Question 1

$\begin{array}{|l|c|c|}
\hline \text { Asset } & \begin{array}{c}
\text { Expected return } \
(\mathrm{E}(\mathrm{R}) \boldsymbol{)}
\end{array} & \begin{array}{c}
\text { Standard deviation } \
\text { (ธ) }
\end{array} \
\hline \mathrm{A} & 10 \% & 9 \% \
\hline \mathrm{B} & 12 \% & 8 \% \
\hline \mathrm{C} & 15 \% & 12 \% \
\hline \mathrm{D} & 13 \% & 11 \% \
\hline
\end{array}$
-Which of the four investments has the highest coefficient of variation?

Accepted Answer

Using equation 1.10,page 18,the coefficient of variation (CV)can be calculated by dividing the standard deviation by the expected return.That is,the CV of asset A is 0.09/0.10 = 0.9,for asset B is 0.08/0.12 = 0.67,for asset C is 0.12/0.15 = 0.8 and for asset D is 0.11/0.13 = 0.85.Therefore,asset A has the highest coefficient of variation.

Question 2

The price at the start of the period for asset A is $5 and for asset B is $6.At the end of the period,asset A has fallen to $4 and asset B has risen to $6.50.Calculate the continuously compounded return on a price-weighted portfolio consisting of shares A and B.

Accepted Answer

Using equation 1.5,page 14,the return on asset A is ln(4/5)= -0.223 and for asset B is ln(6.5/6)= 0.080.The total cost = -0.058 or -5.8%.

Question 3

$\begin{array}{|c|c|c|}
\hline \text { Asset } & \begin{array}{c}
\text { Price at } \
\text { start of } \
\text { period (\$) }
\end{array} & \begin{array}{c}
\text { Price at } \
\text { end of } \
\text { period (\$) }
\end{array} \
\hline \text { A } & \$ 3.25 & \$ 4.25 \
\hline \text { B } & \$ 4.12 & \$ 3.85 \
\hline & & \
\hline
\end{array}$
-Given the prices for A and B at the start and end of the period,calculate the continuously compounded return on an equally weighted portfolio consisting of assets A and B.

Accepted Answer

Using equation 1.5,page 14,the continuously compounded return for share A is ln(4.25/3.25)= 0.268 or 26.8%,and for share B is ln(3.85/4.12)= -0.068 or -6.8%.Using equation 1.15,page 23,the equally weighted (continuously compounded)return on the portfolio is 0.5(26.8)+ 0.5(-6.8)= 10%.

Question 4

$\begin{array}{|l|c|c|}
\hline \text { Asset } & \begin{array}{c}
\text { Expected return } \
(\mathrm{E}(\mathrm{R}) \boldsymbol{)}
\end{array} & \begin{array}{c}
\text { Standard deviation } \
\text { (ธ) }
\end{array} \
\hline \mathrm{A} & 10 \% & 9 \% \
\hline \mathrm{B} & 12 \% & 8 \% \
\hline \mathrm{C} & 15 \% & 12 \% \
\hline \mathrm{D} & 13 \% & 11 \% \
\hline
\end{array}$
-Given no other information other than the expected returns and risk,which asset would a risk-averse investor prefer?

Accepted Answer

For two assets with the same expected return,a risk-averse investor will prefer the investment with the lower standard deviation,and for two investments with the same standard deviation will prefer the investment with the highest return.In this case,since the investments have different risks and expected returns,the coefficient of variation can be used.All other things equal,for a risk-averse investor a lower coefficient of variation is desirable,as this indicates less risk for each unit of expected return.Using equation 1.10,page 20,the coefficient of variation (CV)can be calculated by dividing the standard deviation by the expected return.That is,the CV of asset A is 0.09/0.10 = 0.9,for asset B is 0.08/0.12 = 0.67,for asset C is 0.12/0.15 = 0.8 and for asset D is 0.85.Therefore,asset B has the lowest coefficient of variation,hence will be the most preferred by a risk-averse investor.

Question 5

The returns for two assets,A and B,are 5% and 8% respectively.If an investor holds 25% of her wealth in asset A,what is the continuously compounded return of a value-weighted portfolio comprising assets A and B?

