Chapter 5- Financial Forwards and Futures

Introduction-Financial futures and forwards

On stocks and indexes On currencies On interest rates How are they used? How are they priced? How are they hedged?

Alternative Ways to Buy a StockThe purchase of a stock has three components:1, fixing the price,2. the buyer making payment to the seller, and3. the seller transferring share ownership to the buyer. If we allow for the possibility that payment and physical receipt can occur at different times, say, time 0 to T, then once price is fixed there are four possible purchasing arrangements: Payment can occur at time o or T, and physical receipt can occur at time 0, or T.Four different payment and receipt timing combinations

Outright purchase: ordinary transaction Fully leveraged purchase: investor borrows the full amount Prepaid forward contract: pay today, receive the share later Forward contract: agree on price now, pay/receive later

Payments, receipts, and their timing

Figure 5.1- Four different ways to buy a share of stock that has price S0 at time 0. At time 0 you agree to a price, which is paid either today or at time T. The shares are received either at 0 or T. The interest rate is r.

Pricing Prepaid ForwardsIf we can price the prepaid forward (FP), then we can calculate the price for a forward contractF = Future value of FP

Three possible methods to price prepaid forwards

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1. Pricing by analogy2. Pricing by discounted present value3. Pricing by arbitrage

Pricing Prepaid Forwards

1. Pricing by Analogy

In the absence of dividends, the timing of delivery is irrelevant

Price of prepaid forward contract is the same as current stock price

FP 0,T = S0 where the asset is bought at t=0.

2. Pricing by discounted present value

We can calculate the expected value of the stock at time T and then discount that value at an

appropriate rate of return. Since the expected stock price at time T, E(ST ) is uncertain; we need to use risk –adjusted discount rate based on CAPM.

and Combing the two, we get:

Which for non-dividend stock, the prepaid forward price is the stock

price.

Suppose XYZ the stock price is 50 and the risk free-rate is 6%. The beta of the stock 1.2 and market risk premium is 6%. What is the non-arbitrage forward price for delivery in one year from today?

R=Risk-adjusted discount rate= 6%+1.2(6%) =13.2%

FP 0,T = S0 =$50. 3. Pricing Prepaid Forwards by Arbitrage

Arbitrage: a situation in which one can generate positive cash flow by simultaneously buying and selling related assets, with no net investment and no risk, i.e. free money!!!

But the price of a derivative should be such that no arbitrage exist, no free money.

Let ‘s say at time t=0, the prepaid forward price somehow exceeded the stock price, i.e.,

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FP 0,T > S0

Table below shows cash flows and transaction to undertake arbitrage when the prepaid forward price, F P 0,T exceeds the stock price.

Cash FlowsTransaction Time 0 Time T (expiration)Buy stock @ S0 - S0 + ST

Sell prepaid forward @ FP 0,T + FP 0,T -ST

Total + FP 0,T- S0 0Since this sort of arbitrage profits are traded away quickly, and cannot persist at

equilibrium, then FP 0,T = S0

What happens if the stock paid dividends?

Is FP 0,T = S0 valid. No. Because the holder of the forward will not receive dividends that will

be paid to the holder of the stock holders. FP 0,T > S0

Suppose XYZ pays quarterly dividends of $1. The Company Beta is 1.2 and the continuously compounded risk- free rate is 6%. The Market Risk Premium (MRP) is 6%

a. What is the price of a prepaid forward contract that expires one year from today?

The owner of the stock is entitled to receive dividends. As we will get the stock only in one year, the value of the prepaid forward contract is today’s stock price, less the present value of the four dividend payments:

Present value of dividends= 0.97+0.94+0.91+0.88= $3.69

b. What is the price of a forward contract that expires at the same time?

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The forward price is equivalent to the future value of the prepaid forward. With the risk-adjusted rate of 13.2% an expiration of the forward in one year, we have:

Note that this is equivalent to taking the future value of the initial stock price and subtracting the future value (at 13.2%) of the dividends received.

F0,T =57.06-4.35=52.71Suppose XYZ stock costs $100 today and is expected to pay $1.25 quarterly dividends, with the first coming 3 months from today and the last just prior to the delivery of the stock. Suppose the annual continuously compounded risk-free rate is 10%. The quarterly compounded rate is therefore 2.25%. A 1-year prepaid forward contract for the stock would cost

Continuous Dividends

The prepaid forward price with continuous price is:

Where is the dividend yield.

Example- A $50 stock pays an 8% continuous dividend. The continuous compounded risk-free rate is 6%. What is the price of a prepaid forward contract that expires 1 year from today?

What is the price of a forward contract that expires at same time?

The owner of the stock is entitled to receive dividends. We have to offset the effect of the continuous income stream in form of the dividend yield by tailing the position:

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We see that the value is very similar to the value of the prepaid forward contract with discrete dividends that we have calculated above. we received four cash dividends, with payments spread out through the entire year, totaling $4. This implies a total annual dividend yield of approximately

2. The forward price is equivalent to the future value of the prepaid forward. With an interest rate of 6% and an expiration of the forward in one year we thus have:

We could also use Equation (5.7) of the text, i.e.,

Forward Contracts on Stock

What are the difference between prepaid forward and the forward contract? The difference is only in the timing of the payment for the stock. As we saw in the example above, the forward price is just the future value of the prepaid forward.

