# Lecture 6: Budget Lines, Syndicates, Risky Jobs & Insurance

More Budget Lines:

1) Mean-Variance Utility (Financial Economics) ﻿$M$﻿ ﻿$=$﻿ Return ﻿$\sigma =$﻿ Risk

﻿$\mu (\Mu, \sigma) = \Mu - \sigma ^2$﻿ (Quasi-Linear)

﻿$MRS = \frac{Mux}{Muy} = \frac{-Mu \sigma}{MuM}= -2 \sigma$﻿

Weight

Budget Line: ﻿$M_B = 10 \%$﻿ ﻿$\sigma _B = 0$﻿ ﻿$B \alpha$﻿ Find ﻿$\alpha$﻿ to

﻿$M_S = 30 \%$﻿ ﻿$\sigma_S = 5 \%$﻿ ﻿$S 1^- \alpha$﻿ ﻿$Max \ u$﻿ Slope: Rise Over Run ﻿$﻿= \frac{\mu _S - \mu_ \sigma}{\sigma _S - 0}$﻿﻿$= u$﻿

"Price of Risk"

O-Cost of Risk is ﻿$4 \times$﻿ return on your portfolio

﻿$M = (1- \infty) M_S + \alpha \ M_B \rightarrow$﻿ Return on portfolio

Variance on Portfolio ﻿$=(1- \alpha )^2 \sigma ^2 _S + \alpha ^2 \cancel {\sigma _B ^2}$﻿ ﻿$+2 \alpha (1- \alpha) \sigma _{BS}$﻿

﻿$\sigma^2 = (1- \alpha)^2 \alpha^2_S \rightarrow \sigma = (1- \alpha) \sigma _S \Rightarrow$﻿ Risk ﻿$MRS =$﻿ O-Cost

﻿$2 \sigma = 4 \rightarrow \sigma = 2$﻿ Portfolio with risk of ﻿$2$﻿

﻿$2=(1- \alpha)\sigma _S$﻿

﻿$1 - \alpha = \frac{2}{5}$﻿ ﻿$\alpha = \frac{3}{5}=60$﻿ Hold ﻿$60 \%$﻿ in stock & 40 in bond ﻿$M = 0.4(10) + 0.6(3)$﻿

﻿$= 4+18 =22 \rightarrow$﻿ Return

What if... ﻿$\phi$﻿: ﻿$u = M - \sigma^2$﻿ ﻿$\mu_B = 2$﻿ ﻿$\sigma_B = 0$﻿ ﻿$\alpha$﻿

﻿$\mu _S = 10$﻿ ﻿$\sigma _S = 4$﻿ ﻿$1- \alpha$﻿

a) Find bundle ﻿$A$﻿

b) Financial reforms reduce volatility so ﻿$\sigma _S = 2.5$﻿ SE & IE illustrate Uncertainty: ﻿$G(100, \ 0 \ : \ 0.25, \ 0.75)$﻿ Max willing to pay for this gamble?

Evalue ﻿$= 0.25 \times 100 +0.75 \times 0 = \ 25$﻿ (some would pay, some not)

Risk Averse - Avoid risk (insure) ﻿$WTP < \ 25$﻿

Risk Lover - Casino, Binge ﻿$WTP > \ 25$﻿

Risk Neutral - Statisticians ﻿$WTP = \ 25$﻿

Averse: Diminishing ﻿$Mu$﻿ e.g. ﻿$\sqrt{x}, \ \ln x$﻿  Certainty Equivalent: What amount hold with certainty ﻿$=$﻿ gamble?

﻿$\sqrt{x}= 2.5 \rightarrow x = \ 6.25$﻿ ﻿$\Rightarrow$﻿ Max willing to pay ﻿$EU = 0.25 \times 100^2 + 0.75 \times 0^2 = 2500$﻿ utils

﻿$2500 = x^2 \Rightarrow x = 50$﻿

Max willing to pay even though ﻿$EV = 25$﻿

Syndicates: (investors working together to share the risk)

﻿$u = x^{\frac{1}{2}}$﻿ wealth ﻿$= 225$﻿ ﻿$u(225)=15$﻿ (doing nothing)

