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Econ 134A

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Econ 134A

Test 1

Solution sketches

- 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

- 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

- 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

- 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

- Difference between the two dates
- Easy problem

- 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

- Compounding occurs every 30 months, or 2.5 years
- Hard problem

- 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

- 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

- Compounded monthly
- Medium problem

- …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

- 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

- 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

- 500 (1.08)T = 32,000
- (1.08)T = 64
- T = log1.08 64
- T = log 64/log 1.08 = 54.04
- Easy problem

- 3 easy problems
- 1 easy-medium problem
- 2 medium problems
- 3 hard problems

- 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

- (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

- (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

- (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

- 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

- 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

- 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

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

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

- 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

- 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

99th 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

50th percentile (median)

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