Question

Consider two securities that pay risk-free cash flows over the next two years and that have the current market prices shown here: Security B1 Price Today $90 $72 Cash Flow in one Year Cash Flow in Two Years $100 $0 $0 $80 | B2 a) What is the no-arbitrage price of a security that pays cash flows of $100 in one year and $80 in two years? [2 points] b) What is the no-arbitrage price of a security that pays cash flows of $100 in one year and $600 in two years? [2 points] c) Suppose a security with cash flows of $50 in one year and $160 in two years is trading for a price of $185. Is there an arbitrage opportunity available? If yes, explain in detail how you could exploit this arbitrage opportunity. Assume that you cannot trade fractions of shares in (c). [6 points]

Answer 1

a]

Security B1 pays $100 in one year, and Security B2 pays $80 in two years. Therefore :

No-arbitrage price = price of Security B1 + price of Security B2 = $90 + $72 = $162

b]

Security B1 pays $100 in one year, and Security B2 pays $80 in two years. Therefore :

No-arbitrage price = price of Security B1 + (price of Security B2 * $600 / $80)  

No-arbitrage price = $90 + ($72 * $600 / $80) = $630

c]

Security B1 pays $100 in one year, and Security B2 pays $80 in two years. Therefore :

No-arbitrage price = (price of Security B1 * $50 / $100) + (price of Security B2 * $160 / $80)  

No-arbitrage price = ($90 * $50 / $100) + ($72 * $160 / $80) = $189

However, the security is trading at $185.

The security is underpriced. Hence, we buy the security and sell the required quantity of Securities B1 and B2 so that cash flows are matched.

Buy 2 of the Security. Cash outflow now = $185 * 2 = $370. Cash inflow after 1 year = $50 * 2 = $100. Cash inflow after 2 years = $160 * 2 = $320.

Sell 2 of Security B1. Cash inflow now = $90 * 2 = $180. Cash outflow after 1 year = $50 * 2 = $100.

Sell 4 of Security B2. Cash inflow now = $72 * 4 = $288. Cash outflow after 2 years = $80 * 4 = $320.

The cash inflows after 1 and 2 years are matched by the cash inflows.

Arbitrage profit = cash inflow now - cash outflow now

Arbitrage profit = ($180 + $288) - $370 = $98