Compute Annualized Returns For Given Assets Based On Provide

Compute Annualized Returns for Given Assets Based on Provided Data

The assignment involves calculating the annualized returns for a series of assets based on specific financial data. In order to perform these calculations accurately, it is essential to understand the concepts of return and annualization, which standardize gains or losses over arbitrary time periods into a comparable annual figure. The data provided includes initial or purchase prices, current or ending prices, income received (such as dividends or interest), and the durations of investments, some measured in months, weeks, or years.

To compute the annualized return, the core approach involves calculating the total return over the given period—combining capital appreciation and income—and then adjusting this return to a yearly basis using annualization formulas. The general formula for the annualized return is:

Annualized Return = (Total Return) ^{(Number of periods in a year)} - 1

where Total Return is the sum of capital gains/losses and income received, relative to the initial investment, and the exponent adjusts for the specific duration, converting it into an annualized figure.

Calculating Annualized Returns for the Assets

Part 1: Assets with initial price, price change, and income

Given data:

• Asset A: Initial Price = $6, Price Change = $2, Income = $29, Duration = 15 months
• Asset B: Initial Price = $10, Price Change = $0, Income = $40, Duration = 11 months
• Asset C: Initial Price = $70, Price Change = $50, Income = $30, Duration = 7 years
• Asset D: Initial Price = $8, Price Change = -$3, Income = $20, Duration = 24 months

Note: There appears to be a discrepancy in the provided data for Asset A and Asset C, particularly in the 'Price Change' and 'Initial Price' columns, which may be typographical errors. For the purpose of this calculation, I will interpret the 'Price Change' as the difference between the final and initial prices, and treat the initial price as the starting value, assuming the 'Price Change' correctly reflects the change over the period. Adjustments will be made accordingly.

Calculations:

Asset A

Initial Price: $6

Final Price: $6 + $2 (Price Change) = $8

Income received: $29

Duration: 15 months = 1.25 years

Total return: (Final Price - Initial Price) + Income = ($8 - $6) + $29 = $2 + $29 = $31

Return over period: $31 / $6 ≈ 5.1667

Annualized return: (1 + 5.1667)^{1 / 1.25} - 1 ≈ (6.1667)^0.8 - 1 ≈ 4.01 - 1 = 3.01 or 301%

Asset B

Initial Price: $10

Final Price: $10 + $0 = $10

Income: $40

Duration: 11 months ≈ 0.9167 years

Total return: ($10 - $10) + $40 = $0 + $40 = $40

Return over period: $40 / $10 = 4

Annualized return: (1 + 4)^{1 / 0.9167} - 1 ≈ 5^1.089 - 1 ≈ 5.52 - 1 = 4.52 or 452%

Asset C

Initial Price: $70

Final Price: $70 + $50 = $120

Income: $30

Duration: 7 years

Total return: ($120 - $70) + $30 = $50 + $30 = $80

Return over period: $80 / $70 ≈ 1.1429

Annualized return: (1 + 1.1429)^{1 / 7} - 1 ≈ (2.1429)^0.1429 - 1 ≈ 1.090 - 1 = 0.090 or 9.0%

Asset D

Initial Price: $8

Final Price: $8 - $3 = $5

Income: $20

Duration: 24 months = 2 years

Total return: ($5 - $8) + $20 = -$3 + $20 = $17

Return over period: $17 / $8 ≈ 2.125

Annualized return: (1 + 2.125)^1/2 - 1 ≈ (3.125)^0.5 - 1 ≈ 1.768 - 1 = 0.768 or 76.8%

Part 2: Assets with purchase price, current price, income received, and investment period

Given data:

• Asset A: Purchase Price = $20, Current Price = $26, Income Received = $2, Period = 75 weeks
• Asset B: Purchase Price = $15, Current Price = $18, Income Received = $0.40, Period = 3 months
• Asset C: Purchase Price = $150, Current Price = $130, Income Received = $0, Period = 2 years
• Asset D: Purchase Price = $3.50, Current Price = $3, Income Received = $0.20, Period = 8 months

Calculations:

Asset A

Purchase Price: $20

Current Price: $26

Income: $2

Period: 75 weeks ≈ 1.44 years

Total return: ($26 - $20) + $2 = $6 + $2 = $8

Return over period: $8 / $20 = 0.4

Annualized return: (1 + 0.4)^{1 / 1.44} - 1 ≈ 1.4^0.6944 - 1 ≈ 1.261 - 1 = 0.261 or 26.1%

Asset B

Purchase Price: $15

Current Price: $18

Income: $0.40

Period: 3 months ≈ 0.25 years

Total return: ($18 - $15) + $0.40 = $3 + $0.40 = $3.40

Return over period: $3.40 / $15 ≈ 0.2267

Annualized return: (1 + 0.2267)^{1 / 0.25} - 1 ≈ 1.2267⁴ - 1 ≈ 2.025 - 1 = 1.025 or 102.5%

Asset C

Purchase Price: $150

Current Price: $130

Income: $0

Period: 2 years

Total return: ($130 - $150) + $0 = -$20

Return over period: -$20 / $150 ≈ -0.1333

Annualized return: (1 - 0.1333)^{1 / 2} - 1 ≈ (0.8667)^0.5 - 1 ≈ 0.931 - 1 = -0.069 or -6.9%

Asset D

Purchase Price: $3.50

Current Price: $3

Income: $0.20

Period: 8 months ≈ 0.6667 years

Total return: ($3 - $3.50) + $0.20 = -$0.50 + $0.20 = -$0.30

Return over period: -$0.30 / $3.50 ≈ -0.0857

Annualized return: (1 - 0.0857)^{1 / 0.6667} - 1 ≈ 0.9143^1.5 - 1 ≈ 0.863 - 1 = -0.137 or -13.7%

Conclusion

Calculating annualized returns using standard formulas provides valuable insights into the performance of various assets over different periods. The above calculations demonstrate the variability of returns based on period length, income, and capital appreciation or depreciation. Notably, assets with shorter periods and income components tend to show higher annualized returns, highlighting the importance of considering the time horizon and income streams in portfolio analysis.

References

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