Test 2 solution sketches

Note for multiple-choice questions: Choose the closest

answer

Variable Dividends Natalie buys a stock that pays a $5

dividend today and pays subsequent dividends every year. The dividend will go up by 9% each of the next 3 years, and will go up by 3% every year thereafter.

Variable Dividends How much will the dividend be five

years from today? Div0 = $5 Div1 = $5.45 Div2 = $5.9405 Div3 = $6.47515 Div4 = $6.66940 Div5 = $6.86948

Dividend will be $6.87

9% annual growth

3% annual growth

Standard Deviation Three stocks have annual returns of

0.05, 0.1, and 0.15. The standard deviation of this sample is _____. Average = (.05+.1+.15)/3 = .1 Var = ½ * [(.05-.1)2 + (.1-.1)2 +

(.15-.1)2] Var = ½ * [.0025 + 0 + .0025] Var = .0025 S.D. = (.0025)½ = .05 = 5%

Growing Dividends You buy a stock for $72 today. The

stock’s next dividend of $6 will be paid today. Assume that the growth rate (as a percentage) of the yearly dividend is constant forever, and the effective annual discount rate is 10%.

Growing Dividends What is the annual growth rate of

the stock’s dividend? 72 = 6 + 6(1+g)/(.1-g) 66 = 6(1+g)/(.1-g) 6.60 – 66*g = 6 + 6*g 0.6 = 72*g g = 0.00833 = 0.83%

PV of Perpetuity Emily will receive a perpetuity of

$10,000 every six months, starting one year from now. If the effective annual discount rate is 10%, what is the PV of the payments? 6-month rate = (1.1)½ – 1 = 4.88088%

PV of Perpetuity If the perpetuity started in 6

months: PV = 10,000/.0488088 = $204,880

Since it starts in one year: PV = 204,880 – 10,000/1.0488088 PV = $195,346

Or, PV = (10000/.0488) * (1/1.0488)

Growing Annuity An annuity pays $500 annually,

starting today. Each subsequent payment is 10.25% higher than the previous. The final payment is made 5 years from today. What is the PV of this annuity if the stated annual interest rate is 10%, compounded every 6 months?

Growing Annuity EAIR = (1.05)2 – 1 = 10.25%

So EAIR = g PV0 = 500 PV1 = 500 PV2 = 500 PV3 = 500 PV4 = 500 PV5 = 500

Sum of PV = $3,000

Doubling Dividends A stock is expected to pay a $1

dividend one year from today. Each subsequent dividend will be twice the previous payment, and dividends will be paid forever. What is the PV of this stock if the effective annual discount rate is 150%? r=1.5 and g=1 PV = 1 / (1.5 - 1) = 1/.5 = $2

Profitability Index If Martie buys a new machine, she will

spend $500 today. If purchased, the machine will increase future profits for the company as follows: $300 in 5 years, $400 in 8 years, and $500 in 9 years.

What is the profitability index if the effective annual discount rate is 8%? PV of benefits = 300/(1.08)5 + 400/(1.08)8 +

500/(1.08)9 = $670.41 P.I. = 670.41/500 = 1.34

Minimum Standard Deviation Stock N and Stock B are perfectly

positively correlated. Stock N has an expected return of 0.10 and a standard deviation of 0.08. Stock B has an expected return of 0.15 and a s.d. of 0.16. Which of the following could be the minimum s.d. of a portfolio that includes non-negative combinations of these two stocks?

Minimum Standard Deviation

0.070.080.09 0.1 0.110.120.130.140.150.160.170

0.1

0.2

0.3

Series1; 0.1

0.15

Expected Return of Portfolio

S.D. of Portfolio

M.V. point

Geometric Average Return Suppose that $1 invested 100

years ago is worth $5,000 today. What is the geometric average annual return on this investment? 1 * (1 + r)100 = 5,000 r = Geometric avg = (5000/1)1/100 – 1 Geometric avg = 0.0889043 = 8.89%

Henry Fork’s business Henry Fork must invest $1 million

today in a car business. The only positive cash flow from the products he sells will occur in 2 years as follows: there is a 50% chance he will have a $400,000 cash flow and otherwise he will have a $2 million cash flow.

Henry Fork’s business There is also a 20% probability that

Fork’s business will be bought out for a $5 million payment in 5 years. What is the PV of this business if Fork’s effective annual discount rate is 20%? (In $millions) PV = -1 + .5*(.4/1.22) + .5*(2/1.22)

+ .2*(5/1.25) PV = 0.235211 = $235,211

PV-equivalent Payment Streams Jo Pro has a contract to earn $5

million today, $8 million next year, and $10 million in two years. However, she is renegotiating her contract to instead receive 12 monthly payments of $X, starting 3 years from today. The two contracts have the same PV. Find X if the effective annual discount rate is 15%.

PV-equivalent Payment Streams

Monthly rate = (1.15)1/12 – 1 = 1.17149% PV of original contract (in millions)

= 5 + 8/1.15 + 10/1.152 = 19.5180 (Note that we discount by 35 months

because the 1st payment is in 36 months) Annuity calculation:

19.5180 = 1/(1.0117)35 * X/.0117 * [1-1/(1.0117)12]

19.5180 = 7.40660 X = 2.635218 = $2,635,218

Portfolio Expected Return and S.D. Alexander is investing in Blue

Muffin Jeans stock and a risk-free asset. Blue Muffin Jeans could have returns of -5% or 35%, each with 50% probability. The risk-free asset has an expected return of 5%.

Portfolio Expected Return and S.D.: Part (a) If Blue Muffin Jeans stock has a

beta value of 1.5, what is the expected return of a stock with the same beta value as the market portfolio? Expected Jeans return = (.05+.35)/2

= .15 .15 = .05 + 1.5*(RM - .05) .10 = 1.5*RM - .075 .175 = 1.5*RM RM = 11.6667%

Portfolio Expected Return and S.D.: Part (b) What is the standard deviation of a

portfolio comprised of 40% Jeans stock (asset B) and 60% risk-free asset (asset R)?

State Jeans Return

Deviation from exp. return

Risk-free Return

Deviation from exp. return

Product of deviations

Good .35 .2 .05 0 0Bad -.05 -.2 .05 0 0

Covariance is zero!

Portfolio Expected Return and S.D.: Part (b)

Portfolio Variance= XB

2σB2 + 2XBXRσB,R + XR

2σR2

= XB2σB

2 Portfolio s.d. = XBσB XB = .4, XR = .6 σB

2 = ½ * [(.35-.15)2 + (-.05 -.15)2] = 0.04

σB = 0.2 Portfolio s.d. = 0.4*0.2 = 0.08 = 8%

0 0

Internal Rates of Return If Madison invests in the Quizinoa

Gold mine, she will pay $1 million today, she will receive $3 million in two years, and she will pay $2.05 million in four years. What is the annual internal rate of return for this investment? (Hint: you may want to initially calculate using 2 years as your unit of time.)

Internal Rates of Return Let X be rate of return every 2 years 0 = -1 + 3/(1+X) – 2.05/(1+X)2

Simplifies to: 0 = 20X2 – 20X + 1 X = .0527864 or .9472136 Annual IRR:

(1.0527864)½ – 1 = 2.60538%(1.9472136)½ – 1 = 39.5426%