.. slideconf:: :slide_classes: appear ============================================================================== Rates of Return ============================================================================== Holding Period Return ============================================================================== Consider a stock with beginning price :math:`\smash{P_0}`, ending price :math:`\smash{P_1}` and a dividend payment of :math:`\smash{d}`. .. raw:: - The *holding period return* is .. raw:: .. math:: \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*} .. raw:: 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 :math:`\smash{P_0 = \$547.06}`. .. raw:: - On Nov 12th, Apple payed a dividend of :math:`\smash{d = \$2.65}` per share and the price closed at :math:`\smash{P_1 = \$542.83}`. .. raw:: - What was the HPR? .. raw:: .. math:: \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*} Gross and Net Returns ============================================================================== Forget dividends or cash payouts for a moment. .. raw:: - The *capital gains yield* is .. raw:: .. math:: \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*} Gross and Net Returns ============================================================================== What's the difference between net and gross returns? .. raw:: - The net return is the fraction of your invested money that you gain by holding the asset, excluding the original money. .. raw:: - 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 :math:`\smash{\{r_t\}_{t=0}^{T}}`. Consider two forms of average returns: .. math:: \begin{align*} \text{Arithmetic Average} & = \frac{1}{T} \sum_{t=0}^T r_t \end{align*} .. raw:: and .. math:: \begin{align*} \text{Geometric Average} & = \left(\prod_{t=0}^T (1+r_t)\right)^{\frac{1}{T}}. \end{align*} .. raw:: 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 :math:`\smash{r}` on an asset for each of :math:`\smash{n}` periods in a year. .. raw:: - :math:`\smash{n=12}` is a monthly contract. .. raw:: - :math:`\smash{n=4}` is a quarterly contract. .. raw:: - The Effective Annual Rate (EAR) is .. raw:: .. math:: \begin{align*} 1 + \text{EAR} & = (1 + r)^n. \end{align*} Annualized Returns - APR ============================================================================== Suppose you enter into a contract to pay or receive a net rate of return :math:`\smash{r}` on an asset for each of :math:`\smash{n}` periods in a year. .. raw:: - The Annual Percentage Rate (APR) is .. raw:: .. math:: \begin{align*} \text{APR} & = n \times r. \end{align*} .. raw:: 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. .. raw:: .. math:: \smash{Q1: \$100 \times 1.05 = \$105} .. raw:: .. math:: \smash{Q2: \$105 \times 1.05 = \$110.25} .. raw:: .. math:: \smash{Q3: \$110.25 \times 1.05 = \$115.76} .. raw:: .. math:: \smash{Q4: \$115.76 \times 1.05 = \$121.55} Annualized Returns - Example ============================================================================== .. math:: \smash{EAR: (1.05)^4 - 1 = 0.2155} .. raw:: .. math:: \smash{APR: 0.05 \times 4 = 0.2} .. raw:: .. math:: \smash{HPR: \frac{\$121.55 - \$100}{\$100} = 0.2155.} EAR and APR ============================================================================== What is the relationship between EAR and APR? .. raw:: Since :math:`r = \frac{\text{APR}}{n}` we have .. raw:: .. math:: 1 +\text{EAR} = \left(1 + \frac{APR}{n}\right)^n. .. raw:: We can rearrange the equation above to get .. raw:: .. math:: \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, :math:`\smash{n}`, to become large. .. raw:: - For daily returns, :math:`\smash{n=365}`. .. raw:: - For hourly returns, :math:`\smash{n=8760}`. .. raw:: - For returns each minute, :math:`\smash{n=525,000}`. .. raw:: - For returns each second, :math:`\smash{n=31,536,000}`. Continuous Compounding ============================================================================== Continuous compounding is the limit, when :math:`\smash{n = \infty}`. In this case .. raw:: .. math:: \lim_{n \to \infty} \left(1 + \frac{\text{APR}}{n}\right)^n = e^{\text{APR}}. .. raw:: So, under continuous compounding .. raw:: .. math:: \begin{align*} 1 + \text{EAR} & = e^{\text{APR}} \end{align*} .. raw:: or .. raw:: .. math:: \begin{align*} \text{APR} & = \ln(1+\text{EAR}). \end{align*} Inflation ============================================================================== Inflation is the increase of the general price level over time. .. raw:: - Inflation erodes the purchasing power of a given amount of money over time. .. raw:: - In the presence of inflation, an asset that yields a return of :math:`\smash{r}` doesn't actually generate :math:`\smash{r}` units of additional real purchasing power for each dollar invested. Nominal vs. Real Returns ============================================================================== In the previous slides we computed nominal returns. .. raw:: - Let us momentarily change notation and refer to the nominal return of an asset as :math:`\smash{R}`. .. raw:: - Then the real return of the asset is the nominal return discounted by inflation: .. raw:: .. math:: 1+r = \frac{1+R}{1+\pi}. .. raw:: - :math:`\smash{r}` is the net real return and :math:`\smash{\pi}` is net inflation. Nominal vs. Real Returns ============================================================================== - This relationship is approximated by .. raw:: .. math:: \begin{align*} r & \approx R - \pi. \end{align*} .. raw:: See the proof on the next slide. Nominal vs. Real Returns - Proof ============================================================================== The proof requires an approximation. For some small number :math:`\smash{\varepsilon > 0}`, .. math:: \begin{align*} \ln(1+\varepsilon) & \approx \varepsilon. \end{align*} .. raw:: Thus, .. raw:: .. math:: \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*} 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. .. raw:: - :math:`\smash{R = 0.08}`. .. raw:: - :math:`\smash{\pi = 0.05}`. .. raw:: - :math:`\smash{r \approx 0.08 - 0.05 = 0.03}`. .. raw:: The actual real rate of return is .. raw:: .. math:: 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. .. raw:: - Think of a fixed-income asset. .. raw:: - In this case .. raw:: .. math:: \begin{align*} R & = r + E[\pi]. \end{align*} .. raw:: - :math:`\smash{E[\pi]}` is expected inflation. Expected Inflation ============================================================================== - The returns to typical government bonds are nominal. .. raw:: - In 1997, the U.S. Treasury introduced "Treasury Inflation-Protected Securities" (TIPS). .. raw:: - These have coupon and principle payments that are corrected for observed inflation over time. .. raw:: - The difference between these rates of return on these two instruments can be treated as a measure of expected inflation.