a fixed supply of two risky assets, which we will call equity and bonds. Let the outstanding supplies (that is, the market capitalization weights) be given by e and b, respectively. There is also a risk-free asset, which we refer to as cash. Cash is risk free in the sense that at the end of the period a unit investment in cash will return a known quantity, 1 + r. Equity and bonds are risky in the sense that unit investments return random values, 1 + re and 1 + r&, respectively. We take the risk in this world to be given exogenously. That is, we assume that rs and rb are random variables with known, or estimable, volatilities given by oe, and o&, and a correlation p. In contrast, we do not take the mean returns as given, but rather wish to solve for them in equilibrium. While we don't focus on prices, we do assume that investors will bid the prices for individual stocks and bonds up or down until their prices reach levels such that expected returns clear markets- that is, until the demand for each asset equals the outstanding supply. Let these unknown market-clearing expected returns be |ie and |i&, respectively. At the beginning of the period, each investor must choose a set of portfolio weights that represent the proportion of his or her holdings of cash, bonds, and equity. For a representative investor, we express these portfolio weights as a percentage of beginning of period wealth. Let we and wb represent the portfolio weights in equity and bonds, respectively, for the representative investor. The weight in cash is \ -w -W-. e b The investor chooses asset weights in order to maximize the value of a utility function that rewards higher expected returns and penalizes portfolio risk. In particular, let the expected return on the portfolio be given by ji and the volatility of the portfolio be given by o . Assume the utility function has the simple quadratic form described in Chapter 3 and given by the following equation: U = H-.5.X.o2p (4-1) The parameter X gives the investor's risk aversion, the rate at which he or she will trade off a reduction in expected return for a reduction in variance. The quadratic form of the utility function represents the assumption that as risk increases there is an increasing aversion (in the form of willingness to forgo expected return) to additional increases in risk. Portfolio expected return is given by the asset weights times the expected returns on each asset. \ip = r (1 -we-wb)+we \iE + tvb \ib (4.2) Portfolio variance is also determined by asset weights in the risky assets and the assumed volatilities and correlation between these assets. Letting oe b represent the covariance of equity and bond returns, that is, Os b = Oe ■ 0& ■ p, oj =w2e .oj +wl ,<52b +2.we .wb .cejfc (4-3) Thus, for given weights, ws and wb, the investor has a utility given by: