Hedging continued in futures trading
This is from the options and futures course I have been doing for the last week
See video below and image download for reference
Suppose an S and L agrees to buy mortgages from a building contractor for $1 million in nine months time
At the spot rate the S&L should turn around and sell the mortgage is 6.5%
However if interest rates increase SNL will have to sell the mortgages at a lower price
T-bone contract size is $100,000 so I hedges place by going short 10 contracts
Financial shortage for T-ball and example
Date is November cash 96–03 at 8.5%, basis is 29÷32 futures is at 97-00 action short 10 September
August 92–22 at 9% basis is 10÷32 futures 93–00 Long at 10
Opportunity laws of $34,062 basis is $5937 futures is $40,000 -$750 equaling $39,250
Basis narrowed it $5937
Net outcome: the game in the futures market Rose $39,250 exceeds the up to a loss and the cash market $34,062 due to the favorable basis change
The bank is exposed the interest rate risk. It’s the rates do rise above 8.5%, the S&L will take a loss. Because the mortgage is very long we look at T-bond contracts. The banks will be protected in the futures market if interest rates do rise. The bank is using the futures market as a future sale of those mortgages. The interest rates have risen from a 8.5% in November to August at 9%. Futures price will also go down as well if the interest rates rise. If the futures price falls from 97 to 93 by 400 basis points. This $4000 per contract Times 10 equals 40,000. There is an opportunity loss on the cash side so when the S and L sells the mortgage, because interest rates have risen the price of the mortgage will fall.
The net outcome becomes since there is an opportunity loss on The cash side but the futures market game is greater than the cash loss. The contract size is equal to both the cash side and the future side. The SNL will gain $5937 as the narrowing bases happens. The net outcome is the game in the futures market of $39,250 exceeds the opportunity lost in the cash market of $34,062 due to the favorable basis change.
The hedging in financials is complicated because you can have the interest rate going one way and the price go another. Just follow the framework of the template used in the previous examples.
Suppose a corporation plans to issue some commercial paper in a few months in order to borrow $5 million
Assume the issue will occur on February 1 and have 180 day maturity
If interest rates rise firms financing cost would rise
For emphasized as you can stay rising interest-rate by going short on 10 contracts
Contract size is $1 million and a 90 day maturity
The corporation is borrowing money but also issuing a promissory note.
How many contracts do you think the # of corporation will sell? It Is 10 because it is $5 million for five contracts. It is the time period on the Eurodollar which has a face value of $1 million in the interest is applicable to a 90 day period. If you borrow money for twice as long and the interest rate changes by X percent, you need to double up on the features position relative to the 90 day cycle. If the face value is $1 million in interest rate changes by one
%, we see the value of the instrument changes by $10,000. However it is a 90 day instrument, you need to divide by four which gives us $2500 for a 1% change. If you boil it down to one basis point on the Eurodollar it is $25. If the interest rate increases by one percent the euro dollar contract only changes by $2500 which is fine if you’re borrowing for 90 days. If you borrow $1 million for 180 days and the interest rate changes by 1%,it would affect the value of the loan by $5000 not $2500. To protect that outcome you would need 2 contracts. If they have a cash position of 5 million for it for period twice as long based on the future specification, you need to double up to 10.
The interest rate could change between any point in February. They take a loan out in February so they can sell the paper and get their money back to pay for the building. If they take a long today by interest rates jump up, if they had not hedged a 1% increase on their $5 million loan they take in February. At 1% for the entire year would be $50,000. For half a year, it would be $25000 in the increased cost of the loan. If the rate does go up by one person, if you’re not fully hedged. You’re not fully hedged unless a sufficient number of contracts to cover you for your increased cost on the cash side due to the interest rate change. You are really exposed to the interest rate change. For every X percent increase in interest rate will cost you Y dollar so you have that hedge in the futures market. You also have to double up The difference between 90 to 180 days.
You’re not boring the money until February but you’re using the hedge as a substitute for when you issue the commercial paper and borrow the money. You need to borrow $5 million for half a year. The loan will be priced on February 1. Any interest rate change will affect The pricing depending on the length of the loan.
You short the market today and you hold a position until you issue the commercial paper. It is no different than a soya bean farmer Who has long Soya bean if he sells futures to hold those positions until he delivers his physical commodity. This is When he lifts the hedge. If an airline shorts jet fuel, they buy options or futures to hold a position until they purchase the physical commodity.
In this example we are hedging against an adverse price move between now and February. What Matters is the number of days the loan is for. If the interest rate jumps up 1% between now and February, it will cost you x dollars. When you saw the commercial paper you’re looking for a discount. You’re trying to hedge against a decline in the amount of money you receive when you issue the paper.
Short hedge in Eurodollar example
Date is November cash $91.92 at 8.08% basis is -.42 futures $91.50 action is short 10 September
February cash is $90.77 at 9.73% basis is -.3 futures is 90.47 and long 10.
Opportunity cost is -$28,750 basis is -$3000 futures is $25,750 -$750 equaling $25,000
Cash decreased 115 points at $28,750 loss
Futures decreased by 103 points for $25,000 game
Basis widen at $3000 loss
That outcome is the corporation raised 25,000 more then without the hedge. The opportunity lost in the cash market is $28,750 is almost offset by the futures game of $25,000
On the cash side the interest rate does go out so it does cost you more to take out the loan in February so the opportunity lost becomes $28,750. The basis became more positive but on a short you wanted to narrow. Because it Why didn’t the basis change becomes negative
The cash price decline by 115 basis points becomes a loss of $28,750. Futures price decrease by 103 points so you get a $25,000 game.
