1. IX. EVALUATING TRADING STRATEGIES AND PERFORMANCE

2. 9.1. Evaluating Investment Portfolio Performance
One standard for comparison is the simple buy and hold into a diversified portfolio strategy
Fund net asset value (NAVt) and returns (including the time weighted average return) are computed as follows:

3. Illustration: NAV and Returns

4. Portfolio Benchmarking
Higher returns are generally associated with higher risk, such that appropriate benchmarking is important.
Sharpe Ratio Treynor Ratio
Jensen Measure: Jp = [rp - rf]-[p(rm-rf)]

5. Benchmarking Difficulties
The following represent additional difficulties in using the above risk adjusted portfolio performance measures:
Portfolio managers changing jobs frequently
How frequently can we obtain enough data for statistically significant performance measures?
The Capital Asset Pricing Model (CAPM), a model that decomposes return into compensation for time value of money and compensation for risk, serves as the basis for the Treynor and Jensen Measures:
The CAPM: E[rp] = rf + [p(rm - rf)]
where E[rp], rf and rm are the expected return on the portfolio, the riskless asset and the market as a whole. However, the CAPM is only a 1-time-period model. Multiple time periods and multiple cash flows cause problems in its application. In addition, many analysts will be concerned about the many assumptions that underlie the CAPM, as well as certain statistical tests that cast doubt on the empirical validity of the CAPM.

6. Benchmarking Difficulties: Cont.
Investors holding funds representing only market segments might find that any measure based on the Capital Asset Pricing Model is inappropriate.
The Sharpe Ratio will understate portfolio performance of undiversified portfolios. That is, much of the risk captured in the Sharpe Ratio can be diversified away.
Errors in computing returns will bias measured betas downwards and will "slop" over into unsystematic variances (the part of risk that is unrelated to the market). Even seemingly minor problems can significantly bias beta measures. However, there do exist reasonably good correction procedures for betas measured with error.

7. Portfolio Performance Benchmarking Illustration
The following are portfolio and market returns over a 20-year period:

8. Jensen’s Alpha Scatter Diagram

9. 9.2. Market Timing versus Selection
To what do we attribute superior portfolio performance (or deficiencies)?
Investors with strong timing ability will shift to higher-beta portfolios as market returns rise.
For example, the relationship between portfolio risk premiums and market risk premiums will be concave up for investors with strong timing ability.

10. 9.2. Market Timing versus Selection
The Quadratic Variable Approach

11. The Dummy Variable Approach
rq,t - rf,t = αq + βq(rm,t-rf,t) + γqD(rm,t-rf,t) + et
where: If (rmt-rft) ≥ 0, D = 0
If (rmt-rft) < 0, D = 1

12. 9.3. Trade Evaluation and VWAP
VWAP (Volume Weighted Average Price is calculated by dividing the dollar volume of trading in a stock by the share volume over a given period of time, typically one day.
Arrival Price: The midpoint of the bid-offer spread at the time the order is received (Bid-Ask Midpoint or BAM).
MOC (Market-on-close): the last price obtained by a trader at the end of the day relative to the last price reported by the exchange.
Implementation shortfall: the performance difference between the hypothetical profits realized by a paper or theoretical portfolio replicating an actual portfolio ignoring friction costs and the profits realized by the actual portfolio.

13. VWAP
VWAP can be used as a benchmark to evaluate the quality of the execution provided by the broker.
If the brokerage firm’s purchases were made at a lower VWAP than the market VWAP for the relevant period, the firm presumably handled the order well for the customer.
VWAP, either for the trader or for the market is calculated as follows:

14. VWAP: A Simple Illustration
Suppose that a broker has been instructed to purchase 600 shares at the market. She does so, purchasing them for a total price of 30,011.
The broker's executed transactions were the second through fourth transactions on the table.
The total volume of shares exchanged was 2,500, with a total value of 125,098.
Hence, VWAP for market transactions was 125,098/2,500 =
The broker purchased 200 shares in the first transaction at and 400 shares in the third transaction at
The average share price paid by the broker was 30,010/600 =
Our calculations suggest that the broker beat the market VWAP.

15. 9.4. Implementation Shortfall
Implementation shortfall can be defined as the difference in prices between the ideal transaction and the actual implemented transaction.
The ideal transaction price is typically based on the security price existing when the transaction decision was made, or in the case of an agency broker, at the order arrival.
The difference between this decision price and the actual implemented or execution price is construed to be implementation shortfall.

16. Implementation Shortfall: Friction Costs
The implementation of an investment strategy by the trader or portfolio manager leads to four primary types of friction costs:
Broker, exchange and other explicit fees and commissions. Frequently, brokers bundle exchange, SEC and other fees into their own commissions. Small transactions tend to have higher proportional explicit transactions costs.
Delay costs, based on the price difference between the portfolio manager’s decision price and the broker’s arrival price
Price impact costs associated with transaction executions (slippage). Buy orders will exert upward price pressure on the security; sell orders will exert downward pressure. Larger transactions will tend to have larger impact costs.
Opportunity costs associated with transactions; that is, the opportunities and profits were forgone prior to the trade’s completed execution. Opportunity costs can also include the portion of an order that was canceled due to a limit order restriction.

17. Implementation Shortfall Illustration
Suppose that a portfolio manager makes a decision to purchase 10,000 shares of stock one hour before its open based on its $50.00 closing price the prior day (the decision price) and a limit order at
The stock opened at 9:30 at a price of 50.20, and 1000 shares are purchased at 9:31 at a price of
At 9:32, 5000 shares are purchased for 50.40, and 1000 more for at 10:03.
An additional 1000 shares are purchased for at 12:15, the market price quickly rises to and closes at with 2000 shares in the order unexecuted.
The commissions, including all explicit fees were $0.01 for each of 8,000 shares.

18. Implementation Shortfall Illustration, Continued

19. 9.5. Value at Risk
A 1996 amendment to Basel I permits banks to use their own portfolio models to compute capital requirements.
The Value-at-Risk (VaR) model measures the worst loss size or threshold over a given period of time consistent with a specified probability:
VaR = Asset Value × Daily return standard deviation × Confidence interval factor × the Square Root of time 
VaR = Asset Value ×  × z × t
Asset value is the total value of the bank or relevant component, the daily return standard deviation applies to this asset value, the confidence interval factor represents the maximum acceptable probability that this loss will be exceeded (typically a z-value such as 1% from a normal distribution) and time is measured in days.
Alternative systems are used by banks, including the CreditMetrics system at J.P. Morgan/Chase.

20. Value at Risk Illustration
Suppose that a trader has "borrowed" $900,000 from his employer and invested $100,000. This trader's employer requires that the trader's one-week portfolio VaR not exceed his trading capital of $100,000, with a 99% degree of confidence.
i wi i 1,i 2,i 3,i