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### Subject 4. Protective Puts

A portfolio of stock has a potentially wide range of gains and losses. If all the stocks in the portfolio lost all of their value, the value of the portfolio would also lose all of its value. In other words, it would be possible to lose everything that had been invested. On the other hand, as the stocks in the portfolio increase in value, the value of the portfolio increases. Since the value of a share of stock has (theoretically) no upper limit, the value of a portfolio has no upper limit.

A protective put (portfolio insurance) is an investment management technique designed to protect a stock portfolio from severe drops in value. It involves holding a stock portfolio and buying a put option on the portfolio. At expiration:

• Maximum profit = ST - S0 - p0 = unlimited
• Maximum loss = S0 - X + p0
• Breakeven profit = S0 + p0
• Expiration value = Max(ST, X)
• Profit at expiration = Max(ST, X) - S0 - p0

Portfolio insurance limits the amount of loss on a portfolio by balancing that loss with the gain from a LONG put option. It also reduces the potential gain on a portfolio as a result of the premium paid for the put option. The "insurance" part of portfolio insurance, then, is the limitation of potential loss. The insurance is not free, however. The cost of the insurance is the put option premium.

This strategy is like a long position in a call. A trader can buy a call and invest the extra proceeds in a bond in order to replicate a position in an insured portfolio.

Equivalence to Long Asset/Short Forward Position

As discussed in reading 41, delta measures how the option price changes as the underlying asset price changes.

• Call deltas: 0 to 1
• Put deltas: 0 to -1
• The delta of the underlying: 1
• The delta of an at-the-money option: 0.5 for call and -0.5 for put

That reading shows that three different positions have the same delta: an at-the-money covered call, an at-the-money protective put, and a long stock/short forward position.

#### Practice Question 1

In the protective put and insurance analogies, the insurance premium is similar to the ______ in a put option.

A. time value
B. price of stock
C. exercise price

#### Practice Question 2

An at-the-money protective put position (comprised of owning the stock and the put) ______.

I. protects against loss at any stock price below the strike price of the put
II. has limited profit potential when the stock price rises
III. returns any increase in the stock's value, dollar for dollar, less the cost of the put
IV. provides a pattern of returns similar to a stop-loss order at the current stock price

#### Practice Question 3

An exporter in the U.S. is expecting a payment of BP 5 million in three months. He is planning to hedge the position using options. He should ______.

C. write put options

#### Practice Question 4

A portfolio insurance strategy consists of a ______.

A. LONG stock portfolio and SHORT stock index put
B. LONG stock portfolio and LONG stock index put
C. SHORT stock portfolio and LONG stock index put

#### Practice Question 5

In the protective put and insurance analogies, the face value of an insurance policy is similar to the ______ in a put option.

A. time value
B. price of stock
C. exercise price

#### Practice Question 6

An investor would likely buy a protective put under which of the following conditions?

A. The investor is concerned about downward price movement in the stock.
B. The investor wishes to leverage an anticipated rise in the stock price.
C. The investor does not anticipate a significant change in stock price in the near term.

An investor would likely buy a protective put (portfolio insurance) if he/she were concerned about a downward price movement in the stock.

#### Practice Question 7

An investor owns a stock portfolio that closely follows the Standard & Poor's 500 Index (S&P 500). He purchases one S&P 500 stock index put option. The investor's position now has portfolio insurance with the following characteristics:

Portfolio position: LONG S&P 500
Portfolio purchase price: \$1427.21
Option position: LONG 1 put option
Underlying asset: S&P 500 Index
Exercise price: \$1225
Expiration date: November 30

The expiration-day price of the S&P 500 is ST = \$1500. The expiration-day profit/loss for the portfolio insurance is ______.

A. + 72.79 + 0 - 3 = \$69.79
B. + 72.79 + 0 + 3 = \$75.79
C. - 72.79 + 275 - 3 = \$199.21

Profit at expiration = Max(ST, X) - S0 - p0 = 1500 - 1427.21 - 3.00 = \$69.79

#### Practice Question 8

An investor owns a stock portfolio that closely follows the Standard & Poor's 500 Index (S&P 500). He purchases one S&P 500 stock index put option. The investor's position now has portfolio insurance with the following characteristics:

Portfolio position: LONG S&P 500
Portfolio purchase price: \$1427.21
Option position: LONG 1 put option
Underlying asset: S&P 500 Index
Exercise price: \$1225
Expiration date: November 30

The expiration-day price of the S&P 500 is ST = \$1500. The breakeven point for the portfolio insurance is ______.

A. 1427.21 + 3 = \$1430.21
B. 1500 - 3 = \$1497
C. 1225 + 3 = \$1228

Breakeven point = S0 + p0 = 1427.21 + 3 = \$1430.21

#### Practice Question 9

Suppose the return distribution of a stock is a bell-shaped curve. Buying a put option does which of the following?

