Futures Valuation-A Second Look

1

• PART-X
• Futures Valuation-A Second Look

2
Assets That Pay a Known Income

• If a person receives income from an asset that is
  in his possession, then such a cash inflow will
  obviously reduce the carrying cost
• The carrying cost can now be defined as
• rS I, where I is the future value of the
  income as computed at the time of expiration of
  the forward contract

3
Known Income (Cont)

• Why use the future value of the income and not
  just the income?
• We need to calculate the future value of the
  income, because the interest cost incurred for
  financing the purchase of the asset is payable
  only at the end
• And due to the principle of Time Value of Money,
  all cash flows must be measured at the same point
  in time in order to be comparable

4
Known Income (Cont)

• Consequently in order to rule out cash and carry
  arbitrage we require that
• F S ? rS I ? F ? S(1r) - I
• Similarly from a short sellers perspective, in
  the case of such assets, the effective income
  obtained by investing the proceeds from the short
  sale will be reduced by the amount of income
  received from the asset

5
Known Income (Cont)

• This is because the short seller is required to
  compensate the lender of the asset for the income
  generate by the asset
• Thus in order to preclude reverse cash and carry
  arbitrage, we require that
• F S ? rS I ? F ?S(1r) - I
• Consequently the no arbitrage condition is
• F S(1r) - I

6
Illustration ofCash and Carry Arbitrage

• Consider the case of TISCO once again
• Assume that the share price is Rs 100, and that
  the stock is expected to pay a dividend of Rs 5
  after three months, and another Rs 5 after six
  months
• Forward contracts expiring after six months are
  available at a price of Rs 96 per share

7
Illustration (Cont)

• We will assume that the second dividend payment
  will be made just an instant before the forward
  contract matures
• Take the case of an arbitrageur who can borrow at
  10 per annum
• He can borrow Rs 100 and buy a share of TISCO

8
Illustration (Cont)

• He can simultaneously go short in a forward
  contract to sell the share after six months for
  Rs 96
• After three months he will receive the first
  dividend of Rs 5
• This can be reinvested for the three months that
  remain in the life of the contract at a rate of
  10 per annum

9
Illustration (Cont)

• Finally, just prior to the maturity of the
  forward contract, he will receive the second
  dividend of Rs 5
• Thus at the time of delivery of the share, the
  investors cash inflow is
• 96 5 x (1 0.10) 5 106.025
• -----
• 4

10
Illustration (Cont)

• The rate of return on investment is
• (106.025 100) 0.0625 6.25
• -------------------
• 100
• Which is greater than the borrowing rate of 5
  for six months

11
Illustration (Cont)

• Cash and carry arbitrage was profitable in this
  case since F gt S(1r) - I
• That is, the contract was overpriced

12
Illustration of Reverse Cash and Carry Arbitrage

• Assume that the price of the forward contract is
  Rs 94 and not Rs 96
• An arbitrageur can short sell the stock and
  receive Rs 100
• This can be invested at 10 per annum so as to
  receive Rs 105 after six months
• Simultaneously he can go long in a forward
  contract in order to reacquire the asset after 6
  months

13
Illustration (Cont)

• After 3 months, the company will declare a
  dividend of Rs 5
• Since the arbitrageur has short sold the share,
  he must compensate the lender of the share
• This Rs 5 can be borrowed at the rate of 10 per
  annum
• Similarly, at the end of 6 months, another Rs 5
  will have to be paid when the second dividend is
  declared

14
Illustration (Cont)

• So the inflow at the end of six months is Rs 105
• The outflow at the end of six months is
• 94 5x (1.025) 5 Rs 104.125
• Thus there is clearly an arbitrage profit of
  Rs 0.875
• Reverse cash and carry arbitrage was profitable
  because
• F lt S(1r) - I

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The No-Arbitrage Condition

• The no-arbitrage condition is
• F S(1r) I
• In our example the correct price of the forward
  contract is
• F 100(1.05) 10.125 Rs 94.875

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Physical Assets

• While financial assets like stocks and bonds
  generate cash flows for investors who hold them,
  physical assets entail the incurrence of
  expenditure
• Investors have to bear the costs of storage as
  well as related expenses like insurance premiums

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Physical Assets (Cont)

• A cost is nothing but a negative income
• Hence if we denote the future value of all
  storage related costs as Z, as calculated at the
  time of expiration of the forward contract, then
  Z -I
• Thus the no-arbitrage condition may be expressed
  as F S(1r) (-Z) S(1r) Z

