Test 1 Solution sketches Econ 134A Test 1 Solution sketches

Level of difficulty On multiple choice questions… “Easy” denotes that about 80-100% of students get this question correct “Medium” denotes that about 60-80% of students get this question correct “Hard” denotes that about 40-60% of students get this question correct

Level of difficulty On Problems… I take a 10% sample “Easy” denotes that the average score is 80-100% of the points possible “Medium” denotes that the average score is 60-80% of the points possible “Hard” denotes that that average score is 40-60% of the points possible

Today is Oct. 27, ’11 You invest $2,500 today Find FV on July 27, 2013, 14% SAIR, compounded every three months Quarterly interest rate is 3.5% Difference between the two dates 1.75 years, or 7 quarters FV = $2500(1.035)7 = $3180.70 Easy problem

Today is Oct. 27, ’11 You invest $2,500 today Find FV on Oct. 27, 2041, 2.31% SAIR, compounded continuously Difference between the two dates 30 years To compound continuously… $2500*exp(0.0231*30) = $4,999.26 Easy problem

Today is Oct. 27, ’11 You invest $2,500 today Find FV on April 27, 2019, 5% SAIR, compounded every 30 months Compounding occurs every 30 months, or 2.5 years Add 12.5% interest every 2.5 years Difference between the two dates 7.5 years 3 periods of 2.5 years FV = $2,500(1.125)3 = $3,559.57 Hard problem

Adelle Samuelson, $35,000 real payment 7 years from now Inflation is 3% per year for three years Inflation is 4% per year for four years Nominal payment is $35,000(1.03)3(1.04)4 = $44,741.76 Hard problem

Heath Wells deposits $10,000 in a bank, 9% yearly interest Money in the bank for 5 years How much MORE interest is earned if compounded monthly, relative to simple interest? Compounded monthly $10,000(1.0075)60 = $15,656.81 Simple interest $10,000(1.45) = $14,500 Difference is $1,156.81 Medium problem

Biliana Marks and the will: $3,000 per year forever… …starting 6 months from now EAIR is 8% A perpetuity that pays $3,000 per year forever STARTING ONE YEAR FROM NOW has PV of $3,000 / 0.08 = $37,500 We have to multiply the above value by sqrt(1.08) in order to value each payment six months earlier: $37,500 (1.0392) = $38,971.14 Hard problem

LA Purchase: $15M in 1803; $219M in 2010 with inflation If annual inflation was the same every years over a 207-year period, what is yearly inflation rate? All numbers below in millions of dollars 15(1+r)207 = 219 (1+r)207 = 219/15 1+r = 1.0130  r = 1.30% Easy-medium problem

Rosie buys a robot, $1,000 today; $100 maintenance cost in 1 yr What is EAC if discount rate is 4%/yr.? PV of costs $1,000 + $100/1.04 = $1,096.15 If EAC is X, then X/1.04 + X/1.042 = $1,096.15 1.8661X = $1,096.15 X = $581.17 Medium problem You could also have used the annuity formula here

Taeil Smith deposits $500 today: How long until it grows to $32K? T = log1.08 64 T = log 64/log 1.08 = 54.04 Easy problem

Summary of MC problems 3 easy problems 1 easy-medium problem 2 medium problems 3 hard problems

Problem: You invest $500 today You get back… $214.70 in one year $350.30 in two years (a) What is NPV if EAIR is 15% -500 + 214.7/1.15 + 350.3/1.152 -$48.43 Easy problem

Invest $500 today, get back $214.70 & $350.30 (b) Based on the answer in (a), is the annual IRR >, <, or = to 15% It must be less than 15% Since all future cash flows have positive value, a discount rate that falls will lead to a higher NPV (since these future payments are discounted less) Medium problem

Invest $500 today, get back $214.70 & $350.30 (c) Calculate the annual IRR -500 + 214.7/(1+r) + 350.3/(1+r)2 = 0 There are multiple ways to solve this; I like to solve it by first multiplying both sides by (1+r)2 -500(1+r)2 + 214.7(1+r) + 350.3 = 0 -500r2 – 785.3r + 65 = 0 Use the quadratic formula on the formula sheet to get r = 0.07882 or -1.6494 IRR = 7.882% Hard problem

Invest $500 today, get back $214.70 & $350.30 (d) Suppose that you receive an additional payment in 3 years How much does this payment need to be in order for the NPV of the project to be $100? (Call this payment X in calculations below) -500 + 214.7/1.2 + 350.3/1.22 + X/1.23 = 100 -77.82 + X/1.23 = 100 X/1.23 = 177.82 X = $307.27 Medium problem

Problem: Junk bond with 4 coupon payments of $500 each What is PV of this bond if effective annual discount rate is 6% for 1st 2 years and 14% thereafter? 500 + 500/1.06 + 500/1.062 + 500/[(1.06)2(1.14)] $1,807.05 Medium problem

Problem: Sammy Waffle wins the Wacky Lottery 10 payments $10,000 two years, four years, and six years from today $13,000 eight years from today Payments made in years 10, 12, 14, 16, 18, and 20 (each 3% higher than the previous payment) What is the PV of these payments if the effective annual discount rate is 5%? Medium problem

Sammy Waffle wins the Wacky Lottery Payments in… Yr. 10: $13,390 Yr. 12: $13,791.70 Yr. 14: $14,205.45 Yr. 16: $14,631.61 Yr. 18: $15,070.56 Yr. 20: $15,522.68

Sammy Waffle wins the Wacky Lottery Present value of payments 2 years from today $9,070.29 4 years from today $8,227.02 6 years from today $7,462.15 8 years from today $8,798.91 10 years from today $8,220.30 12 years from today $7,679.73 14 years from today $7,174.72 16 years from today $6,702.91 18 years from today $6,262.13 20 years from today $5,850.33 Total PV of the 10 payments $75,448.50 You can also use the annuity formula to calculate the PV of the last 6 or 7 payments

Summary of problems 2 points of easy problems 24 points of medium problems 6 points of hard problems

Grade distribution Note that the curve is set at the end of the quarter You can look at the syllabus to see what the likely distribution of grades is Even though there are 62 points possible, divide your score by 61 to get your grade Remember that you drop your lowest grade For those of you with low grades, Test 1 may be the score you drop

Grade distribution Average 43.40 points 43.40/61 = 71.1% THE GRADE DISTRIBUTION BELOW IS MEANT TO BE AN APPROXIMATE GUIDELINE Do not use this as an absolute guide for determining your grade

Grade distribution 99th percentile 80th percentile 60th percentile 61 points This is what you divide your score by to determine your grade Those with 62 points will get 100% and be bumped up if on the border of two grades at the end of the quarter 80th percentile 53 points Most students w/53+ points are approximately in the A+ to B+ range 60th percentile 48 points Most students between 48-52 points are approximately in the B+ to B range

Grade distribution 50th percentile (median) 40th percentile 44.5 points The median student at the end of the quarter will likely get a B or B- 40th percentile 42 points Someone here is approximately at a B- or C+ 20th percentile 35 points Students with 35 points or below are in danger to get a C- or below for the class if there is no improvement on the other two tests