Question

The volatility of an asset’s return can be broken into diversifiable (i.e., unsystematic or firm-specific) and non-diversifiable (i.e., systematic) components. Investors who hold the asset should be compensated only for the non-diversifiable risk, which is measured by the “beta” of the asset. A stock that has a zero beta has no systematic risk and its expected rate of return should equal the risk free rate. Suppose that you have two different stocks: • SysVol, which has a beta of 1.00 and 100% of its volatility is non-diversifiable. • IdioVol, which is developing a new cure for cancer, has a zero beta because 100% of its volatility is firm specific and thus diversifiable. • The volatility of their return over the next 3 months is 60% for each stock. Both stocks are currently priced at $50 a share. Neither stock will pay any dividends. Each stock has a futures contract with 3 months to delivery. The riskless (borrowing and lending) interest rate for 3-months is 5%, per 3 months. Assume that transaction costs including short-sale costs are zero, and that there are no margins.

A) Calculate the no-arbitrage futures price of each stock. (1 point)

B) How would the differences in betas affect their current no-arbitrage futures prices? EXPLAIN.(2 points)

C) Now suppose that both stocks have betas of 1.00 and volatilities of 60%. All the other data above remain the same as in (A). In particular, both stocks are currently priced at $50 a share. But, the expected return on SysVol is 10%, whereas the expected return on IdioVol is 20%. How would the differences in expected returns affect the stocks’ current no-arbitrage futures prices? Explain. (2 points)

Answer 1

A. The future price for SysVol stock will be $52.40 which includes $1.50 towards interest cost and $0.90 risk premium. The future price for IdioVol stock will be $51.50 which includes $1.50 towards interest cost. There will be no risk premium in the latter case.

B. The future price of SysVol will include a risk premium of $0.90 to compensate for the non-diversifiable risk. As the risk associated with IdioVol stock is wholly diversifiable, there will not be any risk premium in its future price.

C. As both stocks have betas of 1.00, the future price may not include any risk premium. As the expected rate of return on SysVol stock is 10% per annum, its 3 months' future price will be $51.25. As the expected rate of return on IdioVol stock is 20% per annum, its 3 months' future price will be $52.50.