Valuation of a forward contract for different scenarios: Please attempt

Given Information:

A portfolio manager owns XYZ Inc. which is currently trading a $35 per share.

He plans to sell the stock in 120 days but is concerned about a possible price decline.

He decides to take a short position in a 120 day FC (Forward Contract) on the stock.

The stock will pay a $0.50 per share div. in 35 days & $0.50 again in 125 days.

The risk free rate is 4%.

What is the value of the trader's position in the FC in 45 days, assuming the strike price is $27.50?

The answer for the above question was discussed in my previous post. Now the question is amended for different situations as expressed below:

Scenario 1:

A portfolio manager owns XYZ Inc. which is currently trading a $35 per share.

He plans to sell the stock in 120 days but is concerned about a possible price decline.

He decides to take a short position in a 120 day FC on the stock.

The stock will pay a $0.50 per share div. on 35^th day & another $0.50 again on 75^th day.

The risk free rate is 4%.

What is the value of the trader's position in the FC in 75 days, assuming the strike price is $27.50?

NOW: Scenario (2):

A portfolio manager owns XYZ Inc. which is currently trading a $35 per share.

He plans to sell the stock in 120 days but is concerned about a possible price decline.

He decides to take a short position in a 120 day FC on the stock.

The stock will pay a $0.50 per share div. on 35^th day & another $0.50 again on 75^th day.

The risk free rate is 4%.

What is the value of the trader's position in the FC at the end of FC agreement period (on 120^th day), assuming the strike price is $27.50?

NOW: Scenario (3):

A portfolio manager owns XYZ Inc. which is currently trading a $35 per share.

He plans to sell the stock in 120 days but is concerned about a possible price decline.

He decides to take a short position in a 120 day FC on the stock.

The stock will pay a $0.50 per share div. on 35^th day & another $0.50 again on 75^th day.

The risk free rate is 4%.

What is the value of the trader's position in the FC at the start of FC agreement period (on 1^st  day), assuming the strike price is $27.50?

ONE OF BEST QUESTIONS ON FORWARD CONTRACT VALUATION...

Given Information:

A portfolio manager owns the stock of XYZ Inc. which is currently trading a $35 per share.

He plans to sell the stock in 120 days but is concerned about a possible price decline.

He decides to take a short position in a 120 day Forward Contract (FC) on the stock.

The stock will pay a $0.50 per share div. in 35 days & $0.50 again in 125 days.

The risk free rate is 4%.

What is the value of the trader's position in the FC in 45 days, assuming the stock price in 45 days is $27.50?

In my blog, so far we have not covered valuation: But this question may be used for understanding the concepts on valuation of forward contracts. Because I have covered in my blog only pricing of forward contracts, but not valuation part. 

Solution: The value of a forward contract is given by:

(S₀ e^rt – K) ----Formula (1)

or there is one more formula:

(S₀ – K) x e^-rt   ------Formula (2)

You do not have to mug up the formula, if you understand the logic:

In formula (1), we find out the value of a forward contract on the end of the contract or after certain number of days.

In formula (2), please understand the forward contract value is calculated at time T₀; or in today.

You may ask me a question, how did you understand this point?

When we calculate anything in today’s terms, we need to bring down its value from future value to present value. This is done by e^-rt

We find the value of any agreement on the end of the agreement day or after certain days, we calculate its future value by e^rt

So far, in my blog, I have not discussed about valuation of contracts.

The value of a forward contract is equal to: (price of the forward contract – Strike price)

We know after 45 days, the strike price of the contract = $27.50

But we have to calculate the price of the forward contract after 45 days;

Step-1: first calculate the price of the forward contract after 45 days: $35 x e^{0.04 x (45/360)}

That is equal to: $35.17

Step-2: Now, you reduce the value of dividend yield by 45^th day: The first dividend is received in 35days. Its value has to be calculated as on 45^th day. So, even dividend value needs to be shifted from 35^th day to 45^th day.

