1 The Calculus of Finance Portfolio Diversification, Life Time Savings, and Bankruptcy Bill Scott SAIC May 2003, caveat added May 2005, page 1 notes

2 Calculus of Finance Goal -- Understand several Nobel prize winning finance theories using a scientific understanding of basic calculus to derive the fundamental finance equations with less effort. Focus on portfolio selection as an issue of most importance to scientists and engineers. Later talks can discuss lifetime savings and bankruptcy.

3 Mean-Variance Utility Adjust your portfolio to maximize the utility where expected portfolio growth  P is desired, but portfolio volatility  P   is to be avoided.  is your personal risk aversion. It represents your risk-return tradeoff, how much risk you are willing to bear in order to hope for higher returns. eq 1

4 Growth is a Weighted Average The expected portfolio growth is the weighted-average expected growth over all assets eq 2 where w i is the fraction of wealth held in the ith investment. This assumes frequent rebalance instead of buy-and-hold.

5 Uncertainty Adds like Vectors r1r1 r2r2 r3r3 w11w11 w22w22 w33w33 PP Assuming no covariance.

6 Portfolio Volatility Volatility adds over a portfolio like vectors when there is no covariance. If the performance of i and j are not independent, then where  ij is the covariance between i and j. eq 3

7 Volatility and Covariance Past volatility and past covariance are thought to be the best measures of future volatility and covariance; however, past growth is not thought to be a good measure of future growth.  t is in years so that the rates and variances are annualized.

8 Tobin Two Asset Portfolio Assume a risk free investment that returns f, and a risky asset that you expect to return  with volatility  2. What fraction of wealth w should be invested in the risky asset? Maximize the mean-variance utility by settings its derivative with respect to w to zero. eq 4 Where w * represents an optimized portfolio holding.

9 Importance of Tobin Equation Holdings are proportional to expected excess return. Bigger holdings if optimistic or if interest falls Inversely proportional to volatility and risk aversion Own less to sleep better Reduce holdings when volatility increases Additional risk generally causes selling.

10 Three Asset Diversification Add the general market (S&P 500) with  m and  m 2. These are two equations with two unknowns. Solve with algebra. eq 5 eq 6

11 Three Asset Results Solution to simultaneous equations 5 and 6 eq 7 eq 8 Notice that this is like determinant theory which points to a matrix notation.

12 Markowitz Matrix Notation or eq 9

13 Markowitz Efficient Portfolio Equation 9 is the Markowitz efficient portfolio. Notice that risk aversion  does not effect your allocation between stocks. Your  is determined by how much you desire risk free. The variances and covariances can be measured from historical data, and are thought to be fairly constant into the immediate future. Thus your portfolio allocation depends mostly on , your own expectations of future growth. Actually Markowitz did a lot more work to constrain his portfolios to no short selling and no borrowing. Eq. 9

14 Markowitz Admits Large Holdings w 1 represents volatilities and covariances like SAIC If your expectations of SAIC exceed the market, then strong holdings are recommended by the Markowitz theory.

15 CAPM Capital Asset Pricing Model What  ’s does the market assume? In eq 7, the market must see  1 to be large enough so that people just want to start buying it. All current stock prices must be set so that the numerator of eq 7 is positive but close to zero. define beta CAPM - when you can’t guess  , assume eq 12

16 Observables “I think of expectations as observable, at least in principle.” Fischer Black, 1995 Instead “Honest agents examining the same firm will see a distribution of expectations perhaps centered on CAPM, but with a width of estimates consistent with .” Bill Scott, 1998

17 Arbitrage Pricing Theory APT Usually this is derived from the assumption that an efficient market allows no risk free profits from self-funded hedges; i.e. there are no arbitrage opportunities. Instead, this equation also derives directly from CAPM, in that the current buyers and sellers set the current price. This implies that there is one price regardless of risk aversion or optimism. eq 13

18 Modigliani and Miller Theorem Corporate finance does not affect value to shareholders. Corporations can raise capital several ways Cutting dividends Selling more stock Borrowing by selling bonds Paying employee bonuses with options rather than cash These each have different risk-return effects Such risk-return changes leave share value unchanged - APT All change both the risk and the expected return such that the APT ratio does not change -- thus the stock price is unaffected by finance.

