Solution sketchesTest 2, Fall 2011

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

What is the beta of a stock?

Risk-free rate is 6% A stock has an expected return of 14% Expected return on the market is 11% What is beta?

14% = 6% + ß (11% – 6%) ß = 1.6

Easy question

What is the standard deviation?

Sample of a stock’s returns

14%, 5%, 6% Average is 8.33% Squared deviation of

means (.14 - .0833)2 =

0.00321111 (.05 - .0833)2 =

0.00111111 (.06 - .0833)2 =

0.00054444

Sum of the three numbers above, divided by N – 1 (remember that you lose a degree of freedom with a sample)

Variance is 0.00243333 Standard deviation is the

square root of the variance

0.04933 The closest answer is 5% Easy problem

Geometric average What is the geometric average of

annual returns of 5%, 8%, 10%, –4%, and 15%?

Steps 5th root of (1.05)(1.08)(1.1)(0.96)(1.15)

1.066 Subtract 1

6.6%

Easy problem

Carina Corona, real payment of $50K 10 years from now

Inflation will be 8% per year for the first 4 years

Inflation will be 12% per year after the first 4 years

Nominal payment? $50,000(1.08)4(1.12)6 = $134,268

Easy problem

Perpetuity

A perpetuity makes payments of $1,000 every two years, starting one year from today. If the effective annual interest rate is 9%, what is the present value of this perpetuity?

What is the effective rate every 2 years? 1.092 – 1 = 18.81%

Perpetuity

What is a perpetuity worth if $1,000 is paid every two years, starting two years from today? $1,000/.1881 = $5,316.32

The problem tells us that the perpetuity starts in one year, not two We need to increase the value of each

payment

Perpetuity

We need to find out how much this perpetuity is worth if each payment is made one year sooner $5,316.32(1.09) = $5,794.79

Very hard problem Only about 25% of students got this

correct

IRR problem If I invest $1,000 today in a project and

receive $1,800 five years from today in return, my annual internal rate of return is _____.

For what annual discount rate will the PV of the costs and benefits be equal?

$1,000 (1 + IRR)5 = $1,800 1 + IRR = 1.1247 IRR = 12.47%

Easy problem

You loan a friend $500 Suppose that you loan your friend

$500 and the effective annual interest rate that is agreed on is 12%. Your friend will pay you back $60 per year for the first nine years (starting one year from today). If the final payment will be made 10 years from today, how much will this payment have to be to completely pay back the loan?

You loan a friend $500 Each of the first nine years, you only

pay back the interest (12% of $500), or $60. The balance does not change from year to

year Final payment: $560 total

Interest = 12% of $500, or $60 Pay off the balance: $500

Medium problem

Payback period method In your first job, suppose that you boss tells you

to use the payback period method (undiscounted), with the cutoff date 3 years, 3 months from now. In other words, the payback period is 3 years, 3 months. The effective annual discount rate is 6%. Which of the following offers should you accept if you use this method?

$1,400 every two years forever, with the first payment made two years from today

$40,000 every year forever, with the first payment made four years from today

$450 per year forever, with the first payment made today

$500 per year forever, with the first payment made one year from now

A one-time payment of $1,000,000 five years from today

Payback period method

If we only look at the first 3 years, 3 months, how much is each offer worth? No discounting used in the calculations

Hard problem Not meant to be a hard problem Make sure that you understand the

terminology used in the book and in lectures

Payback period method:How much after 3¼ years? $500 per year forever, starting in 1 year

3 payments: $1,500 A one-time payment of $1M in 5 years

0 payments: $0 $1,400 every 2 years, starting in 2 years

1 payment: $1,400 $40K per year forever, starting in 4

years 0 payments: $0

$450 per year forever, starting today 4 payments: $1,800

Summary of MC questions

5 easy questions 1 medium question 1 hard question 1 very hard question

Problem: Jackie Jackson Jackie Jackson has won the final round of a new fictional

game show, “Pay What You Can Afford.” Jackie gets to take out a 30-year, fixed-rate mortgage for $500,000. The stated annual interest rate of the mortgage is 6%, and is compounded monthly. Jackie makes 360 monthly payments of $1,600 per month, starting one month from today. The producers of the show make an arrangement with the mortgage company so that an additional payment is made 60 months (or 5 years) after Jackie takes out the mortgage. How much will this payment be if we assume that Jackie makes all 360 monthly payments and that the balance of the loan will be zero after 30 years? Round your answer to the nearest dollar.

