effect on the calculation, the betas can be ignores.

– In other cases, the existing or desired beta are not the same as the futures beta and will be given. In such cases, the betas are used and do affect the computation.

– The use of the risk free rate and FV would, in a perfect hedge, mean the synthetic position completely replicated the beginning and ending results that would have been obtained if the desired synthetic position had been actually held.