# Return Concepts

## Holding Period Return (HPR)

The HPR is the return earned from investing in an asset for a specified period, and is given by:

$$\text{r}=\frac{\text{D}_{\text{H}}}{\text{P}_{0}}+\frac{\text{P}_{\text{H}}-\text{P}_{0}}{\text{P}_{0}}=\frac{\text{D}_{\text{H}}+\text{P}_{\text{H}}}{\text{P}_{0}}-1$$

Where:

$$\text{D}_{\text{H}}=$$ Investment income, e.g., dividends.

$$\text{P}_{\text{t}}=$$ Share price at time $$t$$.

$$\text{P}_{0}=$$ Share price at time $$0$$.

The holding period return is the sum of two components: dividend yields $$\bigg(\frac{\text{D}_{\text{H}}}{\text{P}_{0}}\bigg)$$ and price appreciation $$\bigg(\frac{\text{P}_{\text{H}}-\text{P}_{0}}{\text{P}_{0}}\bigg)$$.

This equation assumes the dividends are received at the end of the holding period. Otherwise, the dividends would be reinvested in additional shares on the date and at the price the dividends were received.

Holding period returns are sometimes annualized e.g., a one-day holding return of 0.8% would be 1,732.71%. This is done by compounding the holding period rate in the following way:

$$(1.008)^{365}-1=17.3271 = 1,732.71\%$$

However, annualizing holding periods less than a year may be unrealistic because the reinvestment rate may not be available.

### Realized HPR vs. Expected HPR

The holding period return, when the selling price, $$\text{P}_{\text{H}}$$, and dividend, $$\text{D}_{\text{H}}$$ are known, is called a realized holding period return or a realized return.

In a forward-looking context, an investor forms an expectation about the dividends and selling price. A return calculated using these expected dividends and the selling price is known as the expected holding period return or expected return. Different investors may have different expected returns.

## Required Return

A required return is the minimum level of expected return required by an investor to invest in an asset for a specified period, given its risk level. It represents the opportunity cost (the highest level of expected return available from investments of similar risk) of investing in an asset.

The required return represents a threshold for being fairly compensated for the risk of the asset. If the investor’s expected return exceeds the required return, the asset will appear to be undervalued. If the expected return is less than the required return, the asset will appear to be overvalued.

The required return on common stock and debt are known respectively as cost of equity and cost of debt. The difference between the expected return and required return is the asset’s expected alpha or ex-ante alpha or expected abnormal return, which will be the basis of many calculations in level III.

$$\text{Expected alpha} = \text{Expected return} – \text{Required return}$$

When an asset’s price equals its intrinsic value, the expected return should equal the required return and the expected alpha is zero.

To evaluate the returns of an investment, an analyst examines the realized alpha.

$$\text{Realized alpha} = \text{Actual holding period return} – \text{Required return for the period}$$

## Expected Return Estimates from Intrinsic Value Estimates

An investor’s expected return has two components:

1. The required return earned on the asset’s current market price, ($$\text{r}_{\text{t}}$$).
2. The return from the convergence of price to value, $$\bigg[\frac{\text{V}_{0}-\text{P}_{0}}{\text{P}_{0}}\bigg]$$.

The expected return $$\text{E}(\text{R}_{\text{t}})$$ is, therefore:

$$\text{E}(\text{R}_{\text{t}})=\text{r}_{t}+\frac{\text{V}_{0}-\text{P}_{0}}{\text{P}_{0}}$$

There are risks involved in this:

• The investor’s estimate of intrinsic value may not reflect the asset’s true intrinsic value.
• Convergence of price to value may not occur within the investor’s investment horizon.

## Discount Rate

The discount rate is the rate used to calculate the present values of future cash flows. It equals the risk-free rate plus the necessary compensation for risk. Clearly, the discount rate used to determine the intrinsic value of an investment depends on the inherent characteristics of the investment.

In practice, expected future cash flows may have varying risks and different discount rates may apply to each distinct expected cash flow but practically a single rate is used on all future cash flows.

## Internal Rate of Return

The internal rate of return (IRR) is the discount rate that equates the present values of the future cash flows to the asset’s price.

Assuming a stable growth rate and defining cash flows as dividends, the intrinsic value of a stock can be estimated as:

$$\text{Intrinsic value}=\frac{\text{Next period’s expected dividends}}{\text{Required return}-\text{Expected dividend growth rate}}$$

If an asset’s market price is equal to its intrinsic value, the IRR would equal the required rate of return. The IRR would be estimated as:

$$\text{Required return estimate}=\frac{\text{Next period’s expected dividend}}{\text{Market price}}+\text{Expected dividend growth rate}$$

## Question

Which of the following rates of return most appropriately equates the present values of the future cash flows to the asset’s price?

1. Holding period return.
2. Expected rate of return.
3. Internal rate of return.

#### Solution

The internal rate of return is the rate that equates the present values of the future cash flows to the assets price.

A is incorrect. The holding period return is the rate earned from investing in an asset for a specific period.

B is incorrect. The expected rate of return is a return calculated using the expected dividends and selling price.

LOS 21 (a) Contrast realized holding period return, expected holding period return, required return, return from convergence of price to intrinsic value, discount rate, and internal rate of return.

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