Investment Assignment 1
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Q1 An investor bought a stock and held it for a month. The stock did not pay any dividends during that period. The investor estimated the realized monthly net return, which was 9.05%. If the investor sold the stock at the end of the month for $150.29, what was the purchasing price of the share? Round your answer to two decimals.
To calculate the purchasing price of the share, you can use the formula for calculating the realized monthly net return:
Realized Monthly Net Return = [(Selling Price - Purchasing Price) / Purchasing Price] * 100
Given:
- Realized Monthly Net Return = 9.05% (0.0905)
- Selling Price = $150.29
Let "Purchasing Price" be denoted as P. We can rearrange the formula to solve for P:
0.0905 = [($150.29 - P) / P] * 100
Now, let's solve for P:
0.0905 = [($150.29 - P) / P] * 100
0.0905 = [(150.29 - P) / P] * 100
0.0905 = (150.29 - P) / P
Now, we can isolate P:
P * 0.0905 = 150.29 - P
0.0905P + P = 150.29
1.0905P = 150.29
P = 150.29 / 1.0905
P ≈ 137.59
So, the purchasing price of the share was approximately $137.59 when rounded to two decimals.
Q2 An investor assumes that there are only two possible states of the world for next year, "boom" or "recession", both with equal probability of occurring. If the "recession" scenario occurs, a stock will pay an annual dividend of $0.29, and the stock price will be $11.1 by the end of the year.
If the "boom" scenario occurs, a stock will pay an annual dividend of $0.52, and the stock price will be $18.68 by the end of the year. If the stock price today is $11.04, what is the annual expected net return estimated using this assumptions? Round your intermediate steps (if necessary) and your answer to four decimals. Do not use percentage format! That is, 1.234% should be input as 0.0123
To calculate the annual expected net return based on the given assumptions, we can use the
concept of expected value. The expected annual net return can be calculated as the weighted average of the returns in each scenario, with each scenario weighted by its probability of occurring.
Given:
- Two possible states of the world: "boom" and "recession," each with equal probability.
- In the "recession" scenario, the stock pays a dividend of $0.29, and the end-of-year stock price is $11.1.
- In the "boom" scenario, the stock pays a dividend of $0.52, and the end-of-year stock price is $18.68.
- The current stock price today is $11.04.
Let's calculate the expected annual net return:
1. Calculate the expected dividend (D) and expected end-of-year stock price (P) based on the
two scenarios:
- "Recession" scenario (R):
- Dividend (DR) = $0.29
- End-of-year stock price (PR) = $11.1
- "Boom" scenario (B):
- Dividend (DB) = $0.52
- End-of-year stock price (PB) = $18.68
2. Calculate the expected return (ER) for each scenario, which is the total return including dividends and capital gains, by considering the change in stock price:
- Expected return in the "Recession" scenario (ER_R):
- ER_R = (DR + (PR - P)) / P
- ER_R = ($0.29 + ($11.1 - $11.04)) / $11.04
- ER_R ≈ 0.0054 (rounded to four decimal places)
- Expected return in the "Boom" scenario (ER_B):
- ER_B = (DB + (PB - P)) / P
- ER_B = ($0.52 + ($18.68 - $11.04)) / $11.04
- ER_B ≈ 0.3721 (rounded to four decimal places)
3. Calculate the weighted average of the expected returns using the equal probabilities of the two scenarios:
- Expected Annual Net Return (EANR):
- EANR = (ER_R + ER_B) / 2
- EANR = (0.0054 + 0.3721) / 2
- EANR ≈ 0.18875 (rounded to four decimal places)
So, the annual expected net return, estimated using these assumptions, is approximately 0.1888 when rounded to four decimal places.
Q3 An investor collected data for the past 5 months. The estimated realized monthly net returns for those months are: R 1 = 6.59%, R 2 = 9.11%, R 3 = 9.69%, R 4 = -3% , R 5 = 5.6% Calculate the sample standard deviation of the realized monthly net returns. Round your intermediate steps to four decimals at least (if necessary). Input your answer with four decimals. Do not use percentage format! That is, 1.234%
should be input as 0.0123
To calculate the sample standard deviation of the realized monthly net returns, follow these steps:
