1. Zamboni has never paid any dividends. Seven years ago, the firm’s stock price was $58.34 per share.
a. What was the compound annual return over
... [Show More] the past 7 years if the stock’s price is $36.85?
b. What was the compound annual return over the past 7 years if the stock’s price is $72.99?
(Fall 2014, test 4, question 8)
Compound return = (ending value / starting value)(1/n) – 1
Part a
Ending value = today’s stock price = $36.85
Starting value = stock price seven years ago= $58.34
n = number of periods = 7 years
Compound return = (36.85/58.34)(1/7) = (0.6316)(1/7) – 1
= 0.9365 – 1
= -.0635 = -6.35 percent
Note that $58.34 × (0.9365)7 = $36.86 (different than $36.85 due to rounding)
Starting value × (1 + compound return)n
Part b
Ending value = today’s stock price = $72.99
Starting value = stock price seven years ago= $58.34
n = number of periods = 7 years
Compound return = (72.99/58.34)(1/7) = (1.2511)(1/7) – 1
= 1.0325 – 1
= .0325 = 3.25 percent
Note that $58.34 × (1.0325)7 = $72.98 (different than $72.99 due to rounding)
Starting value × (1 + compound return)n
2. A stock had returns of -23.10% (1 year ago), 12.20% (2 years ago), X (3 years ago), and 24.90% (4 years ago) in each of the past 4 years. Over the past 4 years, the arithmetic average annual return for the stock was 1.35%. What was the geometric average annual return for the stock over the past 4 years?
(Fall 2010, test 4, question 8)
(Fall 2010, final, question 15)
(Fall 2011, final, question 16)
(Spring 2012, final, question 17)
(Spring 2014, test 4, question 8)
(Fall 2015, test 4, question 1)
(Fall 2016, test 4, question 1)
(Spring 2017, final, question 15)
To solve:
1) Find X, the return from 3 years ago
2) Find the geometric return over the past 4 years
1) Find X, the return from 3 years ago
Arithmetic annual return = [(r1) + (r2) + … + (rn)] / n
Over the past 4 years, arithmetic annual return = [(r1) + (r2) + (r3) + (r4)] / 4
So .0135 = [(-.2310) + (.1220) + X + (-.2490)] / 4
So .0135 × 4 = [(-.2310) + (.1220) + X + (-.2490)]
So .0540 = [(-.2310) + (.1220) + X + (-.2490)]
= (-.3580) + X
So .0540 + .3580 = X
X = .4120 = 41.20%
2) Find the geometric return over the past 4 years
Geometric annual return = [(1 + r1)(1 + r2) … (1 + rn)](1/n) – 1
Over the past four years, geometric annual return = [(1 + r1)(1 + r2)( 1 + r3)(1 + r4)](1/4) – 1
= [(1 + (-.2310))(1 + .1220)(1 + .4120))(1 + (-.2490))](1/4) – 1
= [(0.7690)(1.1220)(1.4120)(0.7510)](1/4) – 1
= 0.9780 – 1
= -.0220 = -2.20%
3. A stock had returns of 21.70% (1 year ago), 2.40% (2 years ago), X (3 years ago), and 14.60% (4 years ago) in each of the past 4 years. Over the past 4 years, the geometric average annual return for the stock was 2.85%. What was the arithmetic average annual return for the stock over the past 4 years?
(Spring 2011, test 4, question 7)
(Spring 2011, final, question 17)
(Fall 2011, test 4, question 7)
(Fall 2012, test 4, question 7)
(Spring 2013, final, question 18)
(Fall 2013, final, question 16)
(Spring 2015, test 3, question 7)
(Fall 2015, final, question 15)
(Spring 2016, test 4, question 3)
(Spring 2016, final, question 11)
(Spring 2017, test 4, question 3)
(Fall 2017, test 4, question 1)
(Fall 2017, final, question 15)
Approach
1) Find X, the return from 3 years ago
2) Find the arithmetic return over the past 4 years
1) Find X, the return from 3 years ago
[1+ compound return]n = [(1 + r1)(1 + r2) … (1 + rn)]
In this case:
(1 + return 1 year ago) × (1 + return 2 years ago) × (1 + return 3 years ago) × (1 + return 4 years ago)
= (1 + compound return)4
(1 + 0.2170) × (1 + 0.0240) × (1 + X) × (1 + (-0.1460))
= 1.2170 × 1.0240 × (1 + X) × 0.8540
= (1 + 0.0285)4
= (1.0285)4
So (1 + X) = (1.0285)4 / (1.2170 × 1.0240 × 0.8540)
= 1.0514
So X = return 3 years ago = 1.0514 – 1 = 0.0514 = 5.14%
2) Find the arithmetic return over the past 4 years
Arithmetic annual return
= (return 1 year ago + return 2 years ago + return 3 years ago + return 4 years ago) / 4
= (0.2170 + 0.0240 + 0.0514 + (-0.1460)) / 4
= (0.1464 / 4) = 0.0366 = 3.66%
Answers may differ slightly due to rounding
4. What was the real rate of return over the past year (from one year ago to today) for a stock if the inflation rate over the past year was 5.32%, the risk-free return over the past year was 6.17%, the stock is currently priced at $48.75, the stock was priced at $46.12 one year ago, and the stock just paid a dividend of $2.23?