It’s value as on 45^th day: $0.50 x e^0.04(10/360) = $0.50

Step 3: The price of the forward contract is:

(Price of the contract – dividend income or yield)

$35.17 - $0.50 = $34.67

Step 4:

The value of the forward contract on 45^th day is equal to:

(The price of forward contract on 45^th day minus the strike price on 45^th day)

Therefore, the value of forward contract on 45^th day: $34.67-$27.50 = $7.17

Quiz continued on Forward and Futures Pricing - 2

Proudly I can say that i do not have a habit of copying and pasting the questions from anywhere. I have framed the questions, based on the formulae used in core readings (Hull chapter -5) and (McDonald chapter) on forwards and futures pricing.

Before attempting these questions, please consider that the risk-free rate of interest is quoted in continuous terms and it is quoted per annum.

Understand the difference between each formula and how it changes and why it changes with change of scenarios and circumstances.

A trader who is already holding an asset, entered into a short position in futures contract with an agreed period of 3 months. The opportunity cost to the trader is equal to the risk-free rate of interest of 5.5% (annual terms). The cost of acquisition to the holder is $40. What should be the fair price quoted in the futures market? Solution: S0 x ert 40 x ert 40 x e0.055 x 0.25 40 x 1.013845 $40.55

A trader who is already holding an asset, entered into a short position in futures contract with an agreed period of 6 months. The risk-free rate of interest is 5.5% per annum. The asset was acquired for a price of $80. The trader who is holding the asset is expecting $2 as income on the asset on the first day of agreement. Calculate the fair value of the futures price quoted in the market? Solution: (S0 –I) e(rt) ($80-$2)xe(rt) $78 x e0.055x0.5 $80.175 Here the income of $2 is received on the first day of initiation of the contract. So there is no need to convert it again into present value. This is where a student has to be cautious.

A trader who is already holding an asset, entered into a short position in futures contract with an agreed period of 6 months. The risk-free rate of interest is 5.5% per annum. The asset was acquired for a price of $80. The trader who is holding the asset is expecting $4 as income on the asset on the last day of agreement. Calculate the fair value of the futures price quoted in the market? Solution: The income is expected at the end of the maturity. So we need to calculate its present value at the initiation of the contract. Please understand that the value of the asset is $80 on the first day of the contract. So find the value of $4 (income value also on the first day). For which, we need to calculate its present value using risk-free rate. The present value of $4 is: $4 x e-rt; $4 x e-0.055x0.5; Its value is $3.89. Now, use the formula: (S0 –I) e(rt) ($80-3.89) e(rt) $76.11 x e(0.055x0.5) $78.23

A trader who is already holding an asset, entered into a short position in futures contract with an agreed period of 6 months. The risk-free rate of interest is 5.5% per annum. The asset was acquired for a price of $80. The trader who is holding the asset is expecting $4 as income in next 3 months. Calculate the fair value of the futures price quoted in the market? Solution: The income is expected at the end of the maturity. So we need to calculate its present value at the initiation of the contract. Please understand that the value of the asset is $80 on the first day of the contract. So find the value of $4 (income value also on the first day). For which, we need to calculate its present value using risk-free rate. The present value of $4 is: $4 x e-rt; $4 x e-0.055x0.25; Its value is $3.95 Now, use the formula: (S0 –I) e(rt) ($80-3.95) e(rt) $76.05 x e(0.055x0.5) $78.17

A trader who is already holding an asset, entered into a short position in futures contract with an agreed period of 6 months. The risk-free rate of interest is 5.5% per annum. The asset was acquired for a price of $80. The trader who is holding the asset is expecting 3% as income yield on the asset during the period of holding. Calculate the fair price of the contract? Solution: Here the income is given in the form of yield. The income can either be expressed in dollar terms or also in % terms. When the income is expressed in dollar terms, we use: (S0 –I) e(rt) (Please S0 is at Time-T0 and therefore even income (I) should also be calculated at T0. When the income is expressed in yield or % terms, we use: S0 x e(r-q)T; Here the income while holding the asset is expressed in % terms. We use: S0 x e(r-q)T . This gives us: $80 x e(0.055-0.03)0.5 = $81.00;

Please do not go to the solution directly. First understand yourself, where you are struggling. Do not mug up the formulae. Understand the logic behind that formula.