18
Physical Assets and Arbitrage

• If this relationship is violated, then arbitrage
  profits can be made
• We will first consider an overpriced forward
  contract on gold
• Obviously if the contract is overpriced, then
  Cash and Carry Arbitrage will be profitable

19
Illustration (Cont)

• Let the spot price of gold be 500 per ounce
• Let storage costs be 5 per ounce for a period
• of six months, payable at the end of six
• months
• Let the price of a forward contract for delivery
  of an
• ounce of gold six months hence be 535

20
Illustration (Cont)

• Consider the case of an investor who can borrow
  at 10 per annum
• He can borrow 500 and buy an ounce of gold and
  simultaneously go short in a forward contract
• Six months hence he can deliver the gold for
  535

21
Illustration (Cont)

• His interest cost for six months will be 25 and
  the storage cost will be 5
• Thus the effective carrying cost will be
  30
• The rate of return on investment is
• (535 500) 0.07 7
• --------------
• 500

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Illustration (Cont)

• The effective carrying cost is
• (530 500) 0.06 6
• --------------
• 500
• Hence the cash and carry strategy is profitable
• It is so because F gt S(1r) Z

23
No-Arbitrage

• In order to rule out both cash and carry as well
  as reverse cash and carry arbitrage, it must be
  the case that
• F S(1r) Z

24
The Importance of Short Sales

• In order to carry out reverse cash and carry
  arbitrage, the freedom to short sell is critical
• Thus, if the market is to be free of arbitrage
  opportunities, there must be unfettered freedom
  to short sell
• In practice, short sales need not always be
  feasible

25
Pure versus Convenience Assets

• A Pure asset, also known as an Investment asset,
  is one that is held by the investor as an
  investment
• That is, the investor is holding it purely
  because it is expected to provide some income
  during the holding period, and some capital gain
  at the time of sale
• Of course there could be assets which are
  expected to provide no income, but are being held
  mainly in anticipation of a capital gain

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Pure Assets (Cont)

• Hence, as long as an investor is assured that
  such an asset will be returned to him intact, at
  the end of the period during which he would
  otherwise have held it as an investment, and that
  he will be suitably compensated for any payments
  that he would have received in the interim, then
  he will not mind parting with it

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Pure Assets (Cont)

• In other words, such an investor will be willing
  to lend the asset, to facilitate short selling on
  the part of another
• All financial assets tend to be investment
  assets.
• Precious metals like gold also tend to be
  investment assets

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Convenience Assets

• Consider an agricultural commodity like wheat
• It is often held for reasons other than potential
  returns
• Let us consider the situation from the
  perspective of a person who chooses to hoard it
  before a harvest

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Convenience Assets (Cont)

• Normally prices of commodities rise before
  harvesting is complete and fall thereafter
• Thus a person who hoards wheat during a harvest,
  not only has to incur storage costs, but also
  faces the spectre of a capital loss
• Thus, seen from an investment angle, it makes
  little sense to hold wheat prior to a harvest

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Convenience Assets (Cont)

• However in practice there are investors who
  choose to hold commodities like wheat under such
  circumstances
• Such people are obviously getting some intangible
  benefits from holding the commodity

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Convenience Assets (Cont)

• For instance a wheat mill owner may wish to
  ensure that the mill does not have to be closed
  during an unanticipated shortage due to a cyclone
  or a monsoon failure
• The value of such intangible benefits is called
  the Convenience Value
• If an investor is getting a convenience value
  from an asset he will not part with it to
  facilitate short sales

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Convenience Values

• We can think of the convenience value as an
  implicit dividend
• However, unlike in the case of an explicit
  dividend, a potential short seller cannot
  compensate the owner of such an asset, and induce
  him to part with it
• This is true, firstly because convenience values
  cannot be quantified

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Convenience values (Cont)

• Secondly the perception of such value will differ
  from holder to holder

34
Convenience Assets andNo-Arbitrage

• Thus for assets which are being held for
  consumption purposes, we can only state that
• F S(1r) Z
• The possibility of cash and carry arbitrage will
  ensure that
• F ?S(1r) Z

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Convenience Assets andNo Arbitrage

• However F may be less than S(1r) Z, without
  giving rise to reverse cash and carry arbitrage,
  because facilities for short selling may not exist

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The Mechanics of Reverse Cash and Carry Arbitrage
for Convenience Assets

