Rates of Return¶

Holding Period Return¶

Consider a stock with beginning price $\smash{P_0}$, ending price $\smash{P_1}$ and a dividend payment of $\smash{d}$.

The holding period return is

\[\begin{split}\begin{align*} HPR & = \frac{P_1 - P_0 + d}{P_0} \\ & = \underbrace{\frac{P_1 - P_0}{P_0}}_\text{capital gains yield} + \underbrace{\frac{d}{P_0}}_\text{dividend yield}. \end{align*}\end{split}\]

This definition can be used for assets other than stocks (e.g. a bond with a coupon payment).

Holding Period Return Example¶

On Nov 9th 2012, Apple stock closed at $\smash{P_0 = \$547.06}$.

On Nov 12th, Apple payed a dividend of $\smash{d = \$2.65}$ per share and the price closed at $\smash{P_1 = \$542.83}$.

What was the HPR?

\[\begin{split}\begin{align*} HPR & = \frac{\$542.83 - \$547.06}{\$547.06} + \frac{\$2.65}{\$547.06} \\ & = \frac{-\$4.23}{\$547.06} + \frac{\$2.65}{\$547.06} \\ & = \frac{-\$1.58}{\$547.06} \\ & = -0.00289. \end{align*}\end{split}\]

Gross and Net Returns¶

Forget dividends or cash payouts for a moment.

The capital gains yield is

\[\begin{split}\begin{align*} \underbrace{\frac{P_1 - P_0}{P_0}}_\text{net return}\,\,\, & = \underbrace{\frac{P_1}{P_0}}_\text{gross return} - \,\,\,\,\, 1. \end{align*}\end{split}\]

Gross and Net Returns¶

What’s the difference between net and gross returns?

The net return is the fraction of your invested money that you gain by holding the asset, excluding the original money.

The gross return is the total gain, including your original money. It is the factor by which you multiply your original invested amount to determine the final invested amount.

Multi-period Returns¶

Suppose an asset has net returns $\smash{\{r_t\}_{t=0}^{T}}$. Consider two forms of average returns:

\[\begin{split}\begin{align*} \text{Arithmetic Average} & = \frac{1}{T} \sum_{t=0}^T r_t \end{align*}\end{split}\]

and

\[\begin{split}\begin{align*} \text{Geometric Average} & = \left(\prod_{t=0}^T (1+r_t)\right)^{\frac{1}{T}}. \end{align*}\end{split}\]

The geometric average is the constant return that would have to be earned each period to yield the same final value of the asset.

Annualized Returns - EAR¶

Suppose you enter into a contract to pay or receive a net rate of return $\smash{r}$ on an asset for each of $\smash{n}$ periods in a year.

$\smash{n=12}$ is a monthly contract.

$\smash{n=4}$ is a quarterly contract.

The Effective Annual Rate (EAR) is

\[\begin{split}\begin{align*} 1 + \text{EAR} & = (1 + r)^n. \end{align*}\end{split}\]

Annualized Returns - APR¶

Suppose you enter into a contract to pay or receive a net rate of return $\smash{r}$ on an asset for each of $\smash{n}$ periods in a year.

The Annual Percentage Rate (APR) is

\[\begin{split}\begin{align*} \text{APR} & = n \times r. \end{align*}\end{split}\]

The APR ignores compounding (as seen in the following example).

Annualized Returns - Example¶

You invest $100 in an asset that pays 5% return each quarter for one year.

\[\smash{Q1: \$100 \times 1.05 = \$105}\]

\[\smash{Q2: \$105 \times 1.05 = \$110.25}\]

\[\smash{Q3: \$110.25 \times 1.05 = \$115.76}\]

\[\smash{Q4: \$115.76 \times 1.05 = \$121.55}\]

Annualized Returns - Example¶

\[\smash{EAR: (1.05)^4 - 1 = 0.2155}\]

\[\smash{APR: 0.05 \times 4 = 0.2}\]

\[\smash{HPR: \frac{\$121.55 - \$100}{\$100} = 0.2155.}\]

EAR and APR¶

What is the relationship between EAR and APR?

