## Finance - Ch2 資產報酬計算 Asset Return Calculations

### 一、為何研究報酬率

• 財務市場會被視為接近完全市場 : 參與財務市場 (例如股票市場) 交易的門檻較低，個別投資人對某一資產價格變動的影響是微不足道 (Campbell et al. 1997) ，而報酬率恰可反應這種與投資金額大小無關 (scale-free) 的投資機會。
• 報酬率沒有貨幣單位 : 可以跨使用不同貨幣的不同國家資產之投資成果比較。
• 統計上的定態性質 (stationarity) : 從實證的觀點來看，報酬率較價格更具有統計上的定態性質 (stationarity)，所以在實證的應用較為方便。

### 二、簡單報酬率 simple returns

1, 持有期收益率 Holding period returns，HPR

$R(t_0,t_1) = \frac{P_{t1}-P_{t0}}{P_{t0}}$

2. 單期簡單淨報酬率與簡單毛報酬率 Simple net returns and simple gross returns

$R_t = \frac{P_t - P_{t-1}}{P_{t-1}} = \% \Delta P_t$

$R_t$ 的最小值為 -1 或 -100%，而該資產介於 $t-1$ 至 $t$ 時間之簡單毛報酬則可以表示為：

$1 + R_t = \frac{P_t}{P_{t-1}}$

3. 跨期簡單毛淨報酬率 Multi-period returns

$R_t(2) = \frac{P_t}{P_{t-2}} - 1 = \frac{P_t}{P_{t-1}} \times \frac{P_{t-1}}{P_{t-2}} - 1 = (1+R_t)(1+R_{t-1}) -1$

$1+ R_t(k) = (1+R_t)(1+R_{t-1}) \dots (1+R_{t-k+1})$
$= \prod_{k-1}^{j=0}(1+R_{t-j})$

4. 資產組合報酬率 Portfolio returns

Consider an investment of $\$V$in two assets, A and B. The dollar amounts invested in assets A and B are$\$V \times x_A$ and $\$V \times x_B$,$x_A + x_B = 1$. Then$1 + R_{p,t} = x_A + x_B + x_AR_{A,t} + x_BR_{b,t} = 1 + x_AR_{A,t} + x_BR_{b,t}$The portfolio rate of return is equal to a weighted average of the simple returns on assets A and B, where the weights are the portfolio shares$x_A$and$x_B$. In general,$1 + R_{p,t} = \sum_{i=1}^{n} x_i(1 + R_{i,t})$5. 股利對報酬的調整 Adjusting for dividends$1 + R_t^{total} = \frac{P_t + D_t}{P_{t-1}}$6. 通貨膨脹對報酬的調整 Adjusting for Inflation$1 + R_t^{Real} = \frac{ \frac{P_t}{CPI_t} }{ \frac{P_{t-1}}{CPI_{t-1}}}$Assume we define inflation between periods t-1 and t as$\pi_t = \frac{CPI_t - CPU_{t-1}}{CPI_{t-1}} = \%\Delta CPI_t$Then the above formula can be re-expressed as$R_t^{Real} = \frac{1+R_t}{1+\pi_t} - 1$7. 年化報酬率 Annualizing returns Compute annualized return from one-month return$1 + R_A = 1 + R_t(12) = (1+R)^{12}R_A = (1+R)^{12} - 1$Compute annualized return from two-month return$1 + R_A = 1 + R_t(12) = (1+R(2))^6R_A = (1+R(2))^6 - 1$Compute annualized return from two-year return$(1 + R_A)^2 = 1 + R_t(24)R_A = (1 + R_t(24))^{\frac{1}{2}} - 1$### 三、連續複利報酬率 Continuously compounded returns 1. 單期連續複利報酬率 One-period continuously compounded returns The continuously compounded monthly return is defined as:$e^r_t = 1 + R_t = \frac{P_t}{P_{t-1}}$兩邊同取自然對數得$r_t = ln(1+R_t) = ln(\frac{P_t}{P_{t-1}})= ln(P_t) - ln(P_{t-1}) $hence$r_t$can be computed simply by taking the first difference of the natural logarithms of monthly prices. 2. 跨期連續複利報酬率 Multi-period continuously compounded returns 跨期連續複利報酬率可以由各期的連續複利報酬率相加而得。$r_t(2) = ln(\frac{P_t}{P_{t-1}} \times \frac{P_{t-1}}{P_{t-2}})= ln(\frac{P_t}{P_{t-1}}) + ln(\frac{P_{t-1}}{P_{t-2}})= r_t + r_{t-1}$Hence the continuously compounded two-month return is just the sum of the two continuously compounded one-month returns. Recall, with simple returns the two-month return is a multiplicative (geometric) sum of two one month returns. 3. 資產組合連續複利報酬 Portfolio continuously compounded returns 資產組合連續複利報酬率無加總性質。$r_{p,t} = ln(1+R_{p,t}) = ln(1+\sum_{i=1}^{n}x_iR_{i,t}) \neq \sum_{i=1}^{n}x_ir_{i,t}$If the portfolio return is not too large then$r_{p,t} \approx R_{p,t}$otherwise,$r_{p,t} > R_{p,t}$4. 股利對連續複利報酬的調整 Adjusting for dividends 同simple version, 唯一不同是用ln計算. 5. 通貨膨脹對連續複利報酬的調整 Adjusting for inflation$r_t^{Real} = ln(1+R_t^{Real}) = ln(\frac{P_t}{P_{t-1}} \times \frac{CPI_{t-1}}{CPI_t})=r_t - \pi_t^c$Hence, the real continuously compounded return is simply the nominal continuously compounded return minus the the continuously compounded inflation rate. 6. 年化連續複利報酬率 Annualizing continuously compounded returns$r_A = r_t(12) = r_t + r_{t-1} + \dots + r_{t-11} = 12 \times \bar{r}_m\$

That is, the continuously compounded annual return is twelve times the average of the continuously compounded monthly returns.

### References

http://yaya.it.cycu.edu.tw/course/PPT/ch01-PPT-%E8%B2%A1%E5%8B%99%E8%A8%88%E9%87%8F%E5%B0%8E%E8%AB%96.pdf

http://wiki.mbalib.com/wiki/%E6%8C%81%E6%9C%89%E6%9C%9F%E6%94%B6%E7%9B%8A%E7%8E%87