ECN 134SOLUTION KEY #11. (i) μ = .06 and σ = .10 : Find P (X<0) = P¡Z<0−.06.10¢= P (Z<−.6) .Turning to the normal tables we find that the probability associated with thecritical value of −.6is27.43%. That is 27.43% of the time, and thus approximately13 years , the return on corporate bonds was less than 0%.(ii) μ = .11 and σ = .16 : Find out what the rate of return is for the worst fiveyears of the S&P. That is: what was the rate of return for the worst 10% of the last48 years. We must solve for a : .10 = P (X<a)=P¡Z<a−.11.16¢. Turning to thenormal tables we find that the critical value leading to a .10 outcome is −1.28 . Thusfind a such that−1.28 =a − .11.16a = −.09This tells us (approximately) that in the worst 5 years of the last 48 years the S&Plost9% or more .2. r = 0.12.i) NPV = -75,000 + 140,000/(1+r)5= -75,000 + 79,439.76 = 4,439.76 .ii)Yes , since Venkatesh can borrow against the return from selling the land.iii) Keeping future consumption unchanged means Venkatesh borrows against theentirety of his return, i.e., he borrows 140,000/(1+0.12)5= 79,439.76 at t=0. SoC0=Y0- 75,000 + 79,439.76 = Y0+ 4439.76. The effect on C0is 4439.76 =NPV.iv) Venkatesh is willing to pay up to the PV of his returns on the land, or $79,439.8 .13. Work (39,39) → PVW= 39+39/(1+r);Law school (23,65) → PVL= 23 + 65/(1+r).Note that if the budget constraint is linear, Hang-Ro will allocate half of PV toeach periods consumption.Situation I: r = 30%. Hence, slope = -(1+r) = -1.3 and PVW= 69, PVL=73.Her optimal consumption plan is;Work: C0=69/2 = 34.5, C1= (34.5) (1.3) = 44.9 →Lending=39-34.5=4.5.Law: C0= 73/2 = 36.5, C1= (36.5) (1.3) = 47.5 →Borrowing = 36.5 -23 = 13.5.Since the law school budget constraint is everywhere outside the work budget con-straint, Hang-Ro will clearly c h oose La w sc hool, independent of her preference.Situation II:r=50%.Henceslope=-1.5andPVW=65,PVL= 66.3.i) In this situation, we need to know preferences. If preferences were different, shemigh t choose L ov er W.ii) Work: If borrowing was unrestricted, thenC0= 65/2 = 32.5 , C1=32.5×1.5 = 48.8.Thispointisinfactontherelevantportionoftheconstraint,andHang-Rowillcon-sume at (32.5, 48.8), lending 39.5- 32.5 = 6.5, and attain a utility level of 32.5×48.8=1586 .Law: If borrowing were unrestricted, thenC0= 66.3/2 = 33.2, C1= 33.2 × 1.5 = 49.8.Unfortunately, this point is infeasible. She will end up at the kink, borrowing 3 att=0, and consuming C0=26,C1=60.5, thus enjoying a utility of 26×60.5= 1573 .Now, she is slightly better off working right a
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