Since $r = \frac{\text{APR}}{n}$ we have

\[1 +\text{EAR} = \left(1 + \frac{APR}{n}\right)^n.\]

We can rearrange the equation above to get

\[\text{APR} = \left[(1+\text{EAR})^{\frac{1}{n}} - 1\right] \times n.\]

Continuous Compounding¶

Continuous compounding is what occurs when we allow the number of periods in the year, $\smash{n}$, to become large.

For daily returns, $\smash{n=365}$.

For hourly returns, $\smash{n=8760}$.

For returns each minute, $\smash{n=525,000}$.

For returns each second, $\smash{n=31,536,000}$.

Continuous Compounding¶

Continuous compounding is the limit, when $\smash{n = \infty}$. In this case

\[\lim_{n \to \infty} \left(1 + \frac{\text{APR}}{n}\right)^n = e^{\text{APR}}.\]

So, under continuous compounding

\[\begin{split}\begin{align*} 1 + \text{EAR} & = e^{\text{APR}} \end{align*}\end{split}\]

or

\[\begin{split}\begin{align*} \text{APR} & = \ln(1+\text{EAR}). \end{align*}\end{split}\]

Inflation¶

Inflation is the increase of the general price level over time.

Inflation erodes the purchasing power of a given amount of money over time.

In the presence of inflation, an asset that yields a return of $\smash{r}$ doesn’t actually generate $\smash{r}$ units of additional real purchasing power for each dollar invested.

Nominal vs. Real Returns¶

In the previous slides we computed nominal returns.

Let us momentarily change notation and refer to the nominal return of an asset as $\smash{R}$.

Then the real return of the asset is the nominal return discounted by inflation:

\[1+r = \frac{1+R}{1+\pi}.\]

$\smash{r}$ is the net real return and $\smash{\pi}$ is net inflation.

Nominal vs. Real Returns¶

This relationship is approximated by

\[\begin{split}\begin{align*} r & \approx R - \pi. \end{align*}\end{split}\]

See the proof on the next slide.

Nominal vs. Real Returns - Proof¶

The proof requires an approximation. For some small number $\smash{\varepsilon > 0}$,

\[\begin{split}\begin{align*} \ln(1+\varepsilon) & \approx \varepsilon. \end{align*}\end{split}\]

Thus,

\[\begin{split}\begin{align*} 1+r & = \frac{1+R}{1+\pi} \\ \Rightarrow \ln(1+r) & = \ln\left(\frac{1+R}{1+\pi}\right) \\ \Rightarrow \ln(1+r) & = \ln(1+R) - \ln(1+\pi) \\ \Rightarrow r & \approx R - \pi. \end{align*}\end{split}\]

Nominal vs. Real Returns - Example¶

Suppose you can invest in a CD that pays 8% return over the next year and that inflation is 5% during the same period.

$\smash{R = 0.08}$.

$\smash{\pi = 0.05}$.

$\smash{r \approx 0.08 - 0.05 = 0.03}$.

The actual real rate of return is

\[r = \frac{1.08}{1.05} - 1 = 0.0286.\]

Expected Inflation¶

In practice, future inflation is not known, even though the nominal rate of return may be known with certainty.

Think of a fixed-income asset.

In this case

\[\begin{split}\begin{align*} R & = r + E[\pi]. \end{align*}\end{split}\]

$\smash{E[\pi]}$ is expected inflation.

Expected Inflation¶

The returns to typical government bonds are nominal.

In 1997, the U.S. Treasury introduced “Treasury Inflation-Protected Securities” (TIPS).

These have coupon and principle payments that are corrected for observed inflation over time.

The difference between these rates of return on these two instruments can be treated as a measure of expected inflation.