Calculate Annualized Returns From The Given Information

Given The Information Below Compute Annualized Returnsasset

Given the information below, compute annualized returns for various assets based on their income, price change, initial price, purchase price, current price, income received, and time period. The task involves calculating the annualized return for each asset, converting the provided time periods into years, and then applying the appropriate financial formulas to determine the annualized return rates.

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The process of calculating annualized returns is essential for comparing the profitability of different investments over varying time periods. Annualized return, also known as compound annual growth rate (CAGR), provides a standardized measure that enables investors to evaluate performance as if the return had been realized evenly over each year. This analysis entails understanding the key variables involved: income received, price change, initial price, purchase price, current price, and the duration of the investment.

The first asset (Asset A) has an income of $2, a price change from an initial price of $29 over a period of 15 months. To compute its annualized return, the first step involves calculating the total return, which considers both income and capital appreciation, and then adjusting it to a yearly basis. The formula for CAGR when the total return includes income is:

\[ \text{Annualized Return} = \left( \frac{\text{Ending Value} + \text{Income}}{\text{Beginning Value}} \right)^{\frac{1}{t}} - 1 \]

where \( t \) is the investment period expressed in years.

Converting 15 months into years: \( t = \frac{15}{12} = 1.25 \) years.

The ending value accounts for price change and income received relative to the initial price:

\[ \text{Ending Value} = \text{Price Change} + \text{Initial Price} = 6 + 29 = 35 \]

Thus,

\[ \text{Total return} = \frac{35 + 2}{29} = \frac{37}{29} \approx 1.28 \]

Applying the formula:

\[ \text{Annualized Return} = (1.28)^{1/1.25} - 1 \]

Calculating:

\[ \text{Annualized Return} \approx (1.28)^{0.8} - 1 \approx 1.209 - 1 = 0.209 \text{ or } 20.9\% \]

Proceeding with Asset B, which has a purchase price of $20, a current price of $26, income received of $2, over 75 weeks. First, convert 75 weeks into years:

\[ t = \frac{75}{52} \approx 1.44 \text{ years} \]

Total return:

\[ \frac{\text{Current Price} + \text{Income}}{\text{Purchase Price}} = \frac{26 + 2}{20} = \frac{28}{20} = 1.4 \]

Therefore,

\[ \text{Annualized Return} = (1.4)^{1/1.44} - 1 \]

Calculating:

\[ (1.4)^{0.694} \approx 1.25 \]

\[ \text{Annualized Return} \approx 0.25 \text{ or } 25\% \]

For Asset C, with a time period in months or years but lacking specific monetary values, the same process applies if data are provided. If only the time period is given, similar steps apply: convert the period into years and then apply the CAGR formula based on initial and final prices or incomes.

Asset D's incomplete data indicates the need to clarify the monetary values and period for full computation, but the approach remains consistent: convert the time into years and use the CAGR formula.

In conclusion, accurately computing annualized returns requires converting all time periods into years, calculating total returns considering both income and capital changes, and applying the CAGR formula to standardize performance across different investments. This process allows investors to compare assets effectively, assess growth over different durations, and make informed financial decisions.

References

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