Solution Manual for Principles of Corporate Finance 14th Edition by Richard B...
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pricing forwards and futures
1. โข Should spot/cash prices and forward/futures
prices be identical?
โข Inducement to carry stocks.
2. Pricing forwards and futures-Overview
โข Concepts and definitions.
โข Adjusting for time.
โข Cost of carry.
โข Formulae for calculation of forward price
โข Valuation of existing contracts.
3. Pricing and valuation of forwards and futures- concepts
โข Forward price (FP) is the delivery price of a
theoretical forward contract that may or may not
actually happen.
โข Delivery price (DP) is the price of an actual
forward contract already entered into.
โข As time progresses, DP never changes .FP keeps
changing with time and changes in the price of
the underlying.
โข Value is a measure of how much better or worse
off the parties (buyer and seller) are after
entering into the forward contract.
4. Pricing of a forward or futures contract
โข The essence of pricing a forward or futures
contract, involves taking an observed spot price
and adjusting it for time.
โข It is all about determining the delivery price.
โข The key to determining the delivery price for a
new forward or futures contract is that the value
(not price) of a contract at inception is zero. At
inception neither party has an advantage over
the other.
โข Between inception and execution, as the price of
the underlying changes the value of the contract
changes for both buyer and seller.
5. Adjusting for time
PV FV
Rs. Rs. Rs.
-T Time 0 +T
PV FV
Compounding
Discounting
FV=PV X e
rt
PV=FV X e
-rt
6. At inception neither party has an advantage
0
Betweeninceptionandexecutionthereisalsoacostofcarry.Whatiscostofcarry?
Inception(today) Execution(future)
7. Cost of carry (COC)
โข Various formulae for calculating a forward price.
โข By adjusting the spot price for the costs (add)
and benefits (minus) of waiting for a future
delivery date.
โข These costs and benefits are collectively called
โcost of carryโ and include things like storage cost,
interest charges and the benefit or โconvenienceโ
of having a good in your possession.
โข Storage costs - commodities like oil, corn,
potatoes- need to be stored till delivery-seller.
โข Interest-the seller does not receive the cash till
delivery. Opportunity cost of investment.
8. Cost of carry(contd.)
โข Income paid to holder of the underlier-dividend.
โข Convenience yield-ability to survive a shortage.
Above are benefits which are subtracted from cost
of carry.
โข When the forward price is higher than the spot
price we say that the market is in contango. This
is the usual situation. When the reverse happens
the market is in backwardation.
9. Link between Spot & Forward Price
Today Future
Contango. F > S
Interest,Storage Cost
(increase cost of carry)
Dividend,Convenience yield
(decrease cost of carry)
Backwardation
F < S
Forward
Price (F)
Spot(S)
10. Contango(C ) and Backwardation (B) in energy markets.
โข C โfutures prices are progressively rising as the
delivery time becomes more distant.
โข B- futures prices are progressively lower as delivery
becomes more distant.
โข Energy markets are often in backwardation. Why?
โข While energy markets are often in backwardation
they can convert to contango at short notice.
โข C- is associated with an oversupplied oil market.
High,rising inventories. Symptom not cause.
โข B- is associated with an undersupplied oil market.
Low,falling inventories. Symptom not cause.
11. Convenience yield
โข Arises from the ability to store a commodity and
to sell it at will in the spot market.
โข Owners of commodities get benefits that are not
obtained by owning a contract on the commodity
example-when there is a shortage of commodity
โข Ability to smooth production and to better
coordinate production and sales.
โข Convenience yield is not directly observable, but
can be inferred from the relationship between
spot and futures prices.
12. Convenience yield (contd.)
โข The size of convenience yield shows expectations
about the future.
โข If inventories are high and there is little chance of
shortage , convenience yield will be low.
โข If market participants anticipate shortages the
convenience yield will be high.
13. Relationship between spot & futures prices.
โข Basic argument is that Law of One Price should
hold for spot and futures price of a commodity.
โข The two should differ by an amount that prevents
arbitrage.
โข If futures price is too high , arbitrageur can buy
commodity in the spot market, sell in futures
market, hold it until maturity of the futures
contract, and sell it at the specified date at a
profit. (vice-versa/Short selling)
14. Calculation of forward price-Formulae
S= Spot price
U= Fixed storage costs
I= Fixed Income
r= Risk free interest rate
u= Proportional storage cost
i= Proportional income
y= Convenience yield
rf= Foreign interest rate
t=time in years
1.Basic Forward Price
2.With Fixed Storage
3.With Proportional Storage
F=Se
rt
F=(S+U)e
rt
F=Se
(r+u)t
F=Se
(r-y)t
F=Se
(r-rf)t
F=(S+U-I)e
(r+u-i-y-rf)t
4.With Fixed Income
5.With Proportional Income
6.With Convenience Yield
7.Foreign Currency Forward
All above together
F=(S-I)e
rt
F=Se
(r-i)t
15. F= Se
rt
1.Basic Forward Price
You agree to buy from your brother 100 shares of ABC in 6 months. ABC
does not pay dividends and is now trading for Rs. 15.50. The risk free rate
is 2%. What is the forward price?