• We have provided a detailed illustration of how
  cash and carry arbitrage will take place in the
  case of physical commodities
• The corresponding arguments are valid
  irrespective of whether the asset is a pure asset
  or a convenience asset
• However we have not yet discussed reverse cash
  and carry arbitrage in detail, even for those
  physical commodities which are investment assets
  and not convenience assets

37
Reverse Cash and Carry andConvenience Assets

• It must be remembered that all physical assets
  need not be convenience assets
• However even for those commodities which tend to
  be held for investment purposes, there are some
  finer issues when it comes to reverse cash and
  carry arbitrage

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Reverse Cash and Carry (Cont)

• In the case of financial assets, whenever reverse
  cash and carry arbitrage is undertaken, the
  arbitrageur who is also the short seller, has to
  compensate the lender for any income that he is
  forgoing by parting with the asset

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Reverse Cash and Carry (Cont)

• However, in the case of physical assets, the
  lender is not foregoing any income
• On the contrary he would have incurred storage
  costs had he chosen to hold on to the asset,
  rather than lend it for a short sale
• In this case therefore, reverse cash and carry
  arbitrage will be profitable only if the cost
  savings experienced by the lender are passed on
  to the arbitrageur (short seller)

40
Illustration

• Assume that the spot price of gold is 500 per
  ounce
• Let the price of a six month forward contract be
  525
• The storage cost is 5 per ounce for six months,
  payable at the end of the period.
• The borrowing/lending rate is 10 per annum

41
Illustration (Cont)

• F 525 lt S(1r)Z 500(10.05)5 530
• Take the case of an arbitrageur who short sells
  the asset
• He will receive 500 which he will lend at 10
  per annum
• Simultaneously he will go long in a forward
  contract to acquire the gold after six months at
  525

42
Illustration (Cont)

• At the end of six months his cash inflow will be
  525 which will be the same as his cash outflow
• Thus in order for the arbitrage strategy to be
  profitable, he ought to be compensated by the
  lender of the asset with 5, which is the amount
  of the storage cost saved by him, or at least
  with a fraction of the amount

43
Illustration (Cont)

• In practice such an arrangement may not be
  feasible
• Does this mean that an under priced forward
  contract cannot be exploited even if the
  commodity is being held for investment purposes?
• The answer is no

44
Quasi-Arbitrage

• Consider the situation from the perspective of a
  person who owns one ounce of gold
• He can sell the gold in the spot market and lend
  the proceeds for six months
• Simultaneously he can go long in a forward
  contract to reacquire the gold at 525
• Six months hence his inflow will be 525.

45
Quasi-Arbitrage (Cont)

• This amount will be just adequate to repurchase
  the gold.
• In addition he will have 5 in his possession
  which represents the storage costs saved.

46
Quasi-Arbitrage (Cont)

• Such an investor is not an arbitrageur in the
  conventional sense, although he has clearly
  exploited an arbitrage opportunity
• Such a strategy is called Quasi-Arbitrage
• In derivatives parlance, we say that he has
  replaced a natural spot position with a synthetic
  spot position

47
Synthetic Spot

• What do we mean by a synthetic spot position?
• Notice that this investor gets back his gold at
  the end
• Thus although he has sold the gold, it is
  effectively as if he has not parted with it
• Thus he has sold something without really selling
  it

48
Synthetic Spot (Cont)

• Or put differently, he continues to own the gold
  during the period of six months, without actually
  owning it
• We know that Spot Futures Synthetic T-bill
• Therefore Futures T-bill Synthetic Spot
• Thus in the case of physical commodities that are
  held as investment assets, the possibility of
  cash and carry arbitrage and reverse cash and
  carry quasi-arbitrage, will help ensure that
• F S(1r) Z

49
The Value of a Forward Contract

• When a forward contract is entered into, its
  value to both the parties is zero
• That is, neither the long nor the short has to
  pay any money to get into a forward contract
• Of course both of them have to post margins.
• But a margin is a performance guarantee and not a
  cost

50
Forward Price versus Delivery Price

• The delivery price is the price specified in the
  forward contract
• It is the price at which the short agrees to
  deliver and the long agrees to accept delivery as
  per the contract

51
Forward Price versusDelivery Price (Cont)

• What then is a Forward Price?
• The forward price at a given point in time is the
  delivery price that is applicable for a contract
  being negotiated at that particular instant
• Once a contract is sealed, its delivery price
  will not change

52
Forward Price versusDelivery Price (Cont)

• However, as each new trade is negotiated, the
  forward price will keep changing
• To put things in perspective, if one were to come
  and say that he had entered into a forward
  contract a week ago, we would ask what was the
  delivery price? and not what was the forward
  price then?, although both would mean the same

