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3.12. Payoff with futures (risk profile)

The gains and losses on futures are symmetrical around the difference between the spot price on expiry of the futures contract and the futures price at which the contract was purchased. A simple example may be useful (see Figure 6): one futures contract = one share of ABC Corporation Limited.

On the vertical axis we have the profit or loss scale of the future. On the horizontal axis we have the price of the future at expiry (= spot price). If the long future is bought at LCC70 and the price at expiry is LCC71, the profit is LCC1, i.e. for each LCC1 increase in the price of the future, the profit is LCC1. Thus, if the spot price on maturity is LCC90, the profit is LCC20 (LCC90 - LCC70).

payoff with long futures contract (risk profile)

Figure 6: payoff with long futures contract (risk profile)

It will be apparent that if the spot price on maturity is SPm, and the purchase price is PP, then the payoff on a long position per one unit of the asset is:

It follows that the payoff in the case of a short future (see Figure 7) is:

It will also be clear that the payoff on a future is a total payoff because nothing was paid for the contract (remember: the margin is a deposit that earns interest and is repayable in full).

payoff with short futures contract (risk profile)

Figure 7: payoff with short futures contract (risk profile)

3.13. Pricing of futures (fair value versus trading price)

The reader should at this stage already have a good idea of the principle involved in the pricing of futures contracts. Some elaboration, however, will be useful. All or some of the following factors influences the theoretical price of a future, which is also termed the fair value price (FVP):

• Current (or "spot") price of the underlying asset.

• Financing (interest) costs involved.

• Cash flows (income) generated by the underlying asset.

• Other costs such as storage and transport costs and insurance.

The theoretical price / FVP of a future is determined according to the cost-of-carry model (CCM): the FVP is equal to the spot price (SP) of the underlying asset, plus the cost-of-carry (CC) of the underlying asset to expiry of the contract. Thus:

where:

rfr = risk free rate25 (i.e. the financing cost for the period) I = income earned during the period (dividends or interest) t = days to expiry (dte) of the contract / 365

OC = other costs (which apply in the case of commodities: usually transport, insurance and storage).

Thus, in the case of financial futures:

FVP = SP + CC = SP + {SP x [(rfr - I) x t]} = SP x {1 + [(rfr - I) x t]}.

An example may be handy. The table and graph shown earlier (Table 1 and Figure 3) are expanded to include the fair value prices (FVPs) at the end of each month26 (see Table 3 and Figure 8). Taking April 2010 as an example, we have the following:

SP (index value) = 15357

rfr (assumed) = 8.0% pa

I (assumed dividend yield) = 2.0% pa

t = dte / 365 = 319 / 365

FVP = SP + CC

= SP + {SP x [(rfr - I) x t]}

= SP x {1 + [(rfr - I) x t]}

= 15357 x {1 + [(0.08 - 0.02) x (319 / 365)]}

= 15357 x [1 + (0.06 x 0.873973)]

= 15357 x 1.052438

= 16162.

As can be seen from Table 3, the March 2011 future traded (15870) at lower than its FVP (16162).

Value of index (spot rate)

Market rate (price / value) of future (mark-to-market)

Fair value price

2009

March

13535

13665

15124

April

13733

13860

15277

May

13992

14120

15494

June

14054

14223

15493

July

14177

14525

15557

August

14011

14282

15303

September

13792

14030

14996

October

13916

14252

15060

November

14183

14425

15279

December

14889

15415

15963

2010

January

14754

15262

15744

February

14846

15235

15773

March

14939

15185

15796

April

15870

16162

May

15396

15865

16125

June

15404

15515

16057

July

15651

15865

16235

August

15833

15948

16343

September

15676

15712

16104

October

15724

15862

16073

November

15756

15840

16028

2011

December

15860

15965

16053

January

15054

15165

15160

February

15147

15173

15184

March (15th)

15277

15277

15277

Table 3: March 2011 all share index futures contract

March 2011 all share index (ALSI) future

Figure 8: March 2011 all share index (ALSI) future

It will be apparent that in the above use was made of simple interest. In the case of compound interest, the formula changes to:

FVP = SP x [1 + (rfr - I)]t

Using the above example:

FVP = SP x [1 + (rfr - I)]t = 15357 x 1.060.87397 = 15357 x 1.052244 = 16159.

It is clear that compounding makes little difference in the case of short-term contracts.

 
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