.. slideconf:: :slide_classes: appear ============================================================================== Rates of Return ============================================================================== Holding Period Return ============================================================================== Consider a stock with beginning price :math:`P_0`, ending price :math:`P_1` and a dividend payment of :math:`d`. .. rst-class:: to-build - The *holding period return* is .. rst-class:: to-build .. math:: HPR & = \frac{P_1 - P_0 + d}{P_0} \\ .. rst-class:: to-build .. math:: & = \underbrace{\frac{P_1 - P_0}{P_0}}_\text{capital gains yield} + \underbrace{\frac{d}{P_0}}_\text{dividend yield}. .. rst-class:: to-build 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:`P_0 = \$547.06`. .. rst-class:: to-build - On Nov 12th, Apple payed a dividend of :math:`d = \$2.65` per share and the price closed at :math:`P_1 = \$542.83`. .. rst-class:: to-build - What was the HPR? .. rst-class:: to-build .. math:: HPR & = \frac{\$542.83 - \$547.06}{\$547.06} + \frac{\$2.65}{\$547.06} .. rst-class:: to-build .. math:: & = \frac{-\$4.23}{\$547.06} + \frac{\$2.65}{\$547.06} .. rst-class:: to-build .. math:: & = \frac{-\$1.58}{\$547.06} = -0.00289. Gross and Net Returns ============================================================================== Forget dividends or cash payouts for a moment. .. rst-class:: to-build - The *capital gains yield* is .. rst-class:: to-build .. math:: \underbrace{\frac{P_1 - P_0}{P_0}}_\text{net return}\,\,\, & = \underbrace{\frac{P_1}{P_0}}_\text{gross return} - \,\,\,\,\, 1. Gross and Net Returns ============================================================================== What's the difference between net and gross returns? .. rst-class:: to-build - The net return is the fraction of your invested money that you gain by holding the asset, excluding the original money. .. rst-class:: to-build - 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:`\{r_t\}_{t=0}^{T}`. Consider two forms of average returns: .. rst-class:: to-build .. math:: \text{Arithmetic Average} & = \frac{1}{T} \sum_{t=0}^T r_t .. rst-class:: to-build and .. rst-class:: to-build .. math:: \text{Geometric Average} & = \left(\prod_{t=0}^T (1+r_t)\right)^{\frac{1}{T}}. .. rst-class:: to-build 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:`r` on an asset for each of :math:`n` periods in a year. .. rst-class:: to-build - :math:`n=12` is a monthly contract. .. rst-class:: to-build - :math:`n=4` is a quarterly contract. .. rst-class:: to-build - The Effective Annual Rate (EAR) is .. rst-class:: to-build .. math:: 1 + \text{EAR} & = (1 + r)^n. Annualized Returns - APR ============================================================================== Suppose you enter into a contract to pay or receive a net rate of return :math:`r` on an asset for each of :math:`n` periods in a year. .. rst-class:: to-build - The Annual Percentage Rate (APR) is .. rst-class:: to-build .. math:: \text{APR} & = n \times r. .. rst-class:: to-build 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. .. rst-class:: to-build .. math:: Q1: \$100 \times 1.05 = \$105 .. rst-class:: to-build .. math:: Q2: \$105 \times 1.05 = \$110.25 .. rst-class:: to-build .. math:: Q3: \$110.25 \times 1.05 = \$115.76 .. rst-class:: to-build .. math:: Q4: \$115.76 \times 1.05 = \$121.55 .. rst-class:: to-build .. math:: EAR: (1.05)^4 - 1 = 0.2155 .. rst-class:: to-build .. math:: APR: 0.05 \times 4 = 0.2 .. rst-class:: to-build .. math:: HPR: \frac{\$121.55 - \$100}{\$100} = 0.2155. EAR and APR ============================================================================== What is the relationship between EAR and APR? .. rst-class:: to-build - Since .. rst-class:: to-build .. math:: r = \frac{\text{APR}}{n} .. rst-class:: to-build we have .. rst-class:: to-build .. math:: 1 +\text{EAR} & = \left(1 + \frac{APR}{n}\right)^n. .. rst-class:: to-build We can rearrange the equation above to get .. rst-class:: to-build .. 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:`n`, to become large. .. rst-class:: to-build - For daily returns, :math:`n=365`. .. rst-class:: to-build - For hourly returns, :math:`n=8760`. .. rst-class:: to-build - For returns each minute, :math:`n=525,000`. .. rst-class:: to-build - For returns each second, :math:`n=31,536,000`. Continuous Compounding ============================================================================== Continuous compounding is the limit, when :math:`n = \infty`. In this case .. rst-class:: to-build .. math:: \lim_{n \to \infty} \left(1 + \frac{\text{APR}}{n}\right)^n = e^{\text{APR}}. .. rst-class:: to-build So, under continuous compounding .. rst-class:: to-build .. math:: 1 + \text{EAR} & = e^{\text{APR}} .. rst-class:: to-build or .. rst-class:: to-build .. math:: \text{APR} & = \ln(1+\text{EAR}). Inflation ============================================================================== Inflation is the increase of the general price level over time. .. rst-class:: to-build - Inflation erodes the purchasing power of a given amount of money over time. .. rst-class:: to-build - In the presence of inflation, an asset that yields a return of :math:`r` doesn't actually generate :math:`r` units of additional real purchasing power for each dollar invested. Nominal vs. Real Returns ============================================================================== In the previous slides we computed nominal returns. .. rst-class:: to-build - Let us momentarily change notation and refer to the nominal return of an asset as :math:`R`. .. rst-class:: to-build - Then the real return of the asset is the nominal return discounted by inflation: .. rst-class:: to-build .. math:: 1+r & = \frac{1+R}{1+\pi}. .. rst-class:: to-build - :math:`r` is the net real return and :math:`\pi` is net inflation. Nominal vs. Real Returns ============================================================================== - This relationship is approximated by .. rst-class:: to-build .. math:: r \approx R - \pi. .. rst-class:: to-build See the proof on the next slide. Nominal vs. Real Returns - Proof ============================================================================== The proof requires an approximation. For some small number :math:`\epsilon > 0`, .. math:: \ln(1+\epsilon) & \approx \epsilon. .. rst-class:: to-build Thus, .. rst-class:: to-build .. math:: 1+r & = \frac{1+R}{1+\pi} .. rst-class:: to-build .. math:: \Rightarrow \ln(1+r) & = \ln\left(\frac{1+R}{1+\pi}\right) .. rst-class:: to-build .. math:: \Rightarrow \ln(1+r) & = \ln(1+R) - \ln(1+\pi) .. rst-class:: to-build .. math:: \Rightarrow r & \approx R - \pi. 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. .. rst-class:: to-build - :math:`R = 0.08`. .. rst-class:: to-build - :math:`\pi = 0.05`. .. rst-class:: to-build - :math:`r \approx 0.08 - 0.05 = 0.03`. .. rst-class:: to-build The actual real rate of return is .. rst-class:: to-build .. 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. .. rst-class:: to-build - Think of a fixed-income asset. .. rst-class:: to-build - In this case .. rst-class:: to-build .. math:: R = r + E[\pi]. .. rst-class:: to-build - :math:`E[\pi]` is expected inflation. Expected Inflation ============================================================================== - The returns to typical government bonds are nominal. .. rst-class:: to-build - In 1997, the U.S. Treasury introduced "Treasury Inflation-Protected Securities" (TIPS). .. rst-class:: to-build - These have coupon and principle payments that are corrected for observed inflation over time. .. rst-class:: to-build - The difference between these rates of return on these two instruments can be treated as a measure of expected inflation.