BBA FN415 (18) Lect 08 Hedging Linear Risk.pptx

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Hedging Linear Risk

gi ng L in ea r R is k

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Overview

Risk that can be measured can be

managed. This chapter turns to the

active management of market risks.

 _{Hedging consists of taking position that lowers}

the risk profile of the portfolio.

 _{Farmers can use futures to hedge the price risk of}

their product.

 _{The objective of hedging is to find the optimal}

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Terminology



_{Static hedging}

_{, which consists of}

putting on, and leaving, a position

until the hedging horizon.



_{Dynamic hedging}

_{, which consists of}

continuously rebalancing the

portfolio to the horizon. This can

create a risk profile similar to

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A thought on Hedging

 _{It’s important to note that if the objective}

of hedging is to lower volatility, hedging will eliminate downside risk but also any upside potential.

 _{The objective of hedging is to lower risk,}

not to make profits, so this is a double-edged sword.

 _{Whether hedging is beneficial should be}

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Unitary Hedge

 Consider US exporter expecting to receive 125,000,000 JPY in seven mths.

 OTC is not available  cannot get perfect position.

 Use futures instead. CME lists YEN contract with face value of 12,500,000 YEN which expires in 9 mths.

 _{Exporter places order to short 10 contract, with}

the intention of reversing the position in seven months when the contract will have two mths until maturity.

 _{Since the amount sold is the same as the}

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Futures Hedging

 _{Suppose that the yen depreciates sharply, or}

that the dollar goes up from Y125 to Y150.

 _{This lead to a}_{Loss on the anticipate cash}

position

Y125,000,000 x (0.006667- 0.00800) = -$166,667

 _{This loss, however, is offset by a gain on the}

futures, which is

(-10)x Y12,500,000x {0.006711- 0.00806} = $168,621

 _{The net is a small gain of $1,954.}

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Basis

 _{Define Q as the amount of yen transacted and S and}

F as the spot and futures rates, indexed by 1 at the initial time and by 2 at the exit time.

 The P&L on the unhedged transaction is Q[S₂ – S₁] instead, the hedged profit is

Q[(S₂ – S₁)– (F₂ – F₁)] = Q[(S₂ – F₂)– (S₁ – F₁)] = Q[B₂ – B₁]

 _{Where B = S - F is the basis. The hedged profit}

depends only on the movement in the basis.

 _{Hence, the effect of hedging is to}_{transform price}

risk into basis risk.

 _{A short hedge position is said to be long the basis,}

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Basis Risk

 _{Basis risk arises when the characteristics of the}

futures contract differ from those of the underlying position.

 _{Give an example.}

 _{For most commodities, basis risk is inevitable.}  _{Basis risk is higher with cross-hedging (using}

totally different asset to create the hedge position)

 _{Basis risk is lowest when the underlying position}

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Optimal Hedging

 _{Suppose a fund manager holding a}

portfolio of corporate bond that should out perform the index.

 _{She fears that the interest rate will}

increase.

 _{Due to transaction costs, she cannot sell}

the whole portfolio and buy it back later.

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Optimal Hedging

 _{Define ΔS as the change in the dollar value}

of the inventory and ΔF as the change in the dollar value of one futures contract.

 _{The manager is worried about the future}

movement in the price, ΔS

 _{If the manager goes long N futures}

contracts, the total change in the value of the portfolio is

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Optimal Hedging - MVHR

 _{The variance of the total profit is equal to}

 _{Differentiating with respect to N}

 _{Setting the above equation to zero and solving for}

N we find

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 _{Here S consists of the number of units (shares,}

bonds, bushels, gallons) times the unit price (stock price, bond price, wheat price, fuel price).

 _{Define Q to be the number of units of S and let s}

be the unit price  S = Qs

 Define Q_f to be the number of units of F and let f be the unit price of the future  F = Q_f f

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

_Hence

_N

*

_{can be expressed as}



Where

β

_s,f

is the coefficient in the

regression of

Δs/s

over

Δf/f

.



_{Thus, the best hedge is obtained from}

a regression of the change in the value

of the inventory on the value of the

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Variance of the Profit

 _{Substituting N}*_into

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Hedge Effectiveness

 _{We measure the quality of the optimal}

hedge ratio in terms of the amount by which we decreased the variance of the original portfolio:

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Hedge Effectiveness

 _{We can also express the volatility of the}

hedged position from using the R2_as

 _{This shows that if R}2_{= 1, the regression is a}

perfect fit and the resulting portfolio has zero risk. In this situation, the portfolio has no basis risk.

 _{However if R}2_{is very low, the hedge is not}

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Examples; Airline

 _{An airline knows that it will need to purchase 10,000}

metric tons of jet fuel in 3 months. It wants some protection against an upturn in prices using futures

 _{The company can hedge using heating oil futures}

contracts traded on NYMEX.

 _{The notional for one contract is 42,000 gallons.}  _{As there is no futures contract on jet fuel, the risk}

manager wants to check if heating oil could provide an efficient hedge instead.

 _{The current price of jet fuel is $277 per metric ton. The}

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Application of Optimal Hedging: Duration Hedging

 _{Modified duration can be viewed as a}

measure of the exposure of relative changes in prices to movements in yields

 _D* _{is the Modified Duration and}_D*_P_{is called}

the dollar duration.

 _{We can write the change in the cash position}

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 _{The variance of the cash position is}

 _{The variance of the future position is}

 _{The covariance is given by}

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

_{Alternatively,}

_N

*

_{can be derived using}



_{which is zero when the net exposure,}

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Using future to modified the portfolio’s duration



_{More generally we can use N as a tool}

to modify the total duration of the

portfolio, If we have a target duration

of

D

_V

.

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Application of Optimal Hedging: Beta Hedging

 _{We now turn to equity hedging using stock}

index futures. Beta, or systematic risk can be viewed as a measure of the exposure of the rate of return on a portfolio i to movements in the “market” m:

 _{where β represents the systematic risk, α the}

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Application of Optimal Hedging: Beta Hedging

 _{Thus we can write as an approximation}

 _{Now assume that we have a stock index with a}

beta of 1, then

 _{We can then write the total portfolio payoff as}

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Beta Hedging - Effectiveness

 _{The optimal hedge with stock index futures}

is given by the beta of the cash position times its value divided by the notional of the futures contract.

 _{The quality of the hedge will depend on}

the size of the residual risk in the market model. For large portfolios, the

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