Finance - Ch2 資產報酬計算 Asset Return Calculations

一、為何研究報酬率

大部份研究財務的文獻，都將研究的焦點放在報酬率，而較少研究資產價格。其原因

• 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

持有期收益率是一個重要的投資效率指標，它是指從購入到賣出這段特有期限裡所能得到的收益率。持有期收益率和到期收益率的差別在於將來值的不同。

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

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

令 $P_t$ 代表某一金融資產在 $t$ 時間的價格 (例如某一檔股票在 $t$ 時間的價格)，而 $P_{t−1}$則代表此一金融資產在 $t−1$ 時間的價格。則我們可以計算出「簡單淨報酬率」(simple net return) 為：

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

$R_t$ 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

跨期簡單毛淨報酬可以由各期的簡單毛淨報酬相乘而得。

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

顯然當$R_{t-1}R_t$足夠小時，$R_t(2) = R_{t-1} +R_t$。將上方的算式一般化可表示為 :

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

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

$1 + R_A = 1 + R_t(12) = (1+R(2))^6$
$R_A = (1+R(2))^6 - 1$

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

財務計量導論
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

持有期收益率(Holding Period Return，HPR)
http://wiki.mbalib.com/wiki/%E6%8C%81%E6%9C%89%E6%9C%9F%E6%94%B6%E7%9B%8A%E7%8E%87