Topic 2 Sample Questions - Part 2.5 - 2.8

APPLICATION B (pages 335 and 341) A stock sells for $50 and pays a continuous dividend yield of 1% per year. Assuming a 5% continuously compounded riskless rate, what is the no-arbitrage price of this forward contract with a time to settlement or delivery of 0.25 years?

$50.5025

What are the five derivatives of an option price with respect to the price of the option's underlying asset:

1. Delta 2. Gamma 3. Vega 4. Theta 5. Rho

APPLICATION C (pages 440 and 445) Consider the following information: Market - expected return = 12%; volatility = 14% Portfolio - expected return = 14%; volatility = 28% Riskless Asset - expected return = 2%; volatility = 0% What is the M2?

8%

What are the one-sided z-scores for confidences levels of 90%, 95%, and 99%?

90% is 1.28 95% is 1.65 99% is 2.33

What are the two-sided z-scores for confidences levels of 90%, 95%, and 99%?

90% is 1.645 (≈ 1.64) 95% is 1.960 99% is 2.576 (≈ 2.58)

APPLICATION A (pages 411-412, 416-417) Let's return to the example of JAC Fund's $1 million holding of the ETF with an expected return of zero. Estimating roughly that the daily standard deviation of the ETF is 1.35%, for a 99% confidence interval, what is the 10-day VaR?

= 2.33 × σ × √Days × Value = 2.33 × 1.35% × √10 × $1,000,000 In order to solve this application we need to use Equation 1. In this case, with a 99% confidence interval the z-score is 2.33. Following Equation 1, we multiply 2.33 by 1.35% by the square root of 10 (the days in the period) to get a product of 9.94%. 9.94% represents the percentage change in the value. To complete this solution we multiply 9.94% by $1,000,000 for an answer of $99,469.44. The z-score is a value that is assumed to be provided rather than being memorized or calculated.

What is a covered call?

A covered call combines being long an asset with being short a call option on the same asset. Note from the diagram that a covered call has the same net risk exposure as a naked put.

What is the put-call parity equation?

A long position in both the call option and the bond, combined with a short position in the put option, will have the same value at the expiration date as a long position in the asset that underlies the options. Equation 1 illustrates one arrangement of put-call parity. There are many ways to rearrange Equation 1.

APPLICATION A (pages 331, 339-340) A stock sells for $100 and is certain not to make any cash distributions in the next year. A forward contract on that stock trades with a settlement in one year. Assuming that the interest rate corresponding to one year is 5% compounded continuously, what is the no-arbitrage price of this forward contract?

A one-year forward contract on the stock must trade at $105.13 using Equation 2. At settlement, a long position in the forward contract obligates the holder to pay $105.13 in exchange for delivery of the stock. If the holder of the forward ccontract places the stock's initial value ($100) in an account (or as collateral) offering at 5% continuously compounded return, the $100 investment will enable the purchase of the stock at the end of the year without further cash.

What is a protective put?

A protective put combines being long an asset with a put option on the same asset. Note from the diagram that a protective put has the same net risk exposure as a call option.

What is a naked option?

A short option position that is unhedged is often referred to as a naked option.

What is an option combination?

An option combination contains both calls and puts on the same underlying asset.

What is a bear spread?

An option combination in which the long option position is at the higher of two strike prices is a bear spread, which offers bearish exposure to the underlying asset that begins at the higher strike price and ends at the lower strike price. Note that bear spreads have long positions in the option with the higher strike price whether the spreads are formed with calls or with puts

What is a bull spread?

An option combination in which the long option position is at the lower of two strike prices is a bull spread, which offers bullish exposure to the underlying asset that begins at the lower strike price and ends at the higher strike price. Note that bull spreads have long positions in the option with the lower strike price whether the spreads are formed with calls or puts.

What is an option spread? What is a calendar spread?

An option spread (1) contains either call options or put options (not both), and (2) contains both long and short positions in options with the same underlying asset. Option spreads contain options that differ with regard to strike price, expiration date, or both. Option spreads based on differences only in expiration date are termed calendar spreads, or horizontal spreads.

What is an option straddle? What is a long straddle?

An option straddle is a position in a call and put with the same sign (i.e., long or short), the same underlying asset, the same expiration date, and the same strike price. When the call and put options are both long, the resulting position is a long straddle which is pictured.

What is an option strangle? What is the difference between a long strangle and a short strangle?

