level. In this section, our focus shifts to portfolios and portfolio analytics. These analytics include the definition of portfolio returns and various portfolio measures of exposure. There are four types of portfolios that we are concerned with-the managed portfolio, the benchmark portfolio, the active portfolio, and the market portfolio. 1.    The managed portfolio is directed by the portfolio manager. 2.    The benchmark portfolio is what the portfolio manager manages against. Examples of benchmark portfolios include the S&P 500 and the MSCI World. 3.    The active portfolio is the difference between the managed and benchmark portfolios. 4.    The market portfolio is supposed to be representative of the relevant market. Often, the benchmark and market portfolios are the same. In situations where they are different, risk and return may be calculated relative to the benchmark and market portfolios. At each point in time we have the following quantities: P{t) Closing price of the ?zth asset as reflected in the base currency. qp{t) Quantity-the number of shares held of the nth asset in the managed portfolio; this value can be positive, negative, or zero. posp(t) wth asset's position defined as price times quantity of shares held, that is, Pn{t)xql(t). posp(£) Total market value of the managed portfolio. By definition, posF(t) = ^PosHt) "=1 where there are N assets in the managed portfolio; posp(t) can be positive, negative, or zero. wp(t) nth asset's weight in the managed portfolio. It's defined as posp(£) We define similar quantities for the benchmark, the active, and the market portfolios where we use a superscript "6," "a," and "m" to denote benchmark, active, and market, respectively. That is, qb{t) Quantity-the number of shares held of the nth asset in the benchmark portfolio. qm(t) Number of shares outstanding of common stock, or the number of shares held of the nth asset in the market portfolio. whn{t) nth asset's weight in the benchmark portfolio. Note that this weight is not necessarily a market cap weight. tv*(t) nth asset's weight in the active portfolio. Also known as the active