This is my first post on this website so please forgive me for any mistake or inappropriate use. I am taking a Master level Investments course, in which, amongst the rest, we are deriving $$ \pi(\sigma) = \frac{\alpha(\bar{W})}{2} $$ We started from the following expression $$ \tilde{W} = \bar{W} + \sigma \tilde{\epsilon}, \quad \tilde{\epsilon} \sim (0, 1) $$ where σ represents the risk of the wealth and $$ \tilde{ϵ} $$ the noise associated with randomness. Then we defined risk premium (π) using the following formula: $$ E[u(\tilde{W})] = u(E[\tilde{W}] - \pi) $$ In order to estimate π for small risks, i.e. when σ is very close to 0, we used the 2nd order Taylor expansion: $$ \pi(\sigma) \approx \pi(0) + \pi'(0)\sigma + \frac{1}{2}\pi''(0)\sigma^2 $$ and we proceded to calculate respectively π(0), π′(0) and π′′(0).
- π(0) was trivial as $$ E[u(\bar{W} + \sigma \tilde{\epsilon})] = u(E[\tilde{W}] - \pi(\sigma)) \quad \text{at} \quad \sigma = 0 $$ $$ u(\bar{W}) = u(\bar{W} - \pi(0)) \Rightarrow \pi(0) = 0 $$
- Now, what I don't understand is the way the first and the second derivative were calculated: $$ E[u'(\bar{W} + \sigma \tilde{\epsilon}) \tilde{\epsilon}] = u'[\bar{W} - \pi(0)] * \pi'(0) \text{ at } \sigma = 0 $$ $$ \Rightarrow \pi'(0) = 0 $$
- $$ E[u''(\bar{W} + \sigma \tilde{\epsilon}) \tilde{\epsilon}^2] = u''[\bar{W} - \pi(0)]*[\pi'(0)]^2 \text{ at } \sigma = 0 $$ $$ \Rightarrow \pi''(0) = - \frac{u''(\bar{W})}{u'(\bar{W})} = \alpha(\bar{W}) $$ Can someone please explain? That would help me so much! Thanks