An option strangle is a position in a call and put with the same sign, the same underlying asset, the same expiration date, but different strike prices. When the call and put options are both long, the resulting position is a long strangle; and when the call and put options are both short, the resulting position is a short or strangle. Pictured is a long strangle.

APPLICATION A (pages 267-268) The continuously compounded spot rates corresponding to years 3 and 5 are 6% and 7%, respectively. What is the implied continuously compounded annual interest rate from years 3 to 5?

As it is explained in the textbook, there should be no difference between investing for T years (the longer dated bond) or combining an investment for t years (shorter dated bond) and a forward for T-t years. In this case, the choice is to invest in a 5-year bond or a 3-year bond + 2-year forward contract in year 3 (which gets you to five years). We are given the 5-year (T) and 3-year (t) spot rates, so we must solve for the 2- year (T-t) forward contract. The spot rates are continuously compounded, so we must use Equation 5:

APPLICATION A (pages 300, 304-305) Using the CAPM equation, when the risk-free rate is 2%, the expected return of the market is 10%, and the beta of asset i is 1.25, what is the expected return of asset i?

By placing each of these variables on the right side of Equation 1 and solving the left side, the expected return of asset i is 12%.

What is a risk reversal?

Consider a position similar to the previously discussed synthetic long position (long a call and short a put) but with different strike prices. A long out-of-the-money call combined with a short out-of-the-money put on the same asset and with the same expiration date is termed a risk reversal. Note that the position resembles a synthetic long position except for the level range between the strike prices. Reversing the signs of the option positions (i.e., a short position in a risk reversal) generates a synthetic short position outside of the range between the strike prices.

What are Synthetic Options?

Consider an option combination with a single call option and a single put option with the same underlying asset, the same time to expiration, but opposite signs. If the strike prices of the call and put are the same, the combination is a synthetic position in the underlying asset.

What are diagonal spreads?

Diagonal spreads differ by both expiration date and strike price.

Examples of regulatory constraints that may inhibit competition include?

Examples of regulatory constraints that may inhibit competition include restrictions on short selling and leverage. Fewer regulatory constraints on trading also tend to lead to improved informational market efficiency by expanding competition and trading.

APPLICATION A (pages 276, 289-292) Consider a simplified scenario in which all market interest rates are currently 5%, compounded semiannually. An investor has two bonds in a portfolio: 1. A $1,000 face value one-year, 5% coupon bond (with semiannual coupon payments), and 2. a $2,000 face value, five-year zero-coupon bond. What is the duration of the investor's portfolio?

First, note that the duration of the five-year zero-coupon bond is 5.0. To calculate the duration of the one-year bond, view the bond as a portfolio of two cash flows: a $25 cash flow (coupon) due in six months with a present value (i.e., market value) of $24.39 (found as $25/1.025), and a $1,025 cash flow (principal plus interest) with a value of $975.61 (found as $1,025/1.025/1.025). The duration of the one-year bond is: [($24.39/$1,000) × 0.50] + [($975.61/$1,000) × 1.00] = 0.988. The current value of the portfolio is found by summing the values of the $1,000 coupon bond ($1,000) and the zero-coupon bond [$1,562.40, found as $2,000 × (1.025)²10], for a total value of $2,562.30. The duration of the portfolio is formed as a value-weighted average of the durations of its assets: [($1,000/$2,562.30) × 0.988] + [($1,562.40/$2,562.30)× 5.0] = 3.43 years

Which is more standardized, forward or future contracts?

Forward contracts are ad hoc contracts negotiated between two parties, with flexibility regarding the details to help meet the needs and preferences of each party. As exchange-traded contracts, futures contracts are standardized. Each futures contract trades with a relatively high degree of uniformity with regard to the quantity and quality of the underlying asset and the location and time of delivery.

Which has greater counterparty risk, forwards or futures contracts?

Forwards

APPLICATION E (pages 428 and 436) If a portfolio consistently outperformed its benchmark by 4% per year, but its performance relative to that benchmark typically deviated from that 4% mean with an annualized standard deviation of 10%, then what is its information ratio?

If a portfolio consistently outperformed its benchmark by 4% per year, but its performance relative to that benchmark typically deviated from that 4% mean with an annualized standard deviation of 10%, then its information ratio would be 4%/10%, or 0.40.

What is a Synthetic long position?

If the call option is the long position and the put is short, the result is a synthetic long position in the underlying asset.

What is a Synthetic short position?

If the call option is the short position and the put is long, the result is a synthetic short position in the underlying asset.