Please attempt the remaining questions in the next two days. With that we will be able to complete forward and futures pricing.

This is a query posted in Risk Management Forum in my FB Account

I request all the FRM- 1 aspirants to understand the concepts in core readings. Conceptual understanding makes our life and exam also easy. I will discuss the discrete and continuous rates once I complete the quiz on forward and futures prices.

Please look into the question:

1 – Year zero coupon bond is trading at a price of $95.2381 and the YTM is 5%.

2 – Year Treasury note with 6% coupon is trading at a YTM of 5.5% and its price is $100.9232

3-Year Treasury note with 7% coupon is trading at a YTM of 6% (Price is $102.6730)

Question: Assume annual coupon payment and discrete compounding. Use a bootstrapping method to determine 2Year and 3 Year spot rates

Explanation:

According to John C Hull (Chapter 4), we use already available zero rates of zero coupon bonds to obtain the zero rates of coupon bonds.

Why the question is saying to use discrete? It is quite simple. First to obtain zero rates, generally we first obtain discrete rates and those discrete rates are converted to continuous rates. In this case, no need to convert to continuous rates, as they have asked us to calculate the zero rates using discrete method.

Calculating the zero rate for two years: Use the zero rate available for 1 year to calculate the zero rate of 2-year. The two year T-note is paying a coupon of 6% and its price is $100.9232 ð  (6/1.05) + 106/(1+r)2 = 100.9232 ð  5.7143 + 106/(1+r)2 = 100.9232 ð  [106/(1+r)2] = 95.2089 ð  95.2089 x (1+r)2 = 106 ð  (1+r)2 = 106/95.2089 ð  (1+r)2 = 1.11334 ð  (1+r) = 1.05515 ð  r = 5.515% ð  This is the zero rate of two years

Calculating the zero rate for three years period Use the zero rates available for one year and two years to obtain the zero rate for three year period. The coupon rate of T-note of 3years is 7% and its price is $102.6730; ð  ($7/1.05) + ($7/1.055152) + $107/(1+r)3 = $102.6730 ð  $6.67+$6.2874+$107/(1+r)3 = $102.6730 ð  107/(1+r)3 = 89.7156 ð  89.7156 x (1+r)3 = 107 ð  (1+r)3 = 1.192658 ð  (1+r) = (1.192658)1/3 ð  (1+r) = (1.192658)0.3333 ð  (1+r) = 1.0605 ð  r = 6.05%

Zero rates: 1 year-5%, 2 year- 5.515% and for three years-6.05%

Quiz-1 (Forward pricing)

First of all, sorry to all of you, for the delayed new posting! But I will try to post them regularly.

Which of the following options is most accurate for measuring the forward price of a commodity or an investment asset, assuming that the investment value of the asset grows continuously at risk-free rate of interest?
a.      S₀ x e^rt
b.      S₀ + e^rt
c.       S₀ / e^rt
d.      S₀ - e^rt

Assuming that that the value of investment asset grows at discrete rate of interest, which of the following formulae correctly explain relationship between the current spot price and forward price of an asset?
a.      S₀ (1+r)^t
b.      S₀ +(1+r)^t
c.       S₀ - (1+r)^t
d.      S₀ /(1+r)^t

A trader invested $520 in an investment asset, with an intention to sell it after 6 months. If the risk-free rate of interest is 4.5% per annum and the value of the asset grows continuously, the forward price of the asset is equal to:

a.      530.70
b.      531.83
c.       531.95
d.      532.05

A trader invested $520 in an investment asset, with an intention to sell it after 6 months. If the risk-free rate of interest is 4.5% per annum and the value of the asset grows on discrete basis, the forward price of the asset is equal to:

a.      530.70
b.      531.83
c.       531.95
d.      532.05

Assume that current spot price of an investment asset is $60 and risk-free rate of interest is 4.0% per annum. The forward price of an investment asset, growing at risk-free rate of interest (continuously) is calculated at time- T_{0 ,}  by the following theoretical forward price formula:

F₀ = S₀ x e^rt

When the traders identify that the above formula does not hold good, they attempt for arbitrage opportunities. If the market quoted forward price does not match the theoretical forward price, then there exists an arbitrage opportunity.

