Measures of Return | CFA Level I Quantitative Methods

In this lesson, we’ll explore various ways to measure the returns of a portfolio, starting with single-period measures, multi-period measures, and some other major return measures.

Single-Period Returns

Financial assets typically generate two types of return for investors:

• Periodic income through interest or dividend payments
• A capital gain or loss when the asset is sold or redeemed

Single-period returns are easy to calculate. We’ll use the holding period return (HPR) – the return earned from holding an asset for a specific period. Sum up all the dividends received and the selling price, divide it by the purchase price, minus 1, and you get the holding period return. If the holding period is not exactly one year, we can annualize the return using various formulas depending on the number of years, months, or days in each period.

HPR = (Dividends + Selling Price) / Purchase Price – 1

For example, if a stock was bought for $100 and over a period of 3 years, $5 dividends were received each year, and the stock was sold at $140, the holding period return over the 3 years = (140+15)/100 – 1 = +55%

The annualized gain is (1+0.55)^(⅓) – 1 = 15.7%.

Multi-Period Returns

If there are multiple cash inflows and outflows during the period, we need to break up the returns into multiple periods of the same length, calculate the holding period return for each period, and then calculate the average returns per period.

There are several approaches to determine this average return, including:

• Arithmetic mean return
• Geometric mean return
• Money-weighted return

Arithmetic Mean Return

The arithmetic mean return is simply the simple average, which is to sum all the periodic returns and divide by the number of periods.

Arithmetic mean return = ∑R_n / n

It has the statistical property of being an unbiased estimator of the true mean of the underlying distribution of returns.

In our example, the arithmetic mean is (+25-4.16+31.8)/3 = 17.5%.

Geometric Mean Return

The geometric mean return provides a more accurate representation of the growth in portfolio value over a given time period than does an arithmetic mean return because it accounts for the compounding of returns.

Geometric Mean Return = [(1 + R1) * (1 + R2) * … * (1 + Rn)]^(1/n) – 1

In our example, the geometric mean return is 16.4%.

Money-Weighted Rate of Return (MWRR)

The money-weighted rate of return is the internal rate of return based on all the portfolio’s cash inflows and outflows. We’ll use the cash flow functions in your calculator to solve for the MWRR.

Real, Nominal, and Leveraged Returns

There are other ways to further express the returns to an investor:

• Gross and net returns
• Pretax and after-tax nominal returns
• Real returns
• Leveraged returns

Gross and Net Returns

Gross return refers to the total returns before the deduction of management and administration fees, while net returns are after these fees. Note that commissions on trades and other trading expenses are not part of management fees and should already have been deducted from the gross returns.

Pretax and After-Tax Nominal Returns

Pretax nominal return refers to the return before paying taxes, and after-tax nominal return refers to the return after the tax liability is deducted.

Real Returns

Real return measures the increase in an investor’s purchasing power, that is, how much more goods the investor can purchase at the end of one year due to the increase in the investment value. Nominal returns can be considered to be made up of real returns and an inflation adjustment.

Real Return = (1 + Nominal Return) / (1 + Inflation Rate) – 1

Leveraged Returns

Leveraged return is common in derivative and real estate investments, where some investors purchase on margin. As the portfolio is partly financed through debt, the actual cash outlay by the investor is lower than the portfolio value. When the portfolio value increases, the proportion of increase based on the cash outlay by the investor is larger, and this is known as the leveraged return. The leveraged return should take into account the cost of financing the loan, management fees, admin fees, and commissions and trading fees. The actual calculation of leveraged returns is covered in detail in the equity investments course.

Conclusion

And that wraps up our lesson on measures of return! We’ve learned about single-period and multi-period returns, arithmetic mean, geometric mean, money-weighted returns, and various other return measures.