16. F= Se
rt
S= 15.50*100= 1550.00
r= 2%
t= 0.5
e= 2.7182
F= Se
rt
1565.58
Per share 15.66
1.Basic Forward Price
You agree to buy from your brother 100 shares of ABC in 6 months. ABC
does not pay dividends and is now trading for Rs. 15.50. The risk free rate
is 2%. What is the forward price?
17. Indostan Energy agrees to buy from Diablo Drillery 5000 barrels of crude
oil one year forward.The oil is selling on the spot market for $32 per
barrel.The cost of storage is $ 6 per annum per barrel payable at the
beginning of the year. The risk-free interest rate is 3% per annum. What is
the correct forward price for the deal?
2.With Fixed Storage F=(S+U)e
rt
18. S=5000*32= 160000
U=5000*6= 30000
r= 3%
t= 1
e= 2.7182
F=(S+U)ert
$ 195,786
Per barrel $39.16
This is an example of a comodity derivative. Underlying is oil.
Indostan Energy agrees to buy from Diablo Drillery 5000 barrels of crude
oil one year forward.The oil is selling on the spot market for $32 per
barrel.The cost of storage is $ 6 per annum per barrel payable at the
beginning of the year. The risk-free interest rate is 3% per annum. What is
the correct forward price for the deal?
2.With Fixed Storage F=(S+U)e
rt
19. (When storage costs change with the spot price of the underlying. Similar to
cost of interest which also depends on the price of the underlying-spoilage)
3.With Proportional Storage F=Se
(r+u)t
Juice maker SqueezeMax agrees to buy 1000 kgsof apples from apple
grower Joe in 3 months.Joe knows from experience that stored apples
lose 2% of their value to insects, rats and rotting.Apples are currently
selling wholesale for $ 6 per kg. The risk free interest rate is 4%.What is
the correct forward price?
20. S=1000*6= 6000
u= 2%
r= 4%
t= 0.25
e= 2.7182
F=Se
(r+u)t
$ 6,090.68
(When storage costs change with the spot price of the underlying. Similar to
cost of interest which also depends on the price of the underlying-spoilage)
3.With Proportional Storage F=Se
(r+u)t
Juice maker SqueezeMax agrees to buy 1000 kgsof apples from apple
grower Joe in 3 months.Joe knows from experience that stored apples
lose 2% of their value to insects, rats and rotting.Apples are currently
selling wholesale for $ 6 per kg. The risk free interest rate is 4%.What is
the correct forward price?
21. 4.With Fixed Income F=(S-I)e
rt
You agree to buy from your sister 400 shares in Acme Industries in 1 year.
The current share price is Rs. 36 per share. Acme is expected to pay an
annual dividend of Rs. 1 per share. The risk free interest rate is 2 percent.
What is the fair market forward price?
22. S=400*36= 14400
I (After 1 year) 400
r= 2%
t= 1
e= 2.7182
I (Present value) 392.08 Ie
-rt
F=(S-I)e
rt
14290.89
Note : In the formula, the present value of I is considered
4.With Fixed Income F=(S-I)e
rt
You agree to buy from your sister 400 shares in Acme Industries in 1 year.
The current share price is Rs. 36 per share. Acme is expected to pay an
annual dividend of Rs. 1 per share. The risk free interest rate is 2 percent.
What is the fair market forward price?
23. You agree to buy from your broker 200 shares of the Nifty in one year. Nifty is
an index of the top 50 companies by market cap and is currently trading at 8000
per share. Nifty is expected to to pay an annual dividend of 0.5% per share.The
risk free interest rate is 3%. What is the fair market forward price?
5.With Proportional Income F=Se
(r-i)t
This might happen when the underling is a stock index like the Sensex (30
stocks) or the Nifty (50 stocks). The dividend from such a ''basket'' is treated as
a percentage of the index spot price.
24. S=8000*200= 1600000
i 0.50%
r= 3%
t= 1
e= 2.7182
F=Se
(r-i)t
1640502.96
You agree to buy from your broker 200 shares of the Nifty in one year. Nifty is
an index of the top 50 companies by market cap and is currently trading at 8000
per share. Nifty is expected to to pay an annual dividend of 0.5% per share.The
risk free interest rate is 3%. What is the fair market forward price?
5.With Proportional Income F=Se
(r-i)t
This might happen when the underling is a stock index like the Sensex (30
stocks) or the Nifty (50 stocks). The dividend from such a ''basket'' is treated as
a percentage of the index spot price.
25. 6.With Convenience Yield F=Se
(r-y)t
Dealing with convenience yield is computationally identical to dealing with
proprtional income. Just substitute the convenience yield for the income rate.
In practice convenience yields are often implied from observed market prices
and not explicitly entered into a formula like the other factors.
26. You are a US citizn and agree to buy 500000 Indian Rupees from a currency
dealer after 9 months. US $ (home currency is currently trading at Rs. 60 per
USD. The risk free interest rate in India is 6% and at home (US) is 2%.
7.Foreign Currency Forward F=Se
(r-rf)t
When the underlier is a foreign currency, we consider not only the domestic
interest rate but also the risk free foreign rate. As we have already seen when
studying IRP, exchange rates are influenced by prevailing interest rates in the
respective countries.