(Fall 2012, test 4, question 8)
(Spring 2013, test 4, question 8)
(Fall 2016, final, question 15)
To solve:
1) Find the nominal return over the past year for the stock
2) Find the real rate of return
1) Find the nominal return over the past year for the stock
The nominal return over the past year is the percentage return over the past year
Percentage return over past year
= (dividends paid in period + ending value – initial value) / initial value
= (dividends paid during year + stock price today – stock price 1 year ago) / stock price 1 year ago
= (2.23 + 48.75 – 46.12) / 46.12
= (2.23 + 2.63) / 46.12
= 4.86 / 46.12
= .1054 = 10.54%
2) Find the real rate of return
Real rate
= [(1+nominal rate) ÷ (1+inflation rate)] – 1
= [(1 + .1054) ÷ (1 + .0532)] – 1
= [1.1054 / 1.0532] – 1
= .0496 = 4.96%
5. Over the past year (from one year ago to today), the inflation rate was 4.13%, the risk-free rate was 6.08%, and the real rate of return for a bond was 3.17%. The bond is currently priced at $974.00, pays annual coupons of $84.70, and just made a coupon payment. What was the price of the bond one year ago?
(Fall 2013, test 4, question 9)
(Spring 2017, test 4, question 4)
To solve:
1) Find the nominal return for the bond over the past year
2) Find the price of the bond one year ago
1) Find the nominal return for the bond over the past year
Nominal rate
= [(1 + real rate) × (1 + inflation rate)] – 1
= [(1 + .0317) × (1 + .0413)] – 1
= [(1.0317) × (1.0413)] – 1
= 0.0743 = 7.43%
2) Find the price of the bond one year ago
Percentage return over past year
= (coupons paid in period + ending value – initial value) / initial value
= (coupons paid during year + bond price today – bond price 1 year ago) / bond price 1 year ago
So .0743 = (84.70 + 974.00 – bond price 1 year ago) / bond price 1 year ago
= .0743 = (1,058.70 – bond price 1 year ago) / bond price 1 year ago
So (.0743 × bond price 1 year ago) = 1,058.70 – bond price 1 year ago
So (.0743 × bond price 1 year ago) + bond price 1 year ago = 1,058.70
So (1.0743 × bond price 1 year ago) = 1,058.70
So bond price 1 year ago = 1,058.70 / 1.0743 = $985.48
Confirm:
Percentage return over the past year
= (cash flows from investment + ending value – initial value) / initial value
= (84.70 + 974.00 – 985.48) / 985.48
= 73.22 / 985.48
= .0743
Real rate = [(1+nominal rate) ÷ (1+inflation rate)] – 1
If nominal = .0743 and inflation = .0413
Then real rate = [1.0743 ÷ 1.0413] – 1
= .0317
6. Over the past year (from 1 year ago to today), the inflation rate was 4.17%, the risk-free rate was 6.08%, and the real rate of return for a bond was 3.27%. The bond was priced at $975.00 one year ago and $980.00 two years ago, pays annual coupons of $85.00, and just made a coupon payment. What is the price of the bond today?
(Fall 2011, test 4, question 8)
(Fall 2014, test 4, question 9)
(Spring 2015, final, question 19)
To solve:
1) Find the nominal return for the bond over the past year
2) Find the price of the bond today
Note: Since the relevant period is the past year (from 1 year ago to today), the price from 2 years ago is irrelevant.
