Hedge Portfolio

EBay Inc. is one of the major auction service and one of the worldÐŽ¦s largest online trading community. It provides these services by developing the eBay marketplace, which is an internet based community where buyers and sellers are brought together to buy and sell various products. It enables trade on local, national and international basis. Since the company was founded, it has made many acquisitions; the latest one is buying Skype (a VoIP company).

I used EBay Inc. stock to create a single stock portfolio. The stock price increased from $38.94 to $44.43 (approximately 14%) in the past 14 weeks (12th Sept 2005- 12th Dec 2005)

SINGLE STOCK PORTFOLIO:

I used SML to measure the portfolioÐŽ¦s risk adjusted return.

kp = krf + Ñ"Тp (km ÐŽV krf) + Ñ"С

kp is 14.10%, which is the portfolio return calculated using the geometric average return for the past 14weeks as shown in the excel worksheet.

Ñ"Тp is the beta of my single stock portfolio. EBay Inc has a beta of 2.111 which means that theoretically the stock is about 110% more volatile than the market.

km for the equation is equivalent to S&P 500 return for the 14 weeks, which is (1261.40/1246.60 ÐŽV 1) = 1.187%

krf is the corresponding risk free rate used. It is 1.04% over the past 14 weeks.

.1410 = 0.0104 + 2.11(0.0119- 0.0104) + Ñ"С

The alpha (Ñ"С) term for my security portfolio is 0.1274 or 12.7435%

The basic CAPM formula used above states that the stockÐŽ¦s return is equal to the risk free rate plus the risk premium. Extrapolating from the CAPM formula (also called JensenÐŽ¦s alpha), the unsystematic risk that reflects investment performance over and above the risk associated with the systematic or market wide factor is called alpha (Ñ"С). An alpha (Ñ"С) is risk-adjusted performance measure. Technically a positive (negative) alpha indicates stronger (poorer) fund performance than predicted by the fund's level of risk (measured by beta). A positive alpha does mean that EBay Inc. has outperformed the market.

However for EBay Inc. an alpha of 12.74% over the 14 week period is quite enormous. It shows that there were some events like acquisitions and new developments2 over the past 14 weeks that led to an excess return of 12.74%.

OPTION AND STOCK HEDGE:

The original objective in finding the hedge ratios is to find the approximate number of call options to sell in order to hedge the long stock position. The basic objective is to eliminate risk of price going down.

Here I own 25,686 shares of EBay stock. The delta N(d1) initially was 0.455. Hence I sold 25,686/ 0.455 = 56,453 calls to hedge a long 25,686 stock position and to keep the delta of 0.

Ñ"Ò' = 25686 Ñ"Ò'stock - 56453Ñ"Ò'calls

= 25686 (1.00) ÐŽV 56453 (.455)

= 0 approx

The hedge is "delta neutral" which is what we want, since this means the value of the portfolio does not change with changes in stock price. The aim of any truly hedged position is to stay delta neutral. But this is for the initial covered call only. We have to continuously rebalance the option position in order to maintain the options portfolio beta of zero. Beta is a measure of risk and a perfect hedge will eliminate all risk in theory. Beta of a portfolio could be calculated as a weighted average of the portfolio beta .Technically the beta of the options should have been -2.11, to give us a beta of 0 for the entire portfolio (risk free portfolio will have 0 beta). But it is hard to figure out the real beta due to continuous rebalancing of the portfolio and as we were unable to incorporate small changes in the stock price. This is correct

After hedging the portfolio of $1,000,000 with long shares and short calls, there was a net investment of $909,674 in the 1st week of the hedge. Also the call option, which I sell short, expires in-the-money during the 6th week of the observation. So I had to roll it over to a new option. The overall return on the options hedge was a negative 4.5%, while the return on the stock portfolio was 14%. The total return on the continuous hedged options portfolio should have been similar to the risk free rate, but it was more than the risk free rate (14%-4.5%). So it was not a perfect hedge.

From the excel sheet, I observed that after the rollover to a new option (deep-in-the-money), there has been some evident changes in the