The fund's analyst reports a VaR of $100,000 for position #1 and a VaR of $100,000 for position #2. What would be the the VaR of the combined positions if the two positions are identical or have perfectly positive correlated and identical risk exposures? What about if the two positions have statistically independent risk exposures (zero correlation)? What about if the two positions completely hedge each other's risk exposures (perfect negative correlation)?

If the two positions are identical or have perfectly positive correlated and identical risk exposures, then the VaR of the combination is simply the sum of the squared individual VaRs, $200,000. If the two positions have statistically independent risk exposures, then under some assumptions, such as normally distributed outcomes, the VaR of the combination might be the square root of the sum of the squared individual VaRs, or $141,421, which can be derived from the equation for the variance of uncorrelated normally distributed returns and the formula for parametric VaR based on the normal distribution. If the two positions completely hedge each other's risk exposures, then the VaR of the combination would be $0. (Based on the example; 100,000-100,000 I think)

In an informationally efficient market, when does the market price of an equity, such as Tesla Inc., reflect the value of a new technology developed by the firm? Does the stock price rise when the idea for the technology is created, when the idea is made public, when the firm announces an investment to deploy the technology, when the technology is proved reliable, or when the firm begins receiving cash flows from sales based on the technology?

In an informationally efficient market, the answer is that the stock price reflects all potential cash flows (with their attendant probabilities), the moment the information regarding those cash flows is revealed to the marketplace. In such a market, no investor is able to consistently earn superior risk-adjusted returns based on available information, because the information is instantaneously reflected in market prices when it becomes publicly available.

APPLICATION A (pages 325-327) A three-year riskless security trades at a yield of 3.4%, whereas a forward contract on a two-year riskless security that settles in three years trades at a forward rate of 2.4%. Assuming that the rates are continuously compounded, what is the no-arbitrage yield of a five-year riskless security?

Inserting 3.4% as the shorter-term rate in Equation 1 and 2.4% as the left side of Equation 1, the longer-term rate, RT, can be solved as 3.0%, noting that T = 5 and t = 3. Note that earning 3.0% for five years (15%) is equal to the sum of earning 3.4% for three years (10.2%) and 2.4% for two years (4.8%). The rates may be summed due to the assumption of continuous compounding.

APPLICATION B (pages 440, 443-444) Consider a portfolio with M2 = 4%. The portfolio is expected to earn 10%, whereas the riskless rate is only 2%. What is the ratio of the volatility of the market to the volatility of the portfolio?

Inserting the given rates generates 4% = 2% + [(ratio of volatilities) × 8%]. The ratio of the volatility of the market to the volatility of the portfolio must be 25%. Manipulate the M2 formula to come up with this answer.

APPLICATION B (pages 424, 431-432) Based on the information from the previous question, what are the Sharpe ratios based on semiannual returns and quarterly returns? (Ignoring compounding for simplicity, and assuming statistically independent returns through time)

Note that the Sharpe ratio declines from 0.350 to 0.175, which is a 50% decrease, as the time interval for measurement is reduced by 75%, from annual to quarterly. *See the time changes formula on page 423 for more clarification.

APPLICATION B (pages 284, 292-293) Consider a simplified scenario in which an investor has two bonds in a portfolio: 1. $1,000 of market value in a 10-year zero-coupon bond, and 2. $1,000 of market value in a five-year zero-coupon bond. If the investor wishes to be immunized to a horizon point of 7.0 years, what transactions should be executed?

Note that the investor's current duration is 7.5, found as: (0.50 × 5) + (0.50 × 10). To obtain Whe target duration of 7.0 years, she needs to select portfolio weights based on market values (w) for the five-year bond and (1 − w) for the 10-year bond, such that the protfolio's duration is equal to the 7.0 year time horizon: (w×5) + (1−w) × 10 = 7.0 5w = 10−7 w = 0.60 Thus, $200 of the 10-year bond should be sold to lower its weight to ($800/$2,000 = 40%) and used to purchase more of the five-year bond, to bring its weight to $1,200/$2,000 = 60%, so that the duration is 7.0, found as: (0.6 × 5.0) + (0.4 × 10).

What are vertical spreads?

Option spreads differing only by strike price are often referred to as vertical spreads.

Reduced trading frictions include?

Reduced trading frictions include lower transaction costs, such as brokerage fees, exchange fees, regulatory fees, and taxes.

APPLICATION A (pages 247, 252-256) Find the value of a $50 five-year zero-coupon bond for m = 1, 2, 4, 12, 365, and ∞, given an annual interest rate of 9%.