A. Truncates the left side of the curve and moves the whole bell curve to the right.
B. Truncates the left side of the curve and moves the whole bell curve to the left.
C. Truncates the right side of the curve and moves the whole bell curve to the right.
D. Truncates the right side of the curve and moves the whole bell curve to the left.

#### Practice Question 10

In an insurance policy, the deductible is the amount of loss the insured is willing to bear. In a put option, the deductible is the ______.

B. exercise price minus the option premium
C. stock price minus the exercise price

In general, the higher the deductible, the lower the cost of insurance (option premium).

#### Practice Question 11

Suppose an investor buys one share of stock and a put option on the stock. What will be the value of her investment on the final exercise date if the stock price is below the exercise price?

A. The value of two shares of stock
B. The value of one share of stock minus the exercise price
C. The exercise price.

#### Practice Question 12

An at-the-money call option will have a delta that is about ______.

A. -0.5
B. 0.5
C. 1

#### Practice Question 13

Which one has a delta of 1.0?

A. A long in-the-money call
B. A long out-of-money put
C. A long position in the underlying asset

#### Practice Question 14

The likelihood of loss with an insurance policy is similar to the ______ in a stock put option.

A. time value
B. volatility of stock
C. time until option expiration

#### Practice Question 15

In an insurance policy, the deductible is the amount of loss the insured is willing to bear. A stock is selling at \$21.85. A one-month \$20 strike put is selling for \$1.38. The deductible of the put is ______.

A. \$1.38
B. \$1.85
C. \$0.47

The deductible is the stock price minus the exercise price.

#### Practice Question 16

An investor buys 100 baskets of the Standard and Poor's 500 stock index for \$1,400 each. She insures the portfolio by buying put options. What is the effect of the put options?

A. The upside risk of the insured portfolio is truncated and the downside risk is reduced relative to the uninsured portfolio.
B. The downside risk of the insured portfolio is truncated and the upside return of the insured portfolio is lower relative to the uninsured portfolio.
C. The insured portfolio does better than the uninsured portfolio whenever the market declines.

This truncates the left-hand side of the return distribution.

#### Practice Question 17

The profits and losses from an equity portfolio insured using puts would have the same profit and loss characteristics as they would from ______.

A. being long a call option
B. writing covered puts
C. writing covered calls

Being long a call has limited downside risk, as does portfolio insurance.

#### Practice Question 18

Your research department has just made a sound presentation arguing that the equity markets are due for a severe correction in the short term. Which of the following strategies would be most suitable to safeguard the portfolio that you manage?

A. Long futures contracts on an equity index
B. Buy put options on an equity index
C. Buy call options on an equity index

If you expect stock prices to decline, then you'll want to enter into an agreement that will allow you to sell shares at a pre-determined price. This way, you are effectively locking in the future value of the shares. There are only two strategies that will enable you to carry out this transaction. First, buying a put will give you the "right" to deliver these shares at a pre-determined price. Second, selling stock index futures will "obligate" you to deliver these shares at a pre-determined price.

#### Practice Question 19

Which of the following statements is (are) true with respect to insuring a portfolio by way of a protective strategy?

I. The strategy requires the sale of put options while owning the underlying asset.
II. If the price of the underlying asset increases dramatically, then an insured portfolio will perform much better than an uninsured portfolio.
III. The cost of portfolio insurance will rise as the volatility of the underlying asset increases.
IV. The upside potential for an insured portfolio is unlimited.

A. I and IV
B. III only
C. III and IV

I is incorrect because this strategy requires the purchase of put options while owning the underlying asset.

II is incorrect because if the price of the underlying asset increases dramatically, an insured portfolio will underperform an uninsured portfolio. The cause of this underperformance is that fact that an insured portfolio always lags behind an uninsured portfolio by the amount of the premium.

#### Practice Question 20

A market participant sells a customized over-the-counter put option on the stock when the stock is trading at \$38. The put has an exercise price of \$36 and the put seller receives \$2.25 in premium. The price of the stock is \$35 at expiration. The profit or loss for the put seller at expiration is ______.

A. -\$2.15
B. \$1.25
C. \$2.25

Profit = max (0, -value of put at expiration + premium) = max (0, -(X-S) +premium) = -1+2.25 = \$1.25

#### Practice Question 21

An investor owns a stock portfolio that closely follows the Standard & Poor's 500 Index (S&P 500). He purchases one S&P 500 stock index put option. The investor's position is now portfolio insurance with the following characteristics:

Portfolio position: LONG S&P 500
Portfolio purchase price: \$1427.21
Option position: LONG 1 put option
Underlying asset: S&P 500 Index
Exercise price: \$1225
Expiration date: November 30

The expiration-day price of the S&P 500 is ST = \$1500. The expiration-day value for the portfolio insurance is ______.

A. \$1,500
B. \$1,427.21
C. \$1,225

The expiration value of a protective put is Max(ST, X) = Max (1,500, 1,427.21) = 1,500.

#### Practice Question 22

An at-the-money put option will have a delta that is about ______.

A. -0.5
B. 0.5
C. 1