Conversions and Reverse Conversions for Actuaries: Practice Problems and Solutions

This is Section 3 of the Study Guide. See Section 1 here. See Section 2 here.

In this section, we will explore conversions and reverse conversions - first through conceptual multiple-choice questions and then by means of practice problems that combine these concepts with the formulas for put-call parity discussed in Section 2.

Problem CRC1. What does a conversion involve?

(a) Selling a call, selling a put, buying the stock

(b) Selling a call, buying a put, short-selling the stock

(c) Buying a call, buying a put, short-selling the stock

(d) Selling a call, buying a put, buying the stock

(e) Buying a call, selling a put, short-selling the stock

(f) Buying a call, selling a put, buying the stock

Solution CRC1. A conversion is a transaction that involves selling a call, buying a put, and buying the stock. So the answer is (d). A good way to concisely write this down and memorize it is via an ordered listing of two-letter abbreviations, where the first letter denotes the action and the second denotes the asset being bought or sold. In the first place, "B" stands for "buying" and "S" stands for "selling." In the second place, "C" stands for "call," "P" stands for "put," and "S" stands for "stock." So the way to describe a conversion using this notation is (SC, BP, BS). This is my original notation for it - meant to assist with memorization. Feel free to use it or any other device that helps you.

Problem CRC2. What does a reverse conversion involve?

(a) Selling a call, selling a put, buying the stock

(b) Selling a call, buying a put, short-selling the stock

(c) Buying a call, buying a put, short-selling the stock

(d) Selling a call, buying a put, buying the stock

(e) Buying a call, selling a put, short-selling the stock

(f) Buying a call, selling a put, buying the stock

Solution CRC2. A reverse conversion is, quite intuitively enough, the exact reverse of a conversion and involves buying a call, selling a put, and short-selling the stock. So the answer is (e). Using the notation defined above, a reverse conversion can be described as (BC, SP, SS).

Problem CRC3. What kind of financial vehicle does a conversion synthetically create?

(a) A share of stock

(b) A forward contract

(c) A prepaid forward contract

(d) A T-bill

(e) A call option

(f) A put option

(g) A swap

(h) A certificate of deposit

(i) A futures contract

(j) A Ponzi scheme

Solution CRC3. A conversion synthetically creates a T-Bill. T-Bills require investment but have (practically) no risk involved in the position. By means of selling a call, buying a put, and buying the stock, a conversion hedges all the risk and enables the position's owner to earn the equivalent of risk-free interest income due to the time value of money. So the answer is (d).

Problem CRC4. Amon-Ra wishes to purchase shares of and options contracts on Mythological Industries in order to create a synthetic T-Bill. At a strike price of 23, he sells a call option for 5.43, buys a put for 2.35, and buys the stock for 20. The options expire in one year, and Amon-Ra holds them until expiration, at which time one of them is exercised. Mythological Industries pays no dividends on its stocks. What is the annual continuously compounded rate of return that Amon-Ra earns on his investment?

Solution CRC4. The simplified formula for put-call parity in this situation is

C(K, T) - P(K, T) = S₀ - e^-rTK. We want to find the value of r, so we rearrange the formula as follows:
e^-rTK = S₀ - C(K, T) + P(K, T).

Here, C(K, T) = 5.43; P(K, T) = 2.35; S₀ = 20; T = 1, and K = 23.

Thus, e^-r23 = 20- 5.43 + 2.35

e^-r23 = 16.92

e^-r = 0.7356521739

-ln(0.7356521739) = r = 0.3069978618 = 30.69978618% (I want that as a risk-free rate of return; don't you?)

Problem CRC5. Quetzalcoatl undertakes a reverse conversion using shares of and options on Pyramids and Monuments, Inc. He pays an annual effective rate of interest of 0.03 for this transaction - in which he stays for 5 months. Pyramids and Monuments, Inc. pays no dividends on its stock. Quetzalcoatl short-sells one share of stock at a price of 99 per share. He buys a call option at 6.57. How much money does he get from selling the put option? Both options have a strike price of 96 and expire 5 months from now.

Solution CRC5. The simplified formula for put-call parity in this situation is

C(K, T) - P(K, T) = S₀ - e^-rTK. We want to find P(K, T), so we rearrange the formula as follows:
P(K, T) = C(K, T) - S₀ + e^-rTK. Furthermore, we have T = 5/12 and our interest rate expressed as an annual effective rate rather than a continuously compounded rate. Thus, our present-value factor for K is (1.03)^-5/12 = 0.987759366. Furthermore, C(K, T) = 6.57, S₀ = 99, and K = 96.

Thus, P(K, T) = 6.57 - 99 + 0.987759366*96 = P(K, T) = 2.394899134.

See other sections of The Actuary's Free Study Guide for Exam 3F / Exam MFE.