Accepted Answer

Using equation 1.16,page 23,the value-weighted return on the portfolio (R)is equal to (0.25)(5%)+ (0.75)(8%)= 7.27%.

Question 6

$\begin{array}{|c|c|c|}
\hline \text { Asset } & \begin{array}{c}
\text { Price at } \
\text { start of } \
\text { period (\$) }
\end{array} & \begin{array}{c}
\text { Price at } \
\text { end of } \
\text { period (\$) }
\end{array} \
\hline \text { A } & \$ 3.25 & \$ 4.25 \
\hline \text { B } & \$ 4.12 & \$ 3.85 \
\hline & & \
\hline
\end{array}$
-Given the prices for A and B at the start and end of the period,calculate the price-weighted return on a portfolio consisting of assets A and B.

Accepted Answer

Using equation 1.5,page 14,the continuously compounded return for share A is ln(4.25/3.25)= 0.268 or 26.8%,and for share B is ln(3.85/4.12)= -0.068 or -6.8%.The total cost (in terms of price)of the portfolio is 3.25 + 4.12 = $7.37,of which 3.25/7.37 = 44.1% is invested in asset A,and 55.9% is invested in asset B.Using equation 1.16,page 23,the price-weighted (continuously compounded)return on the portfolio is 0.441(26.8)+ 0.559(-6.8)= 8.02%.

Question 7

An investor purchased an asset for $32 000 at the beginning of the year.This investment is worth $36 000 at the end of the year.What is the return for the year on the investment?

Accepted Answer

Rearranging equation 1.1,page 11,the return for the year = (36000 - 32000)/32000 = 0.125

Question 8

$\begin{array}{|l|r|r|r}
\hline \text { Asset } & \text { Number of shares } & \text { Initial price (\$) } & \text { End price (\$) } \
\hline \text { A } & 2000 & 6.00 & 6.10 \
\hline \text { B } & 5000 & 3.00 & 2.25 \
\hline \text { C } & 3000 & 12.00 & 11.50 \
\hline \text { D } & 12000 & 8.50 & 8.85 \
\hline
\end{array}$
-An investor purchased the investment above.What is the value-weighted return on the portfolio?

Accepted Answer

We must first calculate the continuously compounded return for each of the four shares.Using equation 1.5,page 14,the return for asset A is ln(6.10/6.00)= 0.165,for asset B is ln(2.25/3.00)= -0.288,for asset C is ln(11.50/12.00)= -0.043 and for asset D is ln(8.85/8.5)= 0.040.The total value of the portfolio is (2000 F1F1F1S1F1F1F10 6.00 + 5000 F1F1F1S1F1F1F10 3.00 + 3000 F1F1F1S1F1F1F10 12.00 + 12000 F1F1F1S1F1F1F10 8.50)= 165000.The value weights for each of the assets A,B,C and D,respectively,are for asset A (2000 F1F1F1S1F1F1F10 6.00)/165000 = 0.073,for asset B = (5000 F1F1F1S1F1F1F10 3.00)/165000 = 0.091,for asset C = (3000 F1F1F1S1F1F1F10 12.00)/165000 = 0.218 and for asset D = (12000 F1F1F1S1F1F1F10 8.50)/165000 = 0.618.Using equation 1.14,page 22,the value-weighted return is calculated as R = (0.073)(0.165)+ (0.091)(-0.288)+ (0.218)(-0.043)+ (0.618)(0.040)= 0.012 or 1.2%.

Question 9

$\begin{array}{|c|c|}
\hline \text { Year } & \begin{array}{c}
\text { Return } \
\mathbf{( \% )}
\end{array} \
\hline \text { Year } 1 & 7 \% \
\hline \text { Year } 2 & 17 \% \
\hline \text { Year } 3 & -2.4 \% \
\hline \text { Year } 4 & 2 \% \
\hline
\end{array}$
-The value of an investment at the end of each of four years is shown above.What is the arithmetic cumulative return over the four years?

Accepted Answer

Using equation 1.11,page 18,the arithmetic cumulative return for the investment over the four years is calculated as (0.07 + 0.17 - 0.024 + 0.02)= 0.236 or 23.6%.

