Finance 4335 extra credit opportunity

I have decided to offer the following extra credit opportunity for Finance 4335. You can earn extra credit by attending and reporting on Dr. P. J. Hill’s upcoming lecture entitled “Saving the Environment Through Prices and Property Rights”:

If you decide to take advantage of this opportunity, I will use the grade you earn to replace your lowest quiz grade in Finance 4335 (assuming that your grade on the extra credit is higher than your lowest quiz grade). The report should be in the form of a 1-2 page executive summary in which you provide a critical analysis of Dr. Hill’s lecture. In order to receive credit, the report must be submitted via email to risk@garven.com in either Word or PDF format by no later than Monday, March 26 at 5 p.m.

On the role of replicating portfolios in the pricing of financial derivatives

Replicating portfolios play a central role in terms of pricing financial derivatives. Here is a succinct summary from yesterday’s class meeting:

  1. Buying forward is equivalent to buying the underlying on margin, and selling forward is equivalent to shorting the underlying and lending money. Like options, forwards and futures are priced by pricing the replicating portfolio and invoking the “no-arbitrage” condition. If the forward/futures price it too low, then one can earn positive returns with zero risk and zero net investment by buying forward, shorting the underlying and lending money. Similarly, if the forward futures price is too high, one can earn positive returns with zero risk and zero net investment by selling forward and buying the underlying with borrowed money. This is commonly referred to as “riskless arbitrage”; it’s riskless because you’re perfectly hedged, and it’s arbitrage because you are buying low and selling high.
  2. The replicating portfolio for a call option is a margined investment in the underlying. During yesterday’s class meeting, we priced a one timestep call option where the price of the underlying asset is $100, the exercise price is also $100, u = 1.05, d = .95, the interest rate r = 5%, and the timestep \delta t = 1/12. Given these parameters, the payoff on the call is $5 at the up (u) node and $0 at the down (d) node.  The replicating value consists of half a share that is financed by a margin balance of $47.30; thus the “arbitrage-free” price of the call option is (.5(100) – 47.30) = $2.70.
  3. Since the replicating portfolio for a call option is a margined investment in the underlying, it should come as no surprise that the replicating portfolio for a put option consists of a short position in the underlying combined with lending. Thus, in order to price the put, we need to determine and price the components of the replicating portfolio. We will begin class tomorrow by completing our analysis of the replicating portfolio approach to pricing calls and puts, and move on to other pricing methods such as delta hedging and risk neutral valuation.

Synopsis of today’s Capital Market theory topic

The Capital Market Theory lecture note upon which today’s Finance 4335 class discussion was based provides the following important insights:

  1. Borrowing and lending at the riskless rate of interest in combination with investing in (mean-variance efficient) risky portfolios makes it possible for investors to obtain superior risk-return trade-offs compared with investing only in mean-variance efficient risky portfolios. In the figure below (taken from page 13 of the Capital Market Theory lecture note), investors select portfolios along the Capital Market Line, which is given by the following equation: E({r_p}) = {r_f} + \left[ {\displaystyle\frac{{E({r_m}) - {r_f}}}{{{\sigma _m}}}} \right]{\sigma _p}.Capital Market LineIn the above figure, \alpha corresponds to the optimal level of exposure to the market index which is labled as point M. When \alpha = 0, the investor is fully invested in the riskless asset. When 0 < \alpha < 1, the investor is partially invested in the riskless asset and in the market index; such portfolios are referred to as “lending” portfolios. When \alpha = 1, the investor is fully invested in the market index. Finally, when \alpha> 1, the investor funds her investment in the market index with her initial wealth plus borrowed money; such portfolios are referred to as “borrowing” portfolios.
  2. Given that investors select (based upon their level of tolerance for risk) portfolios that lie on the Capital Market Line, this behavior has implications for the pricing of risk for individual securities. Specifically, the Capital Market Line implies that for individual securities, the Security Market Line must hold. The equation for the Security Market Line (which is commonly referred to as the Capital Asset Pricing Model, or CAPM) is given by the following equation:E({r_i}) = {r_f} + \left[ {E({r_m}) - {r_f}} \right]{\beta _i},where {\beta _i} = {\sigma _{i,m}}/\sigma _m^2.
  3. According to the CAPM, the appropriate measure of risk for an individual stock is its beta, which indicates how much systematic risk the stock has compared with an average risk investment such as the market portfolio. Beta for security i ({\beta _i}) is measured by dividing the covariance between i and the market ({\sigma _{i,m}}) by market variance (\sigma _m^2). If the investor purchases an average risk security, then its beta is 1 and the expected return on such a security is the same as the expected return on the market. On the other hand, if the security is riskier (safer) than an average risk security, then it’s expected return is higher (lower) than the same as the expected return on the market.
  4. If the expected return on a security is higher (lower) than the expected return indicated by the CAPM equation, this means that the security is under-priced (over-priced). Investors will recognize this mispricing and bid up (down) the under-priced (over-priced) security until its expected return conforms to the CAPM equation.
  5. According to the CAPM, only systematic (i.e., non-diversifiable) risk is priced. Systematic risks are risks which are common to all firms (e.g., return fluctuations caused by macroeconomic factors which affect all risky assets). On the other hand, unsystematic (i.e., diversifiable) risk is not priced since its impact on a diversified asset portfolio is negligible. Diversifiable risks comprise risks that are firm-specific (e.g., the risk that a particular company will lose market share or go bankrupt).