1) Find the nominal return for the bond over the past year
(1 + nominal rate) = (1 + real rate) × (1 + inflation rate)
Nominal rate
= [(1 + real rate) × (1 + inflation rate)] – 1
= [(1 + .0327) × (1 + .0417)] – 1
= [(1.0327) × (1.0417)] – 1
= 0.0758 = 7.58%
2) Find the price of the bond today
Percentage return over past year = (coupons paid in period + ending value – initial value) / initial value
= (coupons paid during year + bond price today – bond price 1 year ago) / bond price 1 year ago
So .0758 = (85.00 + bond price today – 975.00) / 975.00
= (bond price today – 890.00) / 975.00
So (.0758 × 975.00) = bond price today – 890.00
So bond price today = (.0758 × 975.00) + 890.00
= $963.91
Confirm:
Percentage return over the past year
= (cash flows from investment + ending value – initial value) / initial value
= (85.00 + 963.91 – 975.00) / 975.00
= 72.44 / 975.00
= .0758
Real rate = [(1+nominal rate) ÷ (1+inflation rate)] – 1
If nominal = .0758 and inflation = .0417
Then real rate = [1.0758 ÷ 1.0417] – 1
= .0327
7. A stock had returns of 21.70% (1 year ago), 2.40% (2 years ago), X (3 years ago), and 14.60% (4 years ago) in each of the past 4 years. Over the past 4 years, the geometric average annual return for the stock was 2.85%. Three years ago, inflation was 3.62% and the risk-free rate was 4.47%. What was the real return for the stock 3 years ago?
(Spring 2012, test 4, question 8)
(Fall 2014, final, question 17)
(Spring 2018, test 4, question 3)
Approach
1) Find X, the return from 3 years ago
2) Find the real return from 3 years ago
1) Find X, the return from 3 years ago
[1+ compound return]n = [(1 + r1)(1 + r2) … (1 + rn)]
In this case:
(1 + return 1 year ago) × (1 + return 2 years ago) × (1 + return 3 years ago) × (1 + return 4 years ago)
= (1 + compound return)4
(1 + 0.2170) × (1 + 0.0240) × (1 + X) × (1 + (-0.1460))
= 1.2170 × 1.0240 × (1 + X) × 0.8540
= (1 + 0.0285)4
= (1.0285)4
So (1 + X) = (1.0285)4 / (1.2170 × 1.0240 × 0.8540)
= 1.0514
So X = return 3 years ago = 1.0514 – 1 = 0.0514 = 5.14%
2) Find the real return from 3 years ago
Real rate = [(1+nominal rate) ÷ (1+inflation rate)] – 1
If nominal = .0514 and inflation = .0362
Then real rate = [1.0514 ÷ 1.0362] – 1
= .0147 = 1.47%
8. A stock had returns of 10.00%, 3.00%, and -7.00% over the past 3 years. What is the mean of the stock’s returns over the past 3 years minus the sample standard deviation of the stock’s returns from the past 3 years?
Approach: Find the mean and then find the standard deviation and then subtract
Mean return = [(r1) + (r2) + … + (rn)] / n
In this case, mean return = [(r1) + (r2) + (r3)] / 3
= [.1000 + .0300 + (-.0700)] / 3
= .0200
Var(R) = {[1/(n-1)] × [R1 – mean(R)]2} + {[1/(n-1)] × [R2 – mean(R)]2} + … + {[1/(n-1)] × [Rn – mean(R)]2}
In this case:
Var(R) = {[1/2] × [R1 – mean(R)]2} + {[1/2] × [R2 – mean(R)]2} + {[1/2] × [R3 – mean(R)]2}
= {[1/2] × [.1000 – .0200]2} + {[1/2] × [.0300 – .0200]2} + {[1/2] × [-.0700 – .0200]2}
= {[1/2] × [.0800]2} + {[1/2] × [.0100]2} + {[1/2] × [-.0900]2}
= {[1/2] × [.00640]} + {[1/2] × [.00010]} + {[1/2] × [.00810]}
= {.003200 + .000050 + .004050]}
= 0.007300
SD(R) = √.007300 = .0854 = 8.54%
The mean of the stock’s returns minus the standard deviation of the stock’s returns
