Mastering the Metrics: Key Return Formulas Every Investor Should Know
Risk and return are two fundamental indicators that characterize financial instruments. They serve as essential input information for various models and are crucial for making informed investment decisions.
What is Return?
Return refers to the gain or loss generated by an investment over a specific period. It measures the profitability of an investment and can come from various sources, such as capital gains or dividends. The return on an investment is typically expressed as a percentage of the initial investment amount.
There are different methods for calculating returns, each suitable for different purposes.
Arithmetic Return (Average Return)
The arithmetic return represents the sum of the returns achieved in each period, divided by the number of periods.
This approach is intuitive and is applied when calculating returns over more than one period. Imagine you buy a financial asset for $100 at the start of the year. By the end of the year, its value has increased to $110, giving you a 10% return. The next year, the asset’s value rises to $130, resulting in an 18.2% return for the second year. To find the average return over these two years, you simply add the two annual returns (10% + 18.2%) and divide by the number of periods, which is 2. This gives you an average return of 14.1%.
Average return is also known as mean return.
Geometric Return
The arithmetic return may not fully capture an investment’s performance because it fails to account for the effects of compounding over multiple periods.
Here’s an explanation of how compounding works using the example:
When you invest money, your returns are typically reinvested, leading to growth based on the accumulated amount rather than just the initial investment.
Example:
Initial Investment: You start with $1,000.
First Year: You earn a 10% return. The calculation is:
\$1{,}000 \times 0.10 = \$100
\]
Adding this to your initial investment:
\$1{,}000 + \$100 = \$1{,}100
\]
So, by the end of the first year, your investment is worth $1,100.
Second Year: You earn an 18.2% return, but this return is applied to the new amount of $1,100, not the original $1,000. The calculation is:
\$1{,}100 \times 0.182 = \$200.20
\]
Adding this to your $1,100:
\$1{,}100 + \$200.20 = \$1{,}300.20
\]
By the end of the second year, your investment has grown to $1,300.20.
Compounding Effect
The key here is that your returns in the second year are based on the total amount from the end of the first year ($1,100) rather than just the original amount ($1,000). This means you earn returns not only on your initial investment but also on the returns you earned in the first year. This process of earning returns on both your initial investment and previously earned returns is known as compounding.
The geometric return accounts for the compounding effect and is a more appropriate method compared to the arithmetic return when the financial instrument is held over time. In some cases, the difference between the arithmetic and geometric returns can be significant.
Geometric Return Calculation
To find the geometric return, you use the formula:
\text{Geometric Return} = \left( \frac{V_{\text{end}}}{V_{\text{start}}} \right)^{\frac{1}{n}} – 1
\]
where Vend is the final value of the investment, Vstart is the initial value, and is the number of periods.
For our example:
\text{Geometric Return} = \left( \frac{1300.20}{1000} \right)^{\frac{1}{2}} – 1 \approx 0.14 \text{ or } 14\%
\]
The geometric return of approximately 14% reflects the true average rate of return per year when considering the effects of compounding, compared to a simple average of 14.1% (from the arithmetic calculation) which doesn’t fully account for compounding.
In summary, while the arithmetic return gives a simple average, the geometric return reflects the true performance of the investment over time.
Money-Weighted Return (MWR)
Money-weighted return (MWR), also known as the Internal Rate of Return (IRR), is a measure of the performance of an investment portfolio that takes into account the timing and amount of cash flows. This method is particularly useful when investors make additional contributions or withdrawals from their investment portfolio, as it accurately reflects the impact of these changes on the overall return.
Calculating Money-Weighted Return
Consider an investor who initially invests $1000 and achieves a return of 20% for the first year. The profit would be $200, making the total value $1200. If the investor withdraws this $200 profit, the investment amount at the beginning of the second year would again be $1000. The return for the second year would then be calculated on $1000, not on $1200.
MWR accounts for such changes in the investment portfolio to provide an accurate assessment of the returns relative to the actual invested capital.
Annualized Return
Annualized return is a way to express the return on an investment as an annual rate. This is useful when the investment horizon is shorter (e.g., a week) or longer (e.g., 10 years) than a year. Investors often want to know the annual return on their investment to compare it with other investments or benchmarks.
For example, if an investment achieves a 1% return over a week, this result can be annualized using the following formula:
R_{\text{annualized}} = (1 + R_{\text{weekly}})^{52} – 1
\]
In this example:
R_{\text{annualized}} = (1 + 0.01)^{52} – 1 = 68\%
\]
Similarly, if we know the annual return (e.g., 10%), we can find the monthly return:
R_{\text{monthly}} = (1 + 0.1)^{\frac{1}{12}} – 1 = 0.8\%
\]
However, annualizing returns from shorter periods (week) to longer periods (year) can be misleading, as it assumes that the weekly return will be repeated consistently over the remaining 51 weeks of the year.
Portfolio Return
When investing in a portfolio of assets, the return for each instrument should be calculated based on its weight in the portfolio.
To formula for portfolio return is:
R_p = \sum_{i=1}^{n} (w_i \times r_i)
\]
Where:
- Rp is the total portfolio return.
- w(i) is the weight of the -th asset in the portfolio.
- r(i) is the return of the -th asset.
For instance, if we have a portfolio of $1000 distributed between two assets (one worth $400 and a yielding 10% annually, and another one priced at $600 and returning 20%), the portfolio return would be:
\[
R_p = (w_1 \times r_1) + (w_2 \times r_2)
\]
\[
R_p = (0.1 \times 400) + (0.2 \times 600) = 40 + 120 = \$160
\]
The achieved return for this portfolio would be 16% (160/1000).
Gross and Net Return
Investors often consider other costs that affect the achieved investment results. Typically, the purchase and sale of a financial instruments is accompanied by costs related to the transaction (broker or exchange commission), management fees, performance fees, administrative fees, and others.
Gross return does not account for investment costs. Alternatively, net return takes into account investment costs and shows the actual achieved investment result. Besides direct costs, some investments are subject to taxes, which can significantly impact realized return.
Other Return Concepts
Holding Period Return (HPR)
Holding Period Return is the total return earned on an investment over the entire period it is held, including both capital gains and any income received, expressed as a percentage of the initial investment.
Realized Return
Realized return refers to the actual profit or loss from an investment over a given time, factoring in both income and capital gains from the asset’s sale or generated cash flow.
Expected Return
The expected return is the anticipated profit or loss from an investment.
Required Return
The required return is the minimum return that an investor would accept to make an investment.
Discount Rate
The discount rate is the rate used to determine the present value of future financial benefits. It is a critical component in various financial models.
Internal Rate of Return (IRR)
The internal rate of return (IRR) is the discount rate that equates the net present value of all expected future cash flows (inflows and outflows) from an investment to zero.
Net Present Value (NPV)
Net present value (NPV) is the difference between the present value of cash inflows and the present value of cash outflows. It is used to assess the profitability of an investment. A positive NPV indicates that the projected earnings (in present dollars) exceed the anticipated costs (also in present dollars), making the investment potentially profitable.
Conclusion
In conclusion, understanding various return calculation methods is crucial for evaluating investment performance accurately. Arithmetic (average) and geometric returns provide insights into the overall performance over time, with the latter adjusting for compounding. Money-weighted returns take into account cash inflows and outflows. Annualized returns standardize performance over a year, making comparisons easier. Lastly, distinguishing between gross and net returns helps investors understand the impact of fees and expenses on their investments.
Next: Portfolio Risk Management