1. Calculate the mean (average) of the returns.
2. Calculate the squared difference between each return and the mean.
3. Calculate the average of the squared differences.
4. Take the square root of the average squared difference.
Let's calculate it step by step:
Given monthly net returns:
R1 = 6.59% (0.0659)
R2 = 9.11% (0.0911)
R3 = 9.69% (0.0969)
R4 = -3% (-0.03)
R5 = 5.6% (0.056)
Step 1: Calculate the mean (average) of the returns:
Mean = (R1 + R2 + R3 + R4 + R5) / 5
Mean = (0.0659 + 0.0911 + 0.0969 - 0.03 + 0.056) / 5
Mean = 0.05678
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Step 2: Calculate the squared difference between each return and the mean:
Squared Difference for R1 = (0.0659 - 0.05678)^2 = 0.00083396
Squared Difference for R2 = (0.0911 - 0.05678)^2 = 0.00120709
Squared Difference for R3 = (0.0969 - 0.05678)^2 = 0.00161529
Squared Difference for R4 = (-0.03 - 0.05678)^2 = 0.00287048
Squared Difference for R5 = (0.056 - 0.05678)^2 = 6.0984e-06 (rounded to 4 decimal places)
Step 3: Calculate the average of the squared differences:
Average of Squared Differences = (0.00083396 + 0.00120709 + 0.00161529 + 0.00287048 + 6.0984e-06) / 5
Average of Squared Differences ≈ 0.000904
Step 4: Take the square root of the average squared difference to get the sample standard deviation:
Sample Standard Deviation = √(0.000904)
Sample Standard Deviation ≈ 0.03006 (rounded to four decimal places)
So, the sample standard deviation of the realized monthly net returns is approximately 0.0301 when rounded to four decimal places.
Q4 An investor held a stock from January 1, 2023 to March 31, 2023 (3 months, or quarterly). On March 31, right before selling the stock, the investor collected $1.46 in dividends from the stock. The investor paid $11.98 for the stock on January 1 and sold it for $11.18 on March 31. Use the realized quarterly net return to calculate the annualized net return for this investment. Round your intermediate steps to at least four decimals (if necessary). Input your answer with four decimals. Do not use percentage format! That is, 1.234% should be input as 0.0123
To calculate the annualized net return for this investment, we'll first calculate the quarterly net return and then annualize it.
The quarterly net return is given by the formula:
Quarterly Net Return = [(Ending Value + Dividends) - Beginning Value] / Beginning Value
Where:
- Ending Value = The value of the investment at the end of the quarter (the selling price).
- Dividends = The dividends collected during the quarter.
- Beginning Value = The initial investment value.
Given:
- Ending Value = $11.18
- Dividends = $1.46
- Beginning Value = $11.98
Now, calculate the quarterly net return:
Quarterly Net Return = [($11.18 + $1.46) - $11.98] / $11.98
Quarterly Net Return = [($12.64) - $11.98] / $11.98
Quarterly Net Return = ($0.66) / $11.98
Quarterly Net Return ≈ 0.0551 (rounded to four decimal places)
Now, to annualize the quarterly net return, we need to account for the fact that there are four quarters in a year. Use the following formula to annualize it:
Annualized Net Return = (1 + Quarterly Net Return)^4 - 1
Annualized Net Return = (1 + 0.0551)^4 - 1
Annualized Net Return = (1.0551)^4 - 1
Annualized Net Return ≈ 0.2340 (rounded to four decimal places)
So, the annualized net return for this investment is approximately 0.2340 (or 23.40% when expressed as a percentage).
Q5 Stock A is expected to pay a dividend of $2.03 per share in a year from now (no other dividends will be paid during this period). An investment with the same risk as Stock A has an expected return of 14.4%. If Stock A's price today is $72.07, what is Stock A's expected price in a year from now, according to the Gordon Growth Model? Round your answer to two decimals.
The Gordon Growth Model, also known as the Gordon-Shapiro Model, is used to estimate the price of a stock based on its expected future dividends and the required rate of return. The formula for the Gordon Growth Model is:
\[P_1 = \frac{D_1}{R - g}\]
Where:
- \(P_1\) is the expected price of the stock in one year.
- \(D_1\) is the expected dividend in one year.
- \(R\) is the required rate of return.
- \(g\) is the expected constant growth rate of dividends.
In this case, we are given:
- \(D_1 = $2.03\) (the expected dividend in one year).
- \(R = 14.4\%\) (the required rate of return).
We need to calculate the expected constant growth rate of dividends (\(g\)). The Gordon Growth Model assumes that dividends grow at a constant rate, so we can use the following relationship:
\[R = \frac{D_0(1+g)}{P_0}\]
Where:
- \(D_0\) is the current dividend.
- \(P_0\) is the current price.
We are given:
- \(D_0 = $2.03\) (the current dividend).
- \(P_0 = $72.07\) (the current price).
Now, we can rearrange the equation to solve for \(g\):
\[g = \frac{(R \cdot P_0)}{D_0} - 1\]
Substitute the values:
\[g = \frac{(0.144 \cdot $72.07)}{$2.03} - 1\]
Now, calculate \(g\):
\[g = \frac($10.39)}{$2.03} - 1\]
\[g ≈ 5.1197 - 1\]
\[g ≈ 4.1197\]
Now that we have the growth rate (\(g\)), we can use it to calculate the expected price (\
(P_1\)) using the Gordon Growth Model:
\[P_1 = \frac{$2.03}{0.144 - 0.041197}\]
Now, calculate \(P_1\):
\[P_1 ≈ \frac{$2.03}{0.102803}\]
\[P_1 ≈ $19.76\]
So, according to the Gordon Growth Model, Stock A's expected price in one year is approximately $19.76 when rounded to two decimals.
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