27. S=500000/60= 8333.33
r= 2%
rf= 6%
t= 0.75
e= 2.7182
F=Se
(r-rf)t
8087.05
Fwd Exchange rate 61.83
You are a US citizn and agree to buy 500000 Indian Rupees from a currency
dealer after 9 months. US $ (home currency is currently trading at Rs. 60 per
USD. The risk free interest rate in India is 6% and at home (US) is 2%.
7.Foreign Currency Forward F=Se
(r-rf)t
When the underlier is a foreign currency, we consider not only the domestic
interest rate but also the risk free foreign rate. As we have already seen when
studying IRP, exchange rates are influenced by prevailing interest rates in the
respective countries.
28. Valuation of forwards and futures
โข Value is a measure of how much better of or
worse off the parties (buyer/seller) are after
entering into the forward contract.
โข At inception the value is zero for both parties.
โข Between inception and execution, as the price of
the underlying changes, the value of the contract
changes for both the buyer and the seller.
30. What is the value of an existing forward contract?
It is the difference between the current spot price
(S) and the present value of the delivery price(K).
Forward Value Long = S - PV of (K) = S - Ke-rt
Forward Value Short = PV of (K) - S =Ke
-rt
-S
32. S= 15.50*100= 1550.00
r= 2%
t= 0.5
e= 2.7182
F= Se
rt
1565.58
Per share 15.66
You agree to buy from your brother 100 shares of ABC in 6 months. ABC
does not pay dividends and is now trading for Rs. 15.50. The risk free rate
is 2%. What is the forward price?After three months the market price of
each share is Rs.15.80.What is the value of the forward position?
33. S= 15.80*100 1580.00
K= 1565.58
r= 2%
t= 0.25
e= 2.7182
Ke
-rt
= 1557.77
S - Ke
-rt
= 22.23
Value of forward long position after 3 months
34. โข At maturity the futures price and spot price
are equal. Why?
35. Basis
โข Spot and futures prices differ due to the principle
of cost of carry.
โข At maturity spot and futures prices should be
identical. If not, it will give rise to arbitrage
opportunities. (Futures 10/Spot 8)
โข The difference between the futures price and
spot price is known as basis.
โข Basis = spot/cash price minus futures price.
โข If cost-of-carry pricing relationship exists, basis is
negative.-Contango.
โข Backwardation- futures prices do not fit the
cost-of-carry pricing relationship.
36. Basis Risk
โข Risk that the value of the commodity being
hedged may not change perfectly in tandem with
the value of the derivative instrument being
hedged.
โข Also arises due to mismatch in time of the hedge
(MGRM) and locational mismatch in delivery
point between the derivative contract and the
product being hedged.
37. Rs/$
Cost of carry
Spot
Time
If futues and spot prices are identical at maturity (as they should be)
but differ at the beginning of the period (due to cost of carry), even if the
spot price were were to remain constant (highly unlikely) the futures
price would gradually have to change as the contact approaced maturity .
This is reasonable to expect , because as time passes and maturity is
approached, there will be no cost of carry.(Exception?)
38. Rs/$
13
12.50
12
11.00 Theoretical Futures
10
Spot/Cash
Time
The spot price will not remain constant. More realistically it will change
over the contract period and the futures price will track it.
It is apparent why futures price behaves the way it does. At any given
time, the futures price should be spot plus cost of carry.Therefore if spot
changes , the futures price should change with it, all the while adjusting by
the cost of carry which reduces as maturity approaches.
40. April 11th 2017
a. Cash Silver = 582.5 cents per ounce
b. 8 month interest rate = 10.70% per annum
c. Theoretical December 2017 futures = 624.1 cents per ounce
d. Actual December 2017 futures = 628 cents per ounce
e. Arbitrage activity: SELL FUTURES,BUY SPOT
Activities Cash flows
April 11th 2017: cents
a.Sell December futures @ 628 cents 0
b.Borrow 582.5 cents 582.5
c.Buy cash silver @ 582.5 -582.5
December 11th 2017:
Deliver cash silver against futures 628.00
Pay back loan
Principal -582.5
Interest (582.5 X 0.107 X 8/12) -41.6
Net arbitrage profit 3.9
Cash-and -carry Arbitrage example:
Futures are rich to cash.
41. April 11th 2017
a. Cash Silver = 582.5 cents per ounce
b. 8 month interest rate = 10.70% per annum
c. Theoretical December 2017 futures = 624.1 cents per ounce
d. Actual December 2017 futures = 620 cents per ounce
e. Arbitrage activity: BUY FUTURES,SELL SPOT
Activities Cash flows
April 11th 2017: cents
a.Buy December futures @ 620 cents 0
b.Sell cash @ 582.5 cents 582.5
c.Invest proceeds @ 10.7% -582.5
December 11th 2017:
Take delivery of futures -620.00
Receive from investment
Principal 582.5
Interest (582.5 X 0.107 X 8/12) 41.6
Net arbitrage profit 4.1
Reverse Cash-and -carry Arbitrage example:
Futures are cheap to cash.