Question 10

$\begin{array}{|c|c|}
\hline \text { Year } & \begin{array}{c}
\text { Return } \
\mathbf{( \% )}
\end{array} \
\hline \text { Year } 1 & 7 \% \
\hline \text { Year } 2 & 17 \% \
\hline \text { Year } 3 & -2.4 \% \
\hline \text { Year } 4 & 2 \% \
\hline
\end{array}$
-The value of an investment at the end of each of four years is shown above.What is the geometric cumulative return over the four years?

Accepted Answer

Using equation 1.12,page 18,the geometric return over the four years is calculated as: [(1 + 0.07)(1 + 0.17)(1 - 0.024)(1 + 0.02)- 1] = 0.2463 or 24.63% (Example 1.9)

Question 11

An investor purchased an asset for $1675.This investment is worth $1742 after three months.What is the effective annualised return for the investment?

Accepted Answer

Using equation 1.1,page 11,the periodic (three-month)return is calculated as (1742 - 1675)/1675 = 0.04.The return for the three-month period is 4%.The effective annual return can be calculated using equation 1.2,page 12 as follows: [(1 + 0.04)⁴ - 1] = 0.17 or 17%.

Question 12

An investor purchased an asset for $1675.This investment is worth $1742 after three months.What is the simple annualised return for the investment (simple annual return)?

Accepted Answer

Rearranging equation 1.1,page 11,the three-month return is equal to (1742 - 1675)/1675 = 0.04.The return for the three-month period is 4%.Hence,the simple annualised return on the investment is 0.04 × 4 = 0.16 or 16%.

Question 13

An investment was purchased for $2985.After seven months this investment was worth $3245.What is the continuously compounded return on the investment over the seven-month period?

Accepted Answer

Using equation 1.5,page 14,the continuously compounded return over the seven-month period is r = ln(3245/2985)= 0.0835 or 8.35%.

Question 14

The returns for two assets,A and B,are 5% and 8% respectively.If a price-weighted portfolio were formed when shares in A were $9 and shares in B were $3,what is the continuously compounded price-weighted return for the portfolio?

Accepted Answer

The total cost (8%)= 5.75%.

Question 15

$\begin{array}{|c|c|}
\hline \text { Year } & \begin{array}{c}
\text { Return } \
\mathbf{( \% )}
\end{array} \
\hline \text { Year } 1 & 7 \% \
\hline \text { Year } 2 & 17 \% \
\hline \text { Year } 3 & -2.4 \% \
\hline \text { Year } 4 & 2 \% \
\hline
\end{array}$
-The value of an investment at the end of each of four years is shown above.What is the geometric average return over the four years?

Accepted Answer

Using the equation in example 1.10,page 19,the geometric average return over the four years is calculated as [(1 + 0.07)(1 + 0.17)(1 - 0.024)(1 + 0.02)]^1/4 - 1 = 0.0566 or 5.66% per annum.

Question 16

$\begin{array}{|c|c|c|}
\hline \text { Asset } & \begin{array}{c}
\text { Price at } \
\text { start of } \
\text { period (\$) }
\end{array} & \begin{array}{c}
\text { Price at } \
\text { end of } \
\text { period (\$) }
\end{array} \
\hline \text { A } & \$ 3.25 & \$ 4.25 \
\hline \text { B } & \$ 4.12 & \$ 3.85 \
\hline & & \
\hline
\end{array}$
-Given the prices for A and B at the start and end of the period,if an investor holds 1000 shares in asset A and 5000 in asset B,what is her value-weighted return on the portfolio?

Accepted Answer

Using equation 1.5,page 14,the continuously compounded return for share A is ln(4.25/3.25)= 0.268 or 26.8%,and for share B is ln(3.85/4.12)= -0.068 or -6.8%.The total value of the portfolio is (1000 F1F1F1S1F1F1F10 3.25 + 5000 F1F1F1S1F1F1F10 4.12)= 23850,of which 3250/23850 = 0.136 is invested in asset A,and 0.864 is invested in asset B.Using equation 1.14,page 22,the value-weighted (continuously compounded)return on the portfolio is 0.136(26.8)+ 0.864(-6.8)= -2.23%.