19 Rational Investing Invest some risk free for your own personal comfort, your  Predict technology and future fads. Search for firms whose  will exceed CAPM Your  estimates may well be better than CAPM! Allocate an optimized portfolio using eq 9, Markowitz. Rebalance the portfolio frequently. Sell the high growth stocks in order to buy more laggers? Reallocate only when you’ve revised your  estimates.

20 Portfolio Diversification If you can’t predict  and use CAPM. –a very small investment in everything However if andcan cause If you think you know what you’re doing, portfolio diversification theory admits to very aggressive holdings. Search for high growth, low volatility investments.

21 Utility and Probability Theory Simple Portfolio Analysis Utility and Probability Bankruptcy Lifetime Savings and Consumption

22 Pratt Constant Relative Utility so that Utility, more wealth is always better, but less is a lot worse. Most employees are  = 2 to 4. Plot of U vs W T with C set so that the utility of present wealth is always 0. Time Separable and State Independent eq 14

23 Log Normal Probabilities Normal distribution of rates of return vs optimism and volatility  Same log normal distributions of stock prices at 5 years

24 Log Normal Probability Math Expectation of wealth grows probabilistically with expected return. Normal probability of possible returns integrates to give so that Expected return is the most likely return plus half the volatility! eq 15

25 Log Normal Utility Math The Pratt utility of wealth can also be integrated under the log normal probability distribution. Maximizing the Pratt utility under lognormal expectations is the same as maximizing the mean-variance utility. eq 16

26 Pratt and simple are the same Pratt and simple mean-variance utilities maximize the same function, thus –Pratt and mean variance  are the same avoiding volatility and losing wealth hurts more than gaining wealth. Both are independent of the time. –Rebalance portfolios over time to maintain w* until you recalculate your  ’s and  ’s.

27 Derivative Pricing Simple Portfolio Analysis Utility and Probability Bankruptcy Lifetime Savings and Consumption

28 Considering Bankruptcy Let b be the annual probability of bankruptcy of the w b asset. Let T be either a fixed period or the bankruptcy time. No Bankruptcy Bankruptcy probability outcome

29 Markowitz with Bankruptcy In limit of low w b *, b acts as risk Which is non linear in w b and saturates Performing the integral and setting derivative to zero

30 Bankruptcy Limits Holdings to 50% w 1 represents volatilities and covariances like SAIC If your expectations of SAIC exceed the market, then strong holdings admit. With any bankruptcy risk, holdings saturate at 50%

31 Lifetime Savings and Consumption Simple Portfolio Analysis Utility and Probability Bankruptcy Lifetime Savings and Consumption

32 Retirement Planning How much can be spent per year during retirement? How much needs to be saved while working? Merton 1971 Annuity equation - wealth P –Can spend R each year –Assumes certainty r –Spends last dollar at D mortgage equation eq 22

33 Inflation Adjusted Retirement Variable annuity equation to keep income adjusted for inflation Live on the amount that interest exceeds inflation, the rest then keeps up with inflation. Thus each year’s spending can grow with inflation.

34 Retirement under Uncertainty Assume that your wealth is efficiently invested by the Markowitz equation which is frequently rebalanced so that expected return is  P and expected volatility is  P 2. Utility of inflation adjusted spent dollars is maximized by the variable annuity equation (Merton) Notice, once the portfolio is maximized for utility, certainty returns are replaced by half the expected excess returns. Volatility isn’t in the equation. eq 23

35 Uncertain Retirement Lognormal random walk of portfolio returns. Variable annuity equation generates steady retirement income likely to adjust to inflation and unlikely to deplete early. Smoothing in early years.

36 Continuous Standard of Living Save C fraction of your salary while working. Salary assumed to grow with inflation. Retire in T years and spend R for D-T more years. Wealth P to finance retirement is in units of today’s annual salary. After retirement spend R Inflation adjusted retirement equals inflation adjusted salary less savings. eq 24