= .0200 – .0854
= -.0654
9. What is the expected standard deviation of stock A’s returns based on the information presented in the table?
Outcome Probability of outcome Return of stock A in outcome
Good 10% 40.0%
Medium 40% 10.0%
Bad ? -10.0%
(Fall 2009, test 4, question 8)
(Fall 2012, test 4, question 9)
(Spring 2013, test 4, question 9)
(Fall 2013, final, question 17)
(Fall 2014, test 4, question 10)
(Spring 2015, test 3, question 8)
(Spring 2016, test 4, question 4)
We know that the sum of the probabilities of the 3 possible outcomes must equal 1, so
p(good) + p(medium) + p(bad) = 1
= .10 + .40 + p(bad) = 1
= .50 + p(bad) = 1
So p(bad) = 1 – .50 = .50
E(R) = [p(1) × R(1)] + [p(2) × R(2)] + [p(3) × R(3)]
= [.10 × 0.400] + [.40 × 0.100] + [.50 × (-0.100)] = .030
Var(R) = σ2 = {p(1) × [R(1) – E(R)]2} + {p(2) × [R(2) – E(R)]2} + {p(3) × [R(3) – E(R)]2}
= {.10 × [.400 – .030]2} + {.40 × [.100 – .030]2} + {.50 × [-0.100 – .030]2}
= {.10 × [.370]2} + {.40 × [.070]2} + {.50 × [-0.130]2}
= {.10 × .13690} + {.40 × .00490} + {.50 × 0.01690}
= .013690 + .001960 + .008450
= .0241000
SD(R) = √.0241000 = .1552 = 15.52%
10. Shares of Rampage are currently priced at $50.00 per share. The following table indicates what could happen with the Rampage stock price and dividend per share over the next year. What is the expected standard deviation of Rampage stock’s returns?
Outcome Probability of outcome Price of Rampage stock in 1 year Dividend paid by Rampage in 1 year
Good .60 $81.27 $3.73
Bad .40 $28.97 $1.03
(Spring 2010, final, question 12)
(Fall 2010, test 4, question 10)
(Spring 2011, final, question 18)
(Fall 2011, final, question 17)
(Spring 2012, final, question 18)
(Fall 2012, final, question 17)
(Spring 2014, test 4, question 9)
(Spring 2015, final, question 20)
(Fall 2015, test 4, question 3)
(Fall 2016, test 4, question 2)
(Fall 2017, test 4, question 2)
(Spring 2018, final, question 15)
Find the return for Rampage for each possible outcome
Return = (dividends + ending price – initial price) / initial price
Return in good = ($3.73 + $81.27 – $50.00) / $50.00 = $35.00 / $50.00 = 0.7000
Return in bad = ($1.03 + $28.97 – $50.00) / $50.00 = -$20.00 / $50.00 = -0.4000
E(R) = [p(1) × R(1)] + [p(2) × R(2)]
= [.60 × 0.7000] + [.40 × (-0.4000)] = .2600
Var(R) = σ2 = {p(1) × [R(1) – E(R)]2} + {p(2) × [R(2) – E(R)]2}
= {.60 × [.7000 – .2600]2} + {.40 × [-.4000 – .2600]2}
= {.60 × [.4400]2} + {.40 × [-.6600]2}
= {.60 × .193600} + {.40 × .435600}
= .116160 + .1742400
= .290400
SD(R) = √.290400 = (.290400).5 = (.290400)(1/2) = .5389 = 53.89%
11. Tony’s portfolio is worth $77,000 and has three stocks. It has $23,000 of stock A, which has an expected return of 19.32%; it has 4,800 shares of stock B, which has a share price of $7.50 and an expected return of 12.66%; and it has some stock C, which has an expected return of 9.93%. What is the expected return of Tony’s portfolio?
(Fall 2015, test 4, question 4)
(Fall 2017, test 4, question 3)
E(Rp) = [xA × E(RA)] + [xB × E(RB)] + [xC × E(RC)]
To solve:
1) find the weight of the stocks in the portfolio
2) find the expected return of the portfolio
1) find the weight of the stocks in the portfolio
Total value of the portfolio = $77,000
Recall that portfolio weight = value of holdings / portfolio value
Value of A = $23,000
Value of B = 4,800 shares × $7.50 per share = $36,000
Value of C: we are not given the value of stock C in Tony’s portfolio, but we know that the Value of stock C equals the value of the portfolio minus the total value of stocks A and B
= 77,000 – (23,000 + 36,000)
= 77,000 – 59,000
= 18,000
xA = weight for A = 23,000 / 77,000
xB = weight for B = 36,000 / 77,000
xC = weight for C = 18,000 / 77,000
2) find the expected return of the portfolio
E(RA) = .1932
E(RB) = .1266
E(RC) = .0993
E(Rp) = [(23k / 77k) × .1932] + [(36k / 77k) × .1266] + [(18k / 77k) × .0993]
.0577 + .0592 + .0232
= .1401 = 14.01%
Answers may vary slightly due to rounding [Show Less]