Problem Set 7 helpful hints

Problem Set 7 is due at the beginning of class on Tuesday, March 20.  Here are some helpful hints:

  1. The least risky combination of Security A and Security B in Problem 1 is found by calculating {w_A} = \displaystyle\frac{{\sigma _B^2 - {\sigma _{AB}}}}{{\sigma _A^2 + \sigma _B^2 - 2{\sigma _{AB}}}} and {w_B} = 1 - {w_A}.
  2. It will always be the case for 2 security portfolios that by following the minimum variance portfolio weighting scheme in the previous bullet point, such a portfolio must have zero variance if {\rho _{AB}} = 1 or -1.
  3. In part B of Problem 2, the Sharpe Ratio for security j is \displaystyle\frac{{E({r_j}) - {r_f}}}{{{\sigma _j}}}.

Important insights from portfolio and capital market theory

The topics covered during the course of the last couple of Finance 4335 class meetings (portfolio and capital market theory) rank among the most important finance topics; after all, the scientific foundations for these topics won Nobel prizes for Markowitz (portfolio theory) and Sharpe (capital market theory). The following outline pretty much summarizes what we covered in class on Tuesday, March 13 and Thursday, March 15:

  • Portfolio Theory (covered on Tuesday, March 13)
      1. Mean-variance efficiency
      2. Portfolio Mean-Variance calculations
      3. Minimum variance portfolio (n = 2 case)
      4. Efficient frontier (n = 2 case under various correlation assumptions)
  • Capital Market Theory (covered on Thursday, March 15)
    1. Efficient frontiers with multiple number (“large” n) of risky assets (aka the “general” case)
    2. Portfolio allocation under the general case
      • degree of risk aversion/risk tolerance determines how steeply sloped indifference curves are
      • indifference curves for investors with high (low) degrees of risk tolerance (aversion) are less steeply sloped than indifference curves than for investors with low (high) degrees of risk tolerance (aversion)).
      • Optimal portfolios (i.e., portfolios that maximize expected utility) occur at points of tangency between indifference curves and efficient frontier.
    3. Introduction of a risk-free asset simplifies the portfolio selection problem since the efficient frontier is now a straight line rather than an ellipse in E({r_p}), {\sigma _p} space. The same selection principle holds as in the previous point (point 2); i.e., investors determine optimal portfolios by identifying the tangency between their indifference curves and the efficient frontier. The point of tangency occurs on the capital market line (CML) where the Sharpe ratio is maximized; everyone chooses some combination of the risk-free asset and the market portfolio, and risk tolerance determines whether the point of tangency involves either a lending (low risk tolerance) or borrowing (high risk tolerance) allocation strategy.
    4. The security market line (SML), aka the CAPM, is deduced by arbitrage arguments. Specifically, it must be the case that all risk-return trade-offs (as measured by the ratio of “excess” return (E({r_j}) - {r_f}) from investing in a risky rather than risk-free asset, divided by the risk taken on by the investor ({\sigma _{j,M}}) are the same. If not, then there will be excess demand for investments with more favorable risk-return trade-offs and excess supply for investments with less favorable risk-return trade-offs). “Equilibrium” occurs when markets clear; i.e., when there is neither excess demand or supply, which is characterized by risk-return ratios being the same for all possible investments. When this occurs, then the CAPM obtains: E({r_j}) = {r_f} + {\beta _j}(E({r_M}) - {r_f}).

     

Finance 4335