Problem: Jackie Jackson

Monthly r = 0.005 = 0.5% PV of 360 monthly payments

($1600/.005) * [1 – 1/(1.005)360] $320,000 (0.833958) $266,866.58

Problem: Jackie Jackson

PV of additional payment $500,000 – $266,866.58 $233,133.42

FV of additional payment (in 5 years) $233,133.42(1.005)60

$314,462 Easy-medium problem

Problem: Prince Pitcher Prince Pitcher is interested in buying a stock. In

order to determine if this stock is worth buying, he assumes an annual effective discount rate of 10%. After a thorough analysis of the stock, he assumes that the company will pay its next dividend of $1 per share one year from today. For two years after the next dividend payment, the dividend will increase by 15.5% per year. After that, the dividend will increase by 8% per year. Based on these assumptions, what is the present value of each share of this stock? Round your answer to the nearest cent.

Problem: Prince Pitcher PV of payment in year 1

1 / 1.1 = $0.909 PV of payment in year 2

1.155 / 1.12 = $0.955 PV of payment in year 3

1.1552 / 1.13 = $1.002 PV of payment in years 4 and beyond

(1.1)-3 * [(1.155)2(1.08)/(0.10 – 0.08) = $54.123

Discount by 3 years

Dividend 4 years from today

Total of all payments in PV terms: $56.99

Easy-medium problem

Problem: 92 days of interest

Today is November 17, 2011. If you deposit $10,000 in the bank today and the stated annual interest rate is 7.3%, how much interest will you earn after 92 days? (Assume 365 days per year in your calculation.) Round your answer to the nearest cent.

Problem: 92 days of interest Daily interest rate

7.3% / 365 = 0.0002 = 0.02% Not 0.2% (common mistake)

Total balance after 92 days $10,000(1.0002)92 = $10,185.68

Interest earned $10,185.68 - $10,000 = $185.68

Easy-medium problem

Problem: Stock A Stock A has an expected annual return of 5% and a standard deviation of

10%. Stock B has an expected annual return of 15% and a standard deviation of 20%. Draw a graph with “expected return on portfolio” on the vertical axis and “standard deviation of portfolio’s return” on the horizontal axis.

(a) (3 points) Make sure that the following are well labeled, including indicating numerical values on each axis for each of the following points: Put an “A” by the point on the graph for Stock A and a “B” by the point on the graph for Stock B. (Hint: You may want to see the graph on the cover sheet to see that it is well labeled.)

(b) (3 points) Draw a line or curve to denote possible expected return-standard deviation combinations of all of the possible portfolios with positive amounts of both Stock A and Stock B. Assume that the correlation between Stock A and Stock B is –0.5. (Note: Be careful how you draw this. However, I am only looking for the shape of the line or curve, so it does not need to be exact.)

(c) (1 point) Label the minimum variance point MV. (Note that you do not necessarily need to calculate the exact point of minimum variance.)

(d) (3 points) Clearly show or explain what the efficient set is. (Note: The efficient set is also referred to as the efficient frontier.)

Problem: Stock A

std. dev. of portfolio’s return

expected return on portfolio

15%

5%

10% 20%

A

B

MV

(d) The points between MV and B are the efficient set

(a): Easy problem

(b): Medium problem

(c): Medium problem

(d): Hard problem

Summary of Problems

3 points of Easy problems 21 points of Easy-Medium

problems 4 points of Medium problems 3 points of Hard problems