In looking over the final exam that I wrote and also my previous ?hints? posting, I would like to offer some further (hopefully helpful) hints.
1. The way that one analyzes how risk aversion changes with respect to changes in wealth and risk is captured by the following definition of the risk premium (source: see the equation for lambda on page 19 of the http://fin4335.garven.com/fall2012/lecture6.pdf lecture note). There, we see that the risk premium depends upon two factors: 1) subjective factor: degree of risk aversion as indicated by the Arrow-Pratt Coefficient of Absolute Risk Aversion, and 2) objective factor: the variance of the risk under consideration.
2. Most of the utility functions (e.g., logarithmic utility, square root utility) we analyzed in Finance 4335 exhibit the property of decreasing absolute risk aversion (DARA). On page 21 of the http://fin4335.garven.com/fall2012/lecture6.pdf lecture note, we show that absolute risk aversion is decreasing in wealth. Behaviorally, this means that as people grow wealthier, the risk premium (in absolute dollar terms) that they assess for a given risk will decline.
3. Absolute risk aversion measures how your risk aversion changes in dollar terms as your wealth changes. On the other hand, relative risk aversion measures how your risk aversion changes in proportional terms as your wealth changes. Decreasing absolute risk aversion implies constant relative risk aversion (CRRA), which in turn implies that one does not become more risk averse in a proportional sense as one grows wealthier. A practical example is that if given a choice between a safe and risky asset, then the relative proportions invested in these two assets won?t change at all for a CRRA investor as that investor?s wealth changes.
4. On using the Black-Scholes-Merton option framework for assessing and pricing credit risk. We learned in our study of corporate limited liability that the value of debt issued by a limited liability corporation is equal to the difference between the present value of the promised payment on a safe bond, minus the value of the limited liability put option; i.e.,
Calculating the present value of the safe bond (
) is quite easy, although calculating the value of the limited liability put option (V(Max(B - V(F))) is a bit more challenging. Look at the Final Exam Formula Sheet, equation (5). This equation gives us the formula for pricing a put option. You can use this equation to calculate the value of the limited liability put option by simply substituting the promised bond payment (B) in place of K, and substituting the value of the firm?s assets (V(F)) in place of S. Once you have calculated V(D), then the bond yield represents the rate of return that one would have to earn in order for V(D) to grow in value to B over the course of T periods. So you can calculate the bond yield by solving the following time value equation for YTM:
If said bond has a high probability of default, then YTM will be substantially higher than the riskless rate of interest r; the difference between YTM and r represents the so-called ?credit risk premium? which compensates investors for having to bear the risk of default.
Source: http://risk.garven.com/2012/12/07/some-more-hopefully-helpful-finance-4335-final-exam-hints/
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