53
Forward Price versusDelivery Price

• However, if we were to negotiate a contract at a
  particular point in time, we would ask what is
  the forward price?
• And if the negotiation were to be successful and
  the contract were to be sealed, then the
  prevailing forward price would become the
  delivery price of the contract being entered into

54
Evolution of Value

• When a contract is first entered into, its value
  to both parties will be zero
• However, as time passes, a pre-existing contract
  will acquire value
• Consider a long forward position that was entered
  into in the past at a time when the forward price
  was K
• Consequently its delivery price as of today will
  be K

55
Evolution of Value (Cont)

• In order to offset this position, the investor
  will have to take a short position, which will
  obviously be executed at the prevailing forward
  price F
• Thus if a counter-position is taken, the investor
  will have a payoff of (F-K) awaiting him at the
  time of expiration of the contract

56
Evolution of value (Cont)

• The value of the original contract is nothing but
  the present value of this payoff

57
Illustration

• Assume that a forward contract exists that
  expires at time T
• Let the delivery price be K
• Let F be the current forward price for a contract
  expiring at time T
• Let r be the risk-less rate of interest for a
  loan between now and time T

58
Illustration (Cont)

• The value of a long forward position is therefore
• F K
• ---------
• (1r)

59
Illustration (Cont)

• The value of a short position will be the
  negative of this, that is
• -(F K) K - F
• ----------- --------
• (1r) (1r)

60
Numerical Illustration

• A long position in a 9 month forward contract was
  entered into 3 months ago
• The delivery price is 100
• Today the forward price for a 6 month contract is
  120
• The risk-less rate of interest for six months is
  10

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Numerical Illustration (Cont)

• The value of a long forward position with a
  delivery price of 100 is therefore
• 120 100
• ------------ 18.18
• 1.10
• The value of a short forward position with a
  delivery price of 100 will be 18.18

62
Value

• As you can see, once a contract is sealed, a
  subsequent increase in the forward price will
  lead to an increase in value for the holder of a
  long position
• A subsequent decline in the forward price will
  lead to an increase in value for the holder of a
  short position

63
Value of a Futures Contract

• The value of a futures contract is zero when the
  contract is initiated
• That is, no money is required to take either a
  long or a short position in futures
• Assume that a futures contract is entered into at
  a price F0
• Let the settlement price at the end of the day be
  F1

64
Value of a Futures Contract (Cont)

• Using the same logic as for forward contracts, if
  this contract were to be offset, the profit for
  the long would be F1 F0
• This is precisely the amount that will be paid
  to/received from the long when the contract is
  marked to market at the end of the day

65
Value of a Futures Contract (Cont)

• Thus the process of marking to market ensures
  that the value of a futures position, whether
  long or short, is reset to zero at the end of the
  day
• Thus between the end of one trading day and the
  next, a futures contract will build up value
• However at the end of the next day, the value
  will revert to zero

66
Forward Price versus Futures Price

• One logical question is, will the price fixed per
  unit of the asset in the case of a forward
  contract be the same as in the case of a futures
  contract on the same asset, if the contracts are
  similar in all other respects?
• It can be shown that under certain conditions,
  this will indeed be the case

67
Pricing

• More specifically, if the risk-less rate of
  interest is a constant, and is the same for all
  the maturities, then forward and futures prices
  will be identical for contracts on the same asset
  and with the same expiration date
• Thus all the no-arbitrage conditions derived
  earlier are valid for futures contracts too

68
Random Interest Rates

• In real life however, interest rates are
  constantly fluctuating and are not constant
• This will therefore have an impact on the
  relationship between the forward price and the
  futures price
• The difference arises because while futures
  contracts are marked to market on a daily basis,
  forward contracts are not

69
Impact of Random Interest Rates

• Let us first consider a situation where interest
  rates and futures prices are positively
  correlated
• That is, when interest rates are high, so are the
  futures prices and vice versa
• Now rising futures prices will lead to cash
  inflows for investors with long positions

70
Impact of Random Rates (Cont)

• Thus the longs will be able to reinvest their
  profits at relatively high rates of interest
• At the same time rising futures prices will lead
  to cash outflows for investors with short futures
  positions
• These investors will have to finance such losses
  at relatively high rates of interest

