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

More Budget Lines:

1) Mean-Variance Utility (Financial Economics)

MM == Return σ=\sigma = Risk

μ(M,σ)=Mσ2\mu (\Mu, \sigma) = \Mu - \sigma ^2 (Quasi-Linear)

MRS=MuxMuy=MuσMuM=2σMRS = \frac{Mux}{Muy} = \frac{-Mu \sigma}{MuM}= -2 \sigma

Weight

Budget Line: MB=10%M_B = 10 \% σB=0\sigma _B = 0 BαB \alpha  Find α\alpha to

MS=30%M_S = 30 \% σS=5%\sigma_S = 5 \% S1αS 1^- \alpha  Max uMax \ u

Slope: Rise Over Run =μSμσσS0 = \frac{\mu _S - \mu_ \sigma}{\sigma _S - 0}=u= u

"Price of Risk"

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


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

Variance on Portfolio =(1α)2σS2+α2σB2=(1- \alpha )^2 \sigma ^2 _S + \alpha ^2 \cancel {\sigma _B ^2} +2α(1α)σBS+2 \alpha (1- \alpha) \sigma _{BS}

σ2=(1α)2αS2σ=(1α)σS\sigma^2 = (1- \alpha)^2 \alpha^2_S \rightarrow \sigma = (1- \alpha) \sigma _S \Rightarrow Risk


MRS=MRS =  O-Cost

2σ=4σ=22 \sigma = 4 \rightarrow \sigma = 2 Portfolio with risk of 22

2=(1α)σS2=(1- \alpha)\sigma _S

1α=251 - \alpha = \frac{2}{5} α=35=60\alpha = \frac{3}{5}=60 Hold 60%60 \% in stock & 40 in bond

M=0.4(10)+0.6(3)M = 0.4(10) + 0.6(3)

=4+18=22= 4+18 =22 \rightarrow Return


What if... ϕ\phi: u=Mσ2u = M - \sigma^2 μB=2\mu_B = 2  σB=0\sigma_B = 0  α\alpha

μS=10\mu _S = 10  σS=4\sigma _S = 4 1α1- \alpha


a) Find bundle AA

b) Financial reforms reduce volatility so σS=2.5\sigma _S = 2.5 SE & IE illustrate

Uncertainty:

G(100, 0 : 0.25, 0.75)G(100, \ 0 \ : \ 0.25, \ 0.75) Max willing to pay for this gamble?

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

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

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

Risk Neutral - Statisticians WTP=$25WTP = \$ 25

Averse: Diminishing MuMu e.g. x, lnx\sqrt{x}, \ \ln x

Certainty Equivalent: What amount hold with certainty == gamble?

x=2.5x=$6.25\sqrt{x}= 2.5 \rightarrow x = \$ 6.25  \Rightarrow Max willing to pay


EU=0.25×1002+0.75×02=2500EU = 0.25 \times 100^2 + 0.75 \times 0^2 = 2500 utils

2500=x2x=502500 = x^2 \Rightarrow x = 50

Max willing to pay even though EV=25EV = 25

Syndicates: (investors working together to share the risk)

u=x12u = x^{\frac{1}{2}} wealth =225= 225 u(225)=15u(225)=15 (doing nothing)

Opportunity to invest 200200. 40%40 \% chance of good \rightarrow +600+600

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

G(225200+600, 25: 0.4, 0.6)G(225-200+600, \ 25: \ 0.4, \ 0.6)

EU=0.625+0.4625EU = 0.6 \sqrt{25}+0.4 \sqrt{625}

=0/65+0.4253+10=0/6 \cdot 5 + 0.4 \cdot 25 - 3 + 10

=1315=13 \neq 15 \Rightarrow  No Invest

b) n=4n=4 G(2252004+6004, 175: 0.4, 0.6)G(225- \frac{200}{4}+ \frac{600}{4}, \ 175: \ 0.4, \ 0.6) \Rightarrow   Joining doesn't change the probability of outcomes

EU=0.6175+0.4325=15.148>15EU = 0.6 \sqrt{175} + 0.4 \sqrt{325}= 15.148 > 15 \Rightarrow   Invest


Risky Jobs:

Safe Job w=$2000w = \$ 2000

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

u=20H10000wu = 20 - H - \frac{10000}{w} What is the lowest willing to accept?

EU=uEU = u  Safe: H=0H = 0  w=2000w=2000

u=205=15u=20-5=15

Risky: u=0.25[201010000w]+0.75[20010000w]]u = 0.25 [20-10 - \frac{10000}{w}]+ 0.75 [20-0-\frac{10000}{w}]]

=17.510000w=17.5 -\frac{10000}{w}

uin=$4000u_{in} = \$ 4000


More Portfolios:

EV1=0.6(10)+0.4(8)=9.2EV_1 = 0.6(10)+0.4(8) = 9.2 EU1=0.6ln10+0.4ln8=2.21EU_1 = 0.6 \ln 10 +0.4 \ln 8 = 2.21

EV2=0.6(12)+0.4(6)=9.6EV_2 = 0.6(12)+0.4(6) = 9.6 EU1=0.6ln12+0.4ln6=2.2EU_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=0.6ln(10α+12(1α))+0.4ln(8α+6(1α))EU = 0.6 \ln (10 \alpha + 12 (1 - \alpha))+ 0.4 \ln (8 \alpha + 6 (1 - \alpha))

=0.6ln(122α)+0.4ln(2α+6)= 0.6 \ln (12-2 \alpha) + 0.4 \ln (2 \alpha + 6)

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


Insurance:

u=x12u = x^{\frac{1}{2}} What is Max WTP for insurance?

Eloss=0.2+300=60E_{loss} = 0.2 + 300 = 60

EU=0.2(100)12+0.8(400)12EU = 0.2(100)^{\frac{1}{2}}+ 0.8(400)^{\frac{1}{2}}

=2+16=18= 2+16 = 18

With Insurance Fee: Certainty equivalent

400fee=18\sqrt{400-fee}= 18 u=Euu = Eu

400fee=182400-fee = 18 ^2

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


Budget Lines:

MRS=MRS =  o-cost

Budget: 0.2Cb+0.8Cg=3400.2 C_b + 0.8 C_g = 340 

TAN: EU=0.8Cg1/2+0.2Cb1.2EU = 0.8 C_g^{1/2} +0.2 C_b ^{1.2}

MRS=MUbMUg=0.1cb1/20.4cg1/2=14(cgcb)1/2MRS = \frac{MUb}{MUg} = \frac{0.1 cb ^{-1/2}}{0.4 cg ^{-1/2}} = \frac{1}{4} (\frac{cg}{cb})^{1/2}

Slope BLBL is 14=PxPy=PbPg=0.20.8=14\frac{1}{4} = \frac{Px}{Py} = \frac {Pb}{Pg} = \frac{0.2}{0.8}=\frac{1}{4} 

TAN: MRS=MRS =  o-cost 14(CgCd)1/2Cg=Cb\rightarrow \frac{1}{4} (\frac{Cg}{Cd})^{1/2} \rightarrow Cg=Cb

Sub: 0.2Cb+0.8Cb=3400.2 Cb + 0.8 Cb =340

Cb=340, Cg=340Cb = 340, \ Cg = 340

400340=60400-340 = 60 340100=240340 - 100 = 240 Fair insurance was $60\$ 60

We gave $60\$ 60 of good yy and got $240\$ 240  good xx. 6014\frac{60}{\frac{1}{4}}=240=240


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





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