Question 17

$\begin{array}{|l|r|r|r}
\hline \text { Asset } & \text { Number of shares } & \text { Initial price (\$) } & \text { End price (\$) } \
\hline \text { A } & 2000 & 6.00 & 6.10 \
\hline \text { B } & 5000 & 3.00 & 2.25 \
\hline \text { C } & 3000 & 12.00 & 11.50 \
\hline \text { D } & 12000 & 8.50 & 8.85 \
\hline
\end{array}$
-An investor purchased the investment above.If continuously compounded returns are used,what is the equal-weighted return on the portfolio?

Accepted Answer

We must first calculate the continuously compounded return for each of the four shares.Using equation 1.5,page 14,the return for asset A is ln(6.10/6.00)= 0.165,for asset B is ln(2.25/3.00)= -0.288,for asset C is ln(11.50/12.00)= -0.043 and for asset D is ln(8.85/8.5)= 0.040.Using equation 1.15,page 23,the equal-weighted return is calculated as R = [(0.165)+ (-0.288)+ (-0.043)+ (0.040)]/4 = -0.0.0315 or -3.2%.

Question 18

$\begin{array}{|c|c|}
\hline \text { Year } & \begin{array}{c}
\text { Return } \
\mathbf{( \% )}
\end{array} \
\hline \text { Year } 1 & 7 \% \
\hline \text { Year } 2 & 17 \% \
\hline \text { Year } 3 & -2.4 \% \
\hline \text { Year } 4 & 2 \% \
\hline
\end{array}$
-The value of an investment at the end of each of four years is shown above.What is the arithmetic average return over the four years?

Accepted Answer

Using equation 1.11,page 18,the arithmetic cumulative return for the investment over the four years is calculated as (0.07 + 0.17 - 0.024 + 0.02)= 0.236 or 23.6%.As per example 1.10,page 18,the arithmetic average return is found by dividing the cumulative return by the number of years,or 0.236/4 = 0.059 or 5.90% per annum.

Question 19

Charles invests in a price-weighted portfolio of two shares,ABC and XYZ.His funds manager sends him a statement of his performance at the end of each year for his tax return.Unfortunately,some of the form was illegible,and all he could discern was that the price-weighted portfolio return was 10%,the returns on ABC and XYZ were 8% and 16%,respectively and that the purchase price of ABC was $10.Given this information,at what price did Charles purchase the shares in company XYZ?

Accepted Answer

Given that there are only two shares in the portfolio,the weights in each asset must sum to unity.That is,if we define the (unknown)price weight in asset B as w_b,then the weight in asset A is 1 - w_b.Using equation 1.16,page 23,we know the price-weighted return on the portfolio is 10%,which can be expressed as 0.10 = (1 - w_b)(8%)+ w_b(16%).Solving yields w_b = 25%.We can now solve for the selling price of asset B.We know that the price-weight of asset B is calculated as the purchase price of asset B divided by the combined purchase prices of both assets A and B,and that the weight must be 25%.Hence if we denote the purchase price of asset A as P,we know that P/(10 + P)= 0.25.Hence,P = 2.5 + 0.25P,which can be solved to yield P = 3.33.To verify the answer is correct,we now know that the price of assets A and B are $10 and $3.33 (and hence a combined cost (16%)= 10.0% as required.

Question 20

Loredana has invested equal amounts in each of the shares of company A,company B and company C,which she purchased one year ago at prices of $10,$2 and $1,respectively.Loredana has recently sold her shares in company B for $2.50 and in company C for $1.15.If Loredana wishes to earn a continuously compounded return of 20% for the year,on her equally weighted portfolio,at what price will she need to sell her shares in company A?

Accepted Answer

Using equation 1.5,page 14,the continuously compounded return for asset B is ln(2.5/2)= 0.223,and for asset C is ln(1.15/1)= 0.140.If we denote x as the return Loredana needs to earn on asset A,and she wishes to earn 20% on the equally weighted portfolio,then using equation 1.15,page 23,we know that the equally weighted portfolio return is: (1/3)(x)+ (1/3)(0.223)+ (1/3)(0.140)= 0.20.Solving the equation for x yields an answer of x = 23.7%.Therefore,Loredana must earn a return on asset A of 23.7%.Denoting P as the selling price,then using equation 1.5,ln(P/$10)= 0.237.Hence,P = 10e^0.237,which equals $12.67.As a check,using equation 1.5,ln(12.67/10)= 23.7.

Quiz 1: The Investment Decision