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PART-X Futures Valuation-A Second Look 1

Assets That Pay a Known Income n If a person receives income from an asset that is in his possession, then such a cash inflow will obviously reduce the carrying cost n The carrying cost can now be defined as r. S – I, where I is the future value of the income as computed at the time of expiration of the forward contract 2

Known Income (Cont…) n Why use the future value of the income and not just the income? n n 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 3

Known Income (Cont…) n n Consequently in order to rule out cash and carry arbitrage we require that F – S r. S – I F S(1+r) - I Similarly from a short seller’s 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 4

Known Income (Cont…) n n n 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 r. S – I F S(1+r) - I Consequently the no arbitrage condition is: 5 F = S(1+r) - I

Illustration of Cash and Carry Arbitrage n Consider the case of TISCO once again n n 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 6

Illustration (Cont…) n n 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 n He can borrow Rs 100 and buy a share of TISCO 7

Illustration (Cont…) n n n 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 8

Illustration (Cont…) n n 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 investor’s cash inflow is 96 + 5 x (1 + 0. 10) + 5 = 106. 025 ----4 9

Illustration (Cont…) n The rate of return on investment is (106. 025 – 100) = 0. 0625 ≡ 6. 25% ---------100 n Which is greater than the borrowing rate of 5% for six months 10

Illustration (Cont…) n n Cash and carry arbitrage was profitable in this case since F > S(1+r) - I That is, the contract was overpriced 11

Illustration of Reverse Cash and Carry Arbitrage n Assume that the price of the forward contract is Rs 94 and not Rs 96 n n n 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 12

Illustration (Cont…) n n 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 13

Illustration (Cont…) n n So the inflow at the end of six months is Rs 105 The outflow at the end of six months is: 94 + 5 x (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 < S(1+r) - I 14

The No-Arbitrage Condition n The no-arbitrage condition is: n n F = S(1+r) – I In our example the correct price of the forward contract is: F = 100(1+. 05) – 10. 125 = Rs 94. 875 15

Physical Assets n n 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 16

Physical Assets (Cont…) n A cost is nothing but a negative income n n 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(1+r) – (-Z) = S(1+r) + Z 17

Physical Assets and Arbitrage n n n 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 18

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 19

Illustration (Cont…) n n n 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 20

Illustration (Cont…) n n n 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 21

Illustration (Cont…) n n n The effective carrying cost is: (530 – 500) = 0. 06 ≡ 6% -------500 Hence the cash and carry strategy is profitable It is so because F > S(1+r) + Z 22

No-Arbitrage n 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(1+r) + Z 23

The Importance of Short Sales n In order to carry out reverse cash and carry arbitrage, the freedom to short sell is critical n n 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 24

Pure versus Convenience Assets n A Pure asset, also known as an Investment asset, is one that is held by the investor as an investment n n 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 25

Pure Assets (Cont…) n 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 26

Pure Assets (Cont…) n n n 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 27

Convenience Assets n Consider an agricultural commodity like wheat n n 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 28

Convenience Assets (Cont…) n n n 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 29

Convenience Assets (Cont…) n n 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 30

Convenience Assets (Cont…) n n n 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 31

Convenience Values n n n 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 32

Convenience values (Cont…) n Secondly the perception of such value will differ from holder to holder 33

Convenience Assets and No-Arbitrage n Thus for assets which are being held for consumption purposes, we can only state that: F ≤ S(1+r) + Z n The possibility of cash and carry arbitrage will ensure that F S(1+r) + Z 34

Convenience Assets and No Arbitrage n However F may be less than S(1+r) + Z, without giving rise to reverse cash and carry arbitrage, because facilities for short selling may not exist 35

The Mechanics of Reverse Cash and Carry Arbitrage for Convenience Assets n We have provided a detailed illustration of how cash and carry arbitrage will take place in the case of physical commodities n n 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 36

Reverse Cash and Carry and Convenience Assets n It must be remembered that all physical assets need not be convenience assets n 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 37

Reverse Cash and Carry (Cont…) n 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 38

Reverse Cash and Carry (Cont…) n n n 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) 39

Illustration n n 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 40

Illustration (Cont…) n n F = 525 < S(1+r)+Z = 500(1+0. 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 41