In general forward price is the future value of prepaid forward:

1. No dividends:

2. Discrete Dividends

3. Continuous dividends

Forward Premium:The difference between current forward price and stock price which can be used to infer the current stock price from forward price. It can be calculated by:

The annualized forward premium=

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For the case of continuous dividend, the AFP is the difference between risk-free rate and dividend yield.

Example- Suppose the stock price is $35 and the continuously compounding interest rate is 5%. A. What is the 6-month forward price, assuming dividends are zero?

B. If the 6-month forward price is $35.50, what is the annualized forward premium?

C. If the 6-month forward price is $35.50, what is the annualized continuous dividend yield?

A. We use the continuously compounded interest rate and the time to expiration in years (6 months is 0.5 year) in Equation (5.7). We have:

B. The annualized forward premium is calculated as:

Notice this is less than the interest rate; hence the index must pay a dividend.

C. We could use and solve for. However, it is easier to use the previous result concerning the annualized forward premium. The forward premium is simply the difference between the risk-free rate and the dividend yield:

Therefore, we can solve:

The annualized dividend yield is 2.16%.

Creating a Synthetic Forward Contract

One can offset the risk of a forward by creating a synthetic forward to offset a position in the actual forward contract

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How can one do this? (assume continuous dividends at rate )

– Recall the long forward payoff at expiration: = ST – F0, T

– Borrow and purchase shares as follows

Table 5.3 Demonstration that borrowing S0e−δT to buy e−δT shares of the index replicates the payoff to a forward contract, ST − F0,T .

• The idea of creating synthetic forward leads to following

– Forward = Stock – zero-coupon bond

– Stock = Forward – zero-coupon bond

– Zero-coupon bond = Stock – forward

• Cash-and-carry arbitrage: Buy the index, short the forward

• Figure 5.6 Transactions and cash flows for a cash-and-carry: A market maker is short a forward contract and long a synthetic forward contract.

Other Issues in Forward Pricing:Does the forward price predict the future price?

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According to the formula the forward price conveys no additional information

about the stock price. Moreover, the forward price underestimates the future stock price.Forward pricing formula and Cost of Carry

Forward Price= Spot Price + (Cost of Carry)

Forward Price= Spot Price + (Interest to carry the asset – asses lease rate)

Futures Contracts

Exchange-traded “forward contracts”

Typical features of futures contracts

Standardized, with specified delivery dates, locations, procedures A clearinghouse

o Matches buy and sell orders

o Keeps track of members’ obligations and payments

o After matching the trades, becomes counterparty

Differences from forward contracts

Settled daily through the mark-to-market process è low credit risk Highly liquid è easier to offset an existing position Highly standardized structure è harder to customize The difference negligible especially for short-lived contracts Can be significant for long-lived contracts and/or when interest rates are correlated with

the price of the underlying assetThe S&P 500 Futures ContractSpecifications for the S&P 500 index futures contract.

Underlying S&P 500 index Where traded CME Size $250 x S&P 500 index Months Mar, Jun, Sep, Dec. Trading ends Business day prior to determination of settlement price. Settlement Cash-settled, based on opening price of S&P500 on third Friday of

expiration month

Example: S&P 500 Futures

Notional value: $250 x Index

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Cash-settled contract Open interest: total number of buy/sell pairs Margin and mark-to-market

o Initial margin

o Maintenance margin (70 – 80% of initial margin)

o Margin call

o Daily mark-to-market

Example- Suppose the future price is 1100 and you wish to acquire a $2.2 million position in the S$P500 index. The notional value of one contract is $250x1100 = $275,000, the amount you are agreeing to pay at expiration per future contracts. To go long you buy 8 futures contracts ($2.2 M /0.275). The notional value of 8 contracts is $2.2million= (8) x (250) x (1100).

Let’s assume there is 10% margin and the mark-to market is weekly. The margin on futures contracts with notional value of $2.2 million is $220,000. If the S&P 500 futures price drops by 1 point, we lose $2,000. Loss= (1199-1100)*(8x250) =-$2,000.

If it drops by 72.01 points to 1027.99 a decline of about 6.5% we lose -$144,020.We have a choice of either paying this loss directly or allowing it to be taken out of the margin balance. It does not matter since we recover the unused margin balance plus interest at any time we close the position. The loss after the price drop is equal to $76, 2333.99 as shown in the table below.

