计算FV时，Return Rate什么时候用到return after realized taxes, 什么时候又直接用return before tax呢（如题）？

问题如下：

Allen is an investment manager, who recently recruited a new customer, Jack. Jack invested $200,000 in an asset for 15 years that promises to offer an annual return of 12% before tax. Assume the cost basis is as same as market value and the investment should be imposed a tax of 10% at the end of the investment period. Calculate the accrual equivalent tax rate on this investment.

选项：

2.09%

5.54%

5.25%

解释：

C is correct.

考点：Accrual equivalent tax rates

解析：首先这题描述的情况是一个递延税的情况，因为税收发生在投资期结束时刻。因此我们运用递延税的计算出投资期期末时刻的资产的终值。

由于投资资产的cost basis和market value相等，所以B=1，

所以$FVIF_{CGT}=\left(1+R\right)^N-\left[\left(1+R\right)^N-1\right]\times T_{CG}$

$FV=200,000\left\{\left(1+12\%\right)^{15}-\lbrack\left(1+12\%\right)^{15}-1\rbrack\times10\%\right\}=1,005,242$

接着我们再套用Effective annual return的公式计算$R_{AE}$

$200,000{(1+R_{AE})}^{15}=1,005,242$ ，反解这个公式得， $R_{AE}=11.37\%$

之后再运用Accrual equivalent tax rates的公式求解最终答案，即：

$R_{AE}=R(1-T_{AE})\rightarrow T_{AE}=1-\frac{R_{AE}}R=1-\frac{11.37\%}{12\%}=5.25\%$