Which of the following statements about the arbitrage opportunities is correct, when the market determined forward price is higher than the theoretical forward price for a maturity period of 6 months?

Answer:

a. Borrow at risk-free rate, buy asset, sell forward

b. Lend at risk-free rate, buy asset, sell forward

c. Borrow at risk-free rate, sell asset and buy forward

d. Lend at risk-free rate, sell asset and buy forward

Frequently asked conceptual queries...

What is lease rate in commodity markets?

That is the rate charged by the lender of commodity, for lending the commodity. In the above case, (discussed in my previous blog) the lease rate is 6%.

What is E₀(S_T)?

We assume that a commodity will reach certain price in future. The assumption or expectation is made today. That is what E₀ refers.  That is the price of the commodity at time T (in future)

What is F_(0,T)?

That is theoretical forward price of the commodity.

Can there be a difference between F_(0,T) and E₀(S_T)?

Yes, there can be.  E₀(S_T) is based on expectations and F_(0,T) is based on mathematical calculation formula of : S₀ x e^rT

What will happen when there is a difference between F_(0,T) and E₀(S_T)?

There might be an arbitrage opportunity, when the theoretical prices deviate from the expected prices. Traders generally generate arbitrage opportunities due to this difference.

Can we discuss two examples of simple arbitrages, due to the difference between F_(0,T) and E₀(S_T)?

Yes, let us discuss these two examples to understand two arbitrages due to the difference between F_(0,T) and E₀(S_T).

S0 (Spot price)	$5.00
Price quotation	Cents per bushel
Contractual commodity	Corn
Time to maturity of the contract	1 year
Risk-free rate	6%
Theoretical forward price formula	S0 x erT
Theoretical forward price	$5.00 x e0.06 x 1 year = $5.31
What the trader can expect?	The trader expects that the market prices will deviate from theoretical prices (due to many reasons) and the trader can expect the prices after one year be either greater than $5.31 or lower than $5.31

Arbitrage example one: When the trader expects the realistic prices do not match with the theoretical prices and realistic prices will be higher than $5.31; Let us say the trader identified another trader who is willing to buy at $5.35.

Some steps are initiated at Time T0	Some steps are initiated at Time TT (on maturity of contract)
Borrow $5.00 @ risk-free rate	Sell  the commodity at $5.35
Buy the commodity in the spot market for $5.00	Repay the loan with interest: -$5.31
Enter into a forward contract to sell at $5.35 (Short forward contract) after one year.	Risk-free profit  due to arbitrage is: $5.35 - $5.31 = $0.04 per bushel
The arbitrage ceases to exist, when either the theoretical prices increase or when the market prices come down. When all the traders start doing the same activity, the price would come down to $5.31 Or, when the risk-free interest increases, the forward price may go up and arbitrage ceases to exist. This is a cash and carry arbitrage (borrow money, buy the asset and sell forward)

Arbitrage example two: When the trader expects the realistic prices do not match with the theoretical prices and realistic prices will be less than $5.31; Let us say the trader identified another trader who is willing to sell at $5.25 after one year.

Some steps are initiated at Time T0	Some steps are initiated at Time TT (on maturity of contract)
Sell the asset today in the cash market at the price of S0	At time T0: We get $5.00 by selling the commodity in cash market.
Invest the proceeds at the risk-free rate of 6%	Invest at 6%; After one year, we will get $5.31 from the investment.
Buy the asset in future through long forward contract at a price of $5.25	Buy the asset $5.25 in the forward market. Create an arbitrage opportunity of $0.06 per bushel.
This is an inverse cash and carry arbitrage: Sell today, invest the proceeds; buy in future or long forward