Simply insert $50 in place of $1,9% in place of r, 5 for t, and the given value for m < ∞ in Equation 2 (and analogously in Equation 4) to produce: $32.497, $32.196, $32.041, $31.935, and $31.883 (and $31.881 for continuous compounding). For continuous compounding you can use equation 4 or see ICONV example on page 255

APPLICATION B (pages 252, 256-259) A six-month zero-coupon bond has a price of $97, while a 12-month 7.00% annual coupon bond (paid semiannually) has a price of $100.50. Both bonds have a face value of $100. Find the 12- month spot rate based on annual compounding, semiannual compounding, and continuous compounding.

Solution: all six-month cash flows are worth 97% of their face value, so the coupon bond's first coupon is worth 0.97 * $3.50 = $3.395, leaving the 12-month cash flow to the coupon bond worth $ 100.50 − $3.395 = $97.105, which, based on a face value of $ 103.50 (including the semiannual coupon) means that 12-month cash flows are worth $97.105 / $103.50 = 93.821% of face value. A 12-month discount factor of 0.93821 implies a 6.59% annually compounded yield, a 6.48% semiannually compounded yield, and a 6.38% continuously compounded yield.

APPLICATION C (pages 287, 293-294) An investor has a $1,000,000 portfolio with long positions that form a duration of 5.0 years. The investor wishes to consider two alternatives: adding $1,000,000 in short positions to hedge the portfolio or adding $500,000 in short positions to hedge the portfolio. What securities would provide immunization under the two scenarios?

Solution: the $1,000,000 long position with a duration of 5.0 years can be hedged with $1,000,000 in short positions if the short positions have a duration of −5.0 years (i.e., short $1,000,000 of five-year zero-coupon bonds or other assets that would have a positive duration of 5.0 if held long). The negative position implicit in the short position will offset the positive duration exposure of the long position for an infinitesimal, parallel, and instantaneous shift in interest rates. In order to form a hedge with only $500,000 of short positions, the positions would have to have a duration of −10.0 years, such as having $500,000 of market value short sold in 10-year zero-coupon bonds. The proceeds of the short sales should be held in cash to avoid introducing further interest rate risk.

Ratio spreads

Spread positions termed ratio spreads can be formed in which the number of options in each position differs. For example, a ratio spread might contain two long call positions at one strike price and one short call position at another strike price, both with the same underlying asset. Ratio spreads tilt the option exposures to provide greater sensitivity (i.e., leverage) in one direction (e.g., bullish) than in the other. Spread ratios serve as an illustration of using greater degrees of leverage through establishing relatively large directional bets.

APPLICATION A (pages 318, 320-321) Nine-month riskless securities trade for $97,000, and 12-month riskless securities sell for $P (both with $100,000 face values and zero coupons). A forward contract on a three-month, riskless, zero-coupon bond, with a $100,000 face value and a delivery of nine months, trades at $99,000. What is the arbitrage-free price of the 12-month zero-coupon security (i.e., P)?

The 12-month bond offers a ratio of terminal wealth to investment of ($100,000/P). The nine month bond reinvested for three months using the forward contract offers ($100,000/$97,000)($100,000/$99,000). Setting the two returns equal and solving for P generates P = $96,030. The 12-month bond must sell for $96,030 to prevent arbitrage.

Consider a three-month FRA (to be settled in several years) with an FRA rate of 5% and a notional value of $1,000,000. At the time of settlement the actual market interest rate (LIBOR) rises to 6%. Does the buyer or seller have to pay? How much cash is exchanged?

The FRA would require the FRA seller to pay the buyer $2,500 because LIBOR, the reference rate, is above the FRA, found as follows: [$1,000,000 × (3 mos./12 mos.) × (6%-5%)]. Had the reference rate fallen to 4%, the buyer would pay the seller $2,500. In practice, the cash settlement amount is usually based on a discounted value (i.e., the $2,500 would be discounted for the length of the loan—three months in this example) and would be payable at the start of the period when the market rate is observed. The example illustrates the important ability of an FRA to allow financial entities to control their borrowing costs and lending revenues.

APPLICATION A (pages 442-443, 429-430) Consider a portfolio that earns 10% per year and has an annual standard deviation of 20% when the risk-free rate is 3%. What is the Sharpe ratio?