Week MultiplierFutures

Price Price ChangeMargin Balance

0 2000 1100 0 $220,0001 2000 1027.99 -72.01 $76,233.992 2000 1037.88 9.89 $96,102.013 2000 1073.23 35.35 $166,912.964 2000 1048.78 -24.45 $118,205.665 2000 1090.32 41.54 $201,422.136 2000 1106.94 16.62 $234,894.677 2000 1110.98 4.04 $243,245.868 2000 1024.74 -86.24 $71,046.699 2000 1007.3 -17.44 $36,248.72

10 2000 1011.65 4.35 $44,990.57

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The 10-week profit on the position is obtained by subtracting from the final margin balance the future value of the original investment. Week 10 profit on the position is

equal to:

If the position had been a forward contract then the profit is equal to:(1011.65-1100) x ($2,000) =-176700

Why do the future and forward profits differ? The reason is the interest earned on the margin balance.

Uses of Index Futures

Why buy an index futures contract instead of synthesizing it using the stocks in the index? Lower transaction costs

Asset allocation: switching investments among asset classes Example: invested in the S&P 500 index and temporarily wish to invest in bonds instead

of index. What to do?

Alternative #1: sell all 500 stocks and invest in bonds

Alternative #2: take a short forward position in S&P 500 index

Effect of owning the stock and selling forward, assuming that S0 = $100 and F0,1= $110.

Uses of Index Futures

Cross-hedging with perfect correlationRp = RF + Bp (R S&P –RF)

To adjust for systematic risk of the portfolio of the stocks, we adjust the naïve formula with its beta:

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Results from shorting 509.09 S&P 500 index futures against a $100 million portfolio with a beta of 1.4.

Cross-hedging with imperfect correlation

Risk management for stock-pickers-

Chapter 5- Additional Materials

1. Demonstration that going long a forward contract at the price F0,T = S0e(r−δ)T and lending the present value of the forward price creates a synthetic share of the index at time T .

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2. Demonstration that buying e−δT shares of the index and shorting a forward creates a synthetic bond.

3. Transactions and cash flows for a reverse cash-and-carry: A market-maker is long a forward contract and short a synthetic forward contract.

4. Synthetic equivalents assuming the asset pays continuous dividends at the rate δ.

Example- Suppose you are a market-maker in S&R Index forward contract. The S&R Index spot price is 1100, the risk-free rate is 5%, and the dividend yield on the index is 0.

a. What is the no-arbitrage forward price for delivery in 9 months?

We use the valuation formula, with a continuously compounded interest rate of a dividend yield of and time to expiration Remember time is in years, hence 9 months is 3/4 of a year. We have:

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b. Suppose a customer wishes to enter a short index futures index. If you take the opposite position, demonstrate how you would hedge your resulting long positions using the index and borrowing or lending.

We engage in a reverse cash and carry strategy. In particular, we do the following:

Description Today In 9 months

Long forward, resulting from customer purchase 0Sell short the indexLendTotal 0

Specifically, with the numbers given in the exercise, and assuming the forward price is the no arbitrage price we determined in Part (1), we have:

Description Today In 9 months

Long forward, resulting from customer purchase 0Sell short the index $1,100Lend $1,100 $1,100

Total 0 0

Therefore, the market maker is perfectly hedged. She does not have any risk in the future, because she has successfully created a synthetic short position in the forward contract.

c. Suppose a customer wishes to enter a long index futures position. If you take the opposite position, demonstrate how you would hedge your resulting short positions using the index and borrowing or lending.

Now, we will engage in cash and carry arbitrage:

Description Today In 9 months

Short forward, resulting from customer purchase 0Buy the indexBorrowTotal 0

Specifically, with the numbers of the exercise, we have:

Description Today In 9 months

Short forward, resulting from customer purchase 0

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Buy the index $1,100Borrow $1,100 $1,100

Total 0 0

Again, the market maker is perfectly hedged. He does not have any index price risk in the future, because he has successfully created a synthetic long position in the forward contract that perfectly offsets his obligation from the sold forward contract.

d. Repeat the problem assuming that the dividend yield is 1.5%.

We use the valuation formula, with a continuously compounded interest rate of a dividend yield of and time to expiration We have:

2 We engage in a reverse cash and carry strategy. We must tail our short index position to Notice shorting the index requires paying the continuous

dividends Hence shorting 0.9888 units requires having to buy back more (i.e., 1 unit) of the index. The specifics of the reverse cash and carry are:

Description Today In 9 months

Long forward, resulting from customer purchase 0Sell short tailed position of the indexLend Total 0

Using the given numbers, we have:

Description Today In 9 months

Long forward, resulting from customer purchase 0Sell short tailed position of the index

Lend

Total 0 0

Therefore, the market maker is perfectly hedged. He does not have any risk in the future, because he has successfully created a synthetic short position in the forward contract.

3.With dividends reinvested, buying units today will grow to 1 unit in T years. The cash and carry table is:

Description Today In 9 months

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Short forward, resulting from customer purchase 0Buy tailed position in indexBorrow Total 0

Specifically, we have:

Description Today In 9 months

Short forward, resulting from customer purchase 0

Buy tailed position in index

Borrow $1,087.69 $1,087.69

Total 0 0

Again, the market maker is perfectly hedged. He does not have any index price risk in the future, because he has successfully created a synthetic long position in the forward contract that perfectly offsets his obligation from the sold forward contract.

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