Opportunity to invest ﻿$200$﻿. ﻿$40 \%$﻿ chance of good ﻿$\rightarrow$﻿ ﻿$+600$﻿

﻿$60 \%$﻿ chance of bad ﻿$\rightarrow +0$﻿

﻿$G(225-200+600, \ 25: \ 0.4, \ 0.6)$﻿

﻿$EU = 0.6 \sqrt{25}+0.4 \sqrt{625}$﻿

﻿$=0/6 \cdot 5 + 0.4 \cdot 25 - 3 + 10$﻿

﻿$=13 \neq 15 \Rightarrow$﻿ No Invest

b) ﻿$n=4$﻿ ﻿$G(225- \frac{200}{4}+ \frac{600}{4}, \ 175: \ 0.4, \ 0.6) \Rightarrow ﻿$﻿ Joining doesn't change the probability of outcomes

﻿$EU = 0.6 \sqrt{175} + 0.4 \sqrt{325}= 15.148 > 15 \Rightarrow ﻿$﻿ Invest

Risky Jobs:

Safe Job ﻿$w = \ 2000$﻿

Risky Job ﻿$25 \%$﻿ getting hurt. Hospital ﻿$= \ 10$﻿

﻿$u = 20 - H - \frac{10000}{w}$﻿ What is the lowest willing to accept?

﻿$EU = u$﻿ Safe: ﻿$H = 0$﻿ ﻿$w=2000$﻿

﻿$u=20-5=15$﻿

Risky: ﻿$u = 0.25 [20-10 - \frac{10000}{w}]+ 0.75 [20-0-\frac{10000}{w}]]$﻿

﻿$=17.5 -\frac{10000}{w}$﻿

﻿$u_{in} = \ 4000$﻿

More Portfolios: ﻿$EV_1 = 0.6(10)+0.4(8) = 9.2$﻿ ﻿$EU_1 = 0.6 \ln 10 +0.4 \ln 8 = 2.21$﻿

﻿$EV_2 = 0.6(12)+0.4(6) = 9.6$﻿ ﻿$EU_1 = 0.6 \ln 12 +0.4 \ln 6 = 2.2$﻿

Key Question - Higher return enough to compensate risk? No

c) What fraction of wealth to hold in which?

﻿$EU = 0.6 \ln (10 \alpha + 12 (1 - \alpha))+ 0.4 \ln (8 \alpha + 6 (1 - \alpha))$﻿

﻿$= 0.6 \ln (12-2 \alpha) + 0.4 \ln (2 \alpha + 6)$﻿

Says: Put ﻿$60 \%$﻿ in stock, ﻿$40 \%$﻿ in bond

Insurance: ﻿$u = x^{\frac{1}{2}}$﻿ What is Max WTP for insurance? ﻿$E_{loss} = 0.2 + 300 = 60$﻿

﻿$EU = 0.2(100)^{\frac{1}{2}}+ 0.8(400)^{\frac{1}{2}}$﻿

﻿$= 2+16 = 18$﻿

With Insurance Fee: Certainty equivalent

﻿$\sqrt{400-fee}= 18$﻿ ﻿$u = Eu$﻿

﻿$400-fee = 18 ^2$﻿

﻿$Fee = \ 76 > 60 \Rightarrow$﻿ Risk Averse

Budget Lines:

﻿$MRS =$﻿ o-cost  Budget: ﻿$0.2 C_b + 0.8 C_g = 340﻿$﻿

TAN: ﻿$EU = 0.8 C_g^{1/2} +0.2 C_b ^{1.2}$﻿

﻿$MRS = \frac{MUb}{MUg} = \frac{0.1 cb ^{-1/2}}{0.4 cg ^{-1/2}} = \frac{1}{4} (\frac{cg}{cb})^{1/2}$﻿

Slope ﻿$BL$﻿ is ﻿$\frac{1}{4} = \frac{Px}{Py} = \frac {Pb}{Pg} = \frac{0.2}{0.8}=\frac{1}{4}﻿$﻿

TAN: ﻿$MRS =$﻿ o-cost ﻿$\rightarrow \frac{1}{4} (\frac{Cg}{Cd})^{1/2} \rightarrow Cg=Cb$﻿

Sub: ﻿$0.2 Cb + 0.8 Cb =340$﻿

﻿$Cb = 340, \ Cg = 340$﻿

﻿$400-340 = 60$﻿ ﻿$340 - 100 = 240$﻿ Fair insurance was ﻿$\ 60$﻿

We gave ﻿$\ 60$﻿ of good ﻿$y$﻿ and got ﻿$\ 240$﻿ good ﻿$x$﻿. ﻿$\frac{60}{\frac{1}{4}}$﻿﻿$=240$﻿

﻿$Max$﻿ willing to pay is ﻿$\ 76$﻿ green budget line. How big of a shift would leave you in different from point of no insurance.