And that outcome is a corporation with sold The commercial paper at a lower rate if they did not hedge because it’s discounted under the $5 million face value but but they Game $25,000 in the futures market. They came very close to offsetting their entire loss
Short hedging Eurodollar example continued
Recall: price equals face value minus discount
So per $100, price=100-100(.0923)(180/360)=95385
The final proceeds from the commercial paper off are: total proceeds equals .9535 times $5 million equaling 4,769,250 Plus $25,000 futures equaling 4,795,250
The initial objective was per hundred dollars equaling 100-8.08×180÷360 equaling 95.96
Total equals .9596 times $5 million equals $4,790,000
What difference of $3750 compared to actual proceeds
For T-bill, when you issue commercial paper there is a discount. You hold it until maturity and redeem it for face value. The difference between what you and the face value is your effective interest. Commercial paper of the face value of $5 million. The corporation sells a piece of paper on the money market. Whoever buys it Will pay lower. They will hold it for a duration as an 180 days. They redeem it to get there 5 million. The difference between what they pay and the 5 million is their interest.
The price on the instrument will equal the face value minus the discount.
In the above example you are using unit dollars of hundred.
So per $100 the prices equal to 100 minus 100 times the interest of .0923×180÷360 =95.385 interest rate is for six months .0923 is a spot rate on February 1. In other words if they want to earn 9.23% on interest they would have to pay $95.385 for every hundred dollars you issue.
To issue $5 million in commercial paper it will equal .9538 5×5,000,000 = 4,769,250 Plus futures profit of 25,000 equaling $4,794,250 . This is the total amount you’ll get for your commercial paper.
Your goal was to make a 8.08% interest. Per $100 equals 100-8.08×180÷360 equaling 95.96
Totals to equal .9596 times $5 million equals $4,798,000. This is the amount you would’ve raised at the current interest rate of at 8.08%.
You could not sell at this rate since the interest rates did go up.
You came very close due to the futures hedge.
difference of 3750 compared Actual proceeds. If you did not head you would allow us the opportunity cost.
Remember that the basis is futures minus cash Price.
Risk averse ledger
Assuming the hedge has $10 less
Investment has a 50-50 chance of returning five dollars versus $15 i.e. expected return it is $10
For a risk-averse and faster, futility of the expected value of the investment,
U greater than .5 times u (five) +.5 times u(15)
See utility versus Wealth
AsSumption is if you have a given cash position you want to hedge an equivalent volume in the futures market. In reality it might not be wise to hedge hundred percent. Of all the airlines to hedge they do not hedge hundred percent. Do you want to choose a hedging ratio like your spot position versus futures position. Is typically less than 100%. This revolves around the Markowitz portfolio Theory. When you talk about optimal hedges, what you are
Doing is if you have a cash
Positon short or long. You are adding to your portfolio by taking a future position you will be Changing the portfolio. A new portfolio is constructed with the cash and a futures position with long or short in each. The question is how do you devisee optimal portfolio.
Risk means you do not accept the fair bet. The certain thought of losing $20 gives a certain negative impression. Winning $20 gives a certain positive utility. The negative futility can exceed the winning utility. This can be risk aversion.
In the chart attached if you’ve utility of 10 for keeping your $10 you become risk-averse versus taking a bit of losing the money with a 50% chance
Risk-averse is the utility of certain equivalent greater than utility of the expected value. As on the chart of utility of $10. This on the utility front tier exceeds running five or .5*u(5)+.5*u(15). This utility becomes concave. Has user consider risk-averse
You can also draw out a indifference curve of expected returns versus risk.there is always a trade-off between risk and expected return.. It is upward sloping becomes move from portfolio a to b, where portfolio way contains low return and low risk. Portfolio be as equally as desirable and which is higher return and higher risk. You would need to take on extra risk to take on higher expected return.
There can also be different levels of risk for the return. When you look at these different indifference curves, they represent a different portfolio mix of assets including your futures versus cash positions.
Future positions may be combined with cash positions to form portfolio
Optimal hedging is the level of the features position relative to a cash resulting in the highest utility relative to risk and returns that are faced by the investor
Up until now we have assumed the optimal it’s ratio was 1.0
Suppose a farmer hedges a fraction h of cash in futures market and leave 1-h unhedged.
Expected return on hedged position is:
E(Rh)=E(Rc) -h*E(Rf) where Rc and Rf is cash and futures return. Rh is hedged ratio. There is a minus sign before the age because you were shorting. Use plus if you’re long.
If the price goes up down be a positive return the cash and negativE return on the futures.
You are interested in The risk of this portfolio. Investors want to diversify to reduce risk. This is done thru variance.
Variance of the return to the hedged portfolio is
Delta ^ h=delta ^ c + h^2 *delta ^2 f – 2h*deltacf (covariance this from financial theory)
Objective is to minimize the variance of cash/futures portfolio
Min(delta^2 h)=delta^2 c + h^2*delta^2 f – 2h*delta c f
Refer to first order of conditions and attached image or refer to time 43:06 In video
Also referred to the variance image attached as explained at 44:21
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