1. Test 1 Solution sketches
Econ 134A
Test 1
Solution sketches

2. Level of difficulty On multiple choice questions…
“Easy” denotes that about % 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

3. Level of difficulty On Problems… I take a 10% sample
“Easy” denotes that the average score is % 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

4. 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 = $
Easy problem

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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 =  r = 1.30%
Easy-medium problem

11. 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/ 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

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

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

14. Problem: You invest $500 today
You get back…
$ in one year
$ in two years
(a) What is NPV if EAIR is 15%
/ /1.152
-$48.43
Easy problem

15. 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

16. Invest $500 today, get back $214.70 & $350.30
(c) Calculate the annual IRR
/(1+r) /(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) (1+r) = 0
-500r2 – 785.3r + 65 = 0
Use the quadratic formula on the formula sheet to get r = or
IRR = 7.882%
Hard problem

17. 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)
/ / X/1.23 = 100
X/1.23 = 100
X/1.23 =
X = $307.27
Medium problem

18. 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?
/ / /[(1.06)2(1.14)]
$1,807.05
Medium problem

19. 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

20. 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

21. 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

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

23. 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

24. 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

25. 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 points are approximately in the B+ to B range

26. 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