The Sharpe ratio is (10% − 3%)/20%, or 0.35. When using annual returns and an annual standard deviation of returns, the Sharpe ratio may be interpreted as the annual risk premium that the investment earned per percentage point in annual standard deviation. In this case, the investment's return exceeded the riskless rate by 35 basis points for each percentage point in standard deviation. In an analysis of past data, the mean return of the portfolio is used as an estimate of its expected return, and the historical standard deviation of the sample is used as an estimate of the asset's true risk. Throughout the remainder of this analysis of performance measures, the analysis may be viewed as interchangeable between using historical estimates and using expectations.

APPLICATION D (pages 427, 434-435) Consider a protfilo that earns 10% per year when the investor's target rate of return in 8% per year. The semistandard deviation based on returns relative to the target is 16% annualized. What is the Sortino ratio?

The Sortino ratio would be (10% − 8%)/16%, or 0.125.

APPLICATION C (pages 426, 433-434) Consider a portfolio that earns 10% per year and has a beta with respect to the market portfolio of 1.5 when the risk-free rate is 3%. What is the Treynor ratio?

The Treynor ratio is (10% − 3%)/1.5, or 0.0467 (4.67%). The Treynor ratio may be interpreted as the risk premium that the investment earns per unit of beta. In this example, the investment's expected return is 4.67% higher then the riskless rate for each unit of beta.

APPLICATION A (pages 438, 441-442) A portfolio is expected to earn 7% annualized return when the riskless rate is 4% and the expected return of the market is 8%. If the beta of the portfolio is 0.5, what is the alpha of the portfolio?

The alpha of the portfolio is 1%, found by substituting into Equation 1 and solving:

In an FRA, if the reference rate falls below the FRA rate, does the buyer or seller have to pay the difference?

The buyer would pay the seller.

APPLICATION B (pages 303, 305-306) Returning to the previous example in which the risk-free rate is 2% and the beta of asset i is 1.25, if the actual return of the market is 22%, what would the ex post CAPM return for the asset be? If the asset actually returned 30%, how much would be attributable to idiosyncratic return, εit?

The ex post CAPM model would generate a return due to non-idiosyncratic effects of 27% for the asset: 2% + [1.25(22% − 2%)]. If the asset's actual return is 30%, then the extra 3% would be attributable to idiosyncratic return, εit.

APPLICATION B (pages 369 and 373) To lock in sales prices for its anticipated production, HiHo Silver Mining Company wishes to take short positions in five silver futures contracts, settling in each quarter for the next four quarters (20 contracts total). If the initial margin requirement is $11,000 per contract, what is the firm's total initial margin requirement?

The firm must have $220,000 of available collateral to establish the positions. The initial margin requirement is $11,000 per contract. HiHo Silver Mining Company is purchasing 20 futures contracts. The product of 20 by $11,000 is $220,000, which is the collateral needed to establish the position.

APPLICATION A (pages 364, 371-372) Futures contracts on crude oil are often denominated in 1,000-barrel sizes. In other words, each contract calls for the holder of a short position at the delivery date of the futures contract to deliver to the long side 1,000 barrels of the specified grade of oil using stated delivery methods. Assume that a trader establishes a long position of five contracts in crude oil futures at the then-current futures market price of $100 per barrel. Both the trader on the long side of the contract and the trader on the short side of the contract post collateral (margin) of, say, $10 per barrel. At the end of the day, the market price of the futures contract falls to $99. How much money will each side of the contract have (assuming that the required collateral was the only cash and that there were no other positions)?

The five contracts call for delivery of 5,000 barrels (five contracts x 1,000 barrels). The long side of the contract loses $5,000 as a result of the decline in price of $1 per barrel. Each side posted collateral of $50,000 (5,000 barrels x $10 per barrel). The long side experiences a decline in collateral position (cash) to $45,000, and the short side experiences an increase in collateral position (cash) to $55,000.

APPLICATION C (pages 337, 342-343) If the spot price of an equity index that pays a dividend yields equal to the riskless rate is $500, what is the one-year forward price on the equity index?

The forward contract of every time to delivery has a forward price of exactly $500. Market participants would be indifferent between buying and selling the index in the spot market with instant delivery or in the forward market with delayed delivery because the interest payments and dividends offset each other.

APPLICATION A (pages 352, 357-358) Consider a six-month forward contract on a commodity that trades at a spot price of $50. The commodity has market-wide convenience yields of 3%, storage costs of 2%, and financing costs (interest rates) of 7%. What is the price of the six-month forward contract on the commodity?

The forward price is $51.52, found by placing 0.5(7% + 2% − 3%) in as the exponent of Equation 1, $50 as Po, and solving for FT.