Illustration (Cont…) n n 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 42

Illustration (Cont…) n n n 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 43

Quasi-Arbitrage n Consider the situation from the perspective of a person who owns one ounce of gold n n n 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. 44

Quasi-Arbitrage (Cont…) n n 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. 45

Quasi-Arbitrage (Cont…) n n n 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 46

Synthetic Spot n What do we mean by a synthetic spot position? n n n 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 47

Synthetic Spot (Cont…) n n 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(1+r) + Z 48

The Value of a Forward Contract n When a forward contract is entered into, its value to both the parties is zero n n n 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 49

Forward Price versus Delivery Price n n 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 50

Forward Price versus Delivery Price (Cont…) n What then is a Forward Price? n n 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 51

Forward Price versus Delivery Price (Cont…) n n 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 52

Forward Price versus Delivery Price n n 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 53

Evolution of Value n When a contract is first entered into, its value to both parties will be zero n n n 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 54

Evolution of Value (Cont…) n n 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 55

Evolution of value (Cont…) n The value of the original contract is nothing but the present value of this payoff 56

Illustration n n 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 57

Illustration (Cont…) n The value of a long forward position is therefore F–K ----(1+r) 58

Illustration (Cont…) n The value of a short position will be the negative of this, that is: -(F – K) K-F ------ = -------(1+r) 59

Numerical Illustration n n 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% 60

Numerical Illustration (Cont…) n n 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 61

Value n n 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 62

Value of a Futures Contract n n 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 F 0 Let the settlement price at the end of the day be F 1 63

Value of a Futures Contract (Cont…) n n Using the same logic as forward contracts, if this contract were to be offset, the profit for the long would be F 1 – F 0 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 64

Value of a Futures Contract (Cont…) n 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 n n 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 65

Forward Price versus Futures Price n n 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 66

Pricing n n 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 67

Random Interest Rates n n n 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 68

Impact of Random Interest Rates n Let us first consider a situation where interest rates and futures prices are positively correlated n n 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 69

Impact of Random Rates (Cont…) n n n 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 70

Impact of Random Rates (Cont…) n On the contrary, if futures prices were to decline, the corresponding interest rates would also be lower n n 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 71

Impact of Random Rates (Cont…) n 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 n Thus compared to such an investor, a person with a long futures position will be better off 72

Impact of Random Rates (Cont…) n n 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 73

Impact of Random Rates (Cont…) n n Viewed from a short’s 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 74

Impact of Random Rates (Cont…) n By a similar argument, if interest rates and futures prices are negatively correlated, then futures prices will be less than the corresponding forward prices 75

The Case of Gold n Assume that interest rates rise because of higher expected inflation n 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 76

The Case of T-bonds n Interest rates are negatively related to bond prices n So T-bond futures prices should be negatively correlated with interest rates 77

Conclusion n n 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 78

Conclusive Evidence? n n 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 n Firstly the transactions costs could be different in the two markets 79

Conclusive ? n n 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 80

Net Carry n The term `Net Carry’ refers to the net carrying cost of the underlying asset, expressed as a fraction of the current spot price n n If the risk-less rate is r, and the future value of income from the asset is I, then Net Carry = r. S – I = r – I ------- -S S 81

Net Carry (Cont…) n For physical assets which entail the payment of storage costs Net Carry = r + Z --S 82

Net Carry (Cont…) n n n For financial assets: F = S(1+r) – I = S + Net Carry x S For physical assets which are held for investment purposes: F = S(1+r) + Z = S + Net Carry x S However in the case of convenience assets: F ≤ S(1+r) + Z F = S(1+r) + Z 83 Y

Net Carry (Cont…) n The variable Y which equates the two sides of the relationship, is the marginal convenience value n n 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 84

Net Carry (Cont…) n n 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 85

Net Carry (Cont…) n 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 86

Illustration of a Contango Market Contract Price Spot 500 March Futures 510 June Futures 520 September Futures 525 December Futures 540 87

Illustration of a Backwardation Market Contract Price Spot 500 March Futures 485 June Futures 470 September Futures 450 December Futures 440 88

Net Carry n n For financial assets, the net carry can either be positive or negative, depending on the relationship between the financing cost, r. S, 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 89

Net Carry (Cont…) n n 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 90