71
Impact of Random Rates (Cont)

• On the contrary, if futures prices were to
  decline, the corresponding interest rates would
  also be lower
• Declining prices will lead to losses for the
  longs and profits for the shorts
• Thus the longs can finance their losses at low
  rates of interest while the shorts will have to
  invest their profits at low rates

72
Impact of Random Rates (Cont)

• An investor with a long position in a forward
  contract on the same asset, will not be affected
  by such interest movements in the interim, since
  he will have no intermediate cash flows
• Thus compared to such an investor, a person with
  a long futures position will be better off

73
Impact of Random Rates (Cont)

• By the same logic, a person with a short futures
  position will be worse off as compared to an
  investor with a short forward position
• Thus a person taking a long futures position
  should be required to pay more for this advantage

74
Impact of Random Rates (Cont)

• Viewed from a shorts angle, a person with a
  short futures position should receive more for
  this disadvantage
• Hence if interest rates and futures prices are
  positively correlated, futures prices will exceed
  forward prices

75
Impact of Random Rates (Cont)

• By a similar argument, if interest rates and
  futures prices are negatively correlated, then
  futures prices will be less than the
  corresponding forward prices

76
The Case of Gold

• Assume that interest rates rise because of higher
  expected inflation
• Gold is widely perceived as a hedge against
  inflation
• So gold prices will be expected to rise if
  inflation is expected to rise
• Hence hold prices and interest rates should be
  positively correlated

77
The Case of T-bonds

• Interest rates are negatively related to bond
  prices
• So T-bond futures prices should be negatively
  correlated with interest rates

78
Conclusion

• We would expect a gold futures contract to be
  priced higher than a comparable gold forward
  contract
• And a T-bond futures contract to be priced lower
  than a comparable T-bond forward contract

79
Conclusive Evidence?

• If we observe a difference between the futures
  price and the price of a comparable forward
  contract, for an asset, can we conclude that it
  is due to a relationship between futures prices
  and interest rates?
• The answer is no
• Firstly the transactions costs could be different
  in the two markets

80
Conclusive ?

• Secondly forward markets are usually much less
  liquid than futures markets
• Thirdly futures contracts carry a lower risk of
  default due to the role of the Clearinghouse and
  the Marking to Market mechanism
• To test the interest rate correlation hypothesis
  we need to look at an asset for which these other
  factors are insignificant
• It has been argued that FOREX markets offer an
  appropriate testing ground

81
Net Carry

• The term Net Carry refers to the net carrying
  cost of the underlying asset, expressed as a
  fraction of the current spot price
• If the risk-less rate is r, and the future value
  of income from the asset is I, then
• Net Carry rS I r I
• ------- --
• S S

82
Net Carry (Cont)

• For physical assets which entail the payment of
  storage costs
• Net Carry r Z
• ---
• S

83
Net Carry (Cont)

• For financial assets
• F S(1r) I S Net Carry x S
• For physical assets which are held for investment
  purposes
• F S(1r) Z S Net Carry x S
• However in the case of convenience assets
• F S(1r) Z ? F S(1r) Z - Y

84
Net Carry (Cont)

• The variable Y which equates the two sides of the
  relationship, is the marginal convenience value
• If Y 0, then we say that the market is at full
  carry
• Thus investment assets, which includes all
  financial assets and certain physical assets,
  will always be at full carry

85
Net Carry (Cont)

• However, futures markets for convenience assets
  will not be at full carry
• If the futures price of an asset exceeds the spot
  market price, or if the price of a near month
  contract is less than the price of a far month
  contract, then we say that the market is in
  Contango

86
Net Carry (Cont)

• However if the futures price is less than the
  spot price, or if the price of the near month
  contract is more than the price of a far month
  contract, then we say that the market is in
  Backwardation

87
Illustration of a Contango Market
Contract Price
Spot 500
March Futures 510
June Futures 520
September Futures 525
December Futures 540
88
Illustration of a Backwardation Market
Contract Price
Spot 500
March Futures 485
June Futures 470
September Futures 450
December Futures 440
89
Net Carry

• For financial assets, the net carry can either be
  positive or negative, depending on the
  relationship between the financing cost, rS, and
  the future value of the income from the asset, I
• A positive net carry will manifest itself as a
  Contango market, whereas a negative net carry
  will reveal itself as a market in Backwardation

90
Net Carry (Cont)

• In the case of physical commodities, if the
  market is at full carry, then we will have a
  Contango market
• However if the market is not at full carry, then
  we may have either a Backwardation or a Contango
  market, depending on the relative magnitudes of
  the net carry and the convenience yield