Consider the stock with no dividend in Application A (with a riskless rate of 5%). Let's change the dividend assumption to having the stock distribute a continuous dividend at an annual rate of 3% (i.e., the stock's annual dividend yield). What is the forward price?

The forward price would be $102.02 using Equation 4.

APPLICATION A (pages 276 and 278) A stock currently selling for $10 will either rise to $30 or fall to $0 in three months. How much would a three-month call sell for if its strike price were $20?

The key piece to understand in this application is that the payoff of the call ($10 or $30 − $20) is 1/3 ($10/$30) of the ending stock price of $30. To solve this application, we need to find the payoff of the call by subtracting the future stock price of $30 by the strike price of $20 for a difference of $10. Then divide $10 (pay off the call) by $30 (future stock price), which equals 1/3. Then multiply the 1/3 by $10 (current stock price) for a product of $3.33, which is the price of the call.

APPLICATION C (pages 370, 374-375) Returning to the previous example of an oil trader with a long position of five contracts established at an initial futures price of $100 per barrel, the five contracts call for delivery of 5,000 barrels (five contracts x 1,000 barrels). The trader posts exactly the required initial margin of $50,000 ($10,000 per contract). Suppose that the maintenance margin requirement is $25,000 ($5,000 per contract) and that the price of oil drops $6 per barrel. What is the trader's margin balance after the price decline? Also, describe any margin call that might be made and what it would require.

The long side of the contract loses $6,000 per contract ($30,000) as a result of the decline in price of $6 per barrel. The initial collateral of $50,000 falls to a remaining margin balance of $20,000 ($4,000 per contract). The trader receives a margin call, since the remaining margin is less than the maintenance margin requirement. The amount of the margin call is $30,000 to bring the margin back to the initial margin requirement.

APPLICATION E (pages 338, 346-347) Assuming a continuously compounded annual interest rate of 2%, if the spot price of an equity index with 3% continuously paid dividends is $500, what would be the forward price of a contract with settlement in three months? Six months? 12 months?

The price of every forward contract of every time to delivery would be $500e^(−0.01)T, with (r−q) = − 1%. The three-month forward price would be $500e − 0.01 × 0.25, or $498.75. Six-month and 12-month forward prices would be $497.51 and $495.02, respectively (found by inserting 0.50 and 1.00 for T).

APPLICATION D (pages 338, 344-345) Assuming a continuously compounded annual interest rate of 5%, if the spot price of an equity index with 2% continuously paid dividends is $500, what would be the forward price on the equity index with settlement in three months? Six months? 12 months?

The price of every forward contract on that index for every time to settlement would be $500e(0.05−0.02)T. The three-month forward price would be $500e(0.03 × 0.25), or $503.76. Six-month and 12-month forward prices would be $507.56 and $515.23, respectively (found by inserting 0.50 and 1.00 for T, and 0.03 for r−q).

What are the five variables that determine the price of an option on a non-dividend stock according to the Black-Scholes option pricing model?

The price of the underlying asset The strike price The return volatility of the underlying asset The time to the option's expiration The risk-less rate

APPLICATION B (pages 357, 359-360) Consider a forward contract with three-month delivery on a commodity that currently trades at a spot price of $110. The commodity has a current marketwide convenience yield of 2%, storage costs of 4%, and financing costs (interest rates) of 6%. If the forward price in the contract is $101.00, what is the value of the contract to the long side assuming that the underlying commodity can be readily short sold?

The value of the contract to the long side is found using Equation 2, the forward price of $101.00, and using 0.25(6% + 4% − 2%) as the exponent of Equation 2. Noting that the time remaining (T − t) is 0.25 and using $110 as P0 solves for the contract's value, $11.22, to the long side of the contract (and a value of −$11.22 to the short side).

APPLICATION D (pages 287-288, 295-296) An investor has a $1,000,000 portfolio with long positions that form a duration of 5.0 years. The investor's goal for the protfolio is to have a duration of 4.0 years because the protfolio is to be liquidated at that time to fund a project. The investor wishes to add short positions to hedge the portfolio to a duration of 4.0 years.

There are many solutions to this problem. For example, the investor could short $200,000 of five-year zero-coupon bonds and hold the proceeds from the short sale in cash. Any combination of dollar amount V and duration D that solves the following equation would lower the duration to 4.0 years: ($1,000,000×5.0) − (V×D) = $4,000,000 V×D = ($1,000,000×5.0) − $4,000,000 = $1,000,000

What is a short straddle?

When the call and put options are both short, the resulting position is a short straddle