1. Internal Rate of Return 內部報酬率 張智星 (Roge Jang) jang@mirlab.org http://mirlab.org/jang 台大資工系 多媒體檢索實驗室

2. Scientific Computing: Compounding Intro to IRR Internal rate of return (IRR) Definition: The rate at which an investment breaks even Also known as Annualized effective compounded return rate … Application: Comparison of 2 investment project Reference Wiki 綠角財經筆記

3. Scientific Computing: Compounding IRR Computation IRR satisfies the following equation: Or equivalently:

4. Scientific Computing: Compounding IRR Example An simple example of IRR Cash flow table Equation Year (i)Cash flow (ci) 0-1234 1362 2548 3481

5. Scientific Computing: Compounding IRR Computed by MATLAB Case 1 Case 2 fun='-1234+362/(1+x)+548/(1+x)^2+481/(1+x)^3'; x0=0; r=fzero(fun, x0) fun='-1234+362/(1+x/12)^12+548/(1+x/12)^24+481/(1+x/12)^36'; x0=0; r=fzero(fun, x0)

6. Scientific Computing: Compounding Display Intermediate Results fun='-1234+362/(1+x/12)^12+548/(1+x/12)^24+481/(1+x/12)^36'; x0=0; opt=optimset('Display', 'iter'); r=fzero(fun, x0, opt) Search for an interval around 0 containing a sign change: Func-count a f(a) b f(b) Procedure 1 0 157 0 157 initial interval 3 -0.0282843 241.978 0.0282843 77.7252 search 5 -0.04 278.954 0.04 46.4578 search 7 -0.0565685 333.118 0.0565685 3.72982 search 9 -0.08 413.646 0.08 -53.8509 search Search for a zero in the interval [-0.08, 0.08]: Func-count x f(x) Procedure 9 0.08 -53.8509 initial 10 0.0615696 -8.83435 interpolation 11 0.0580284 0.0466363 interpolation 12 0.058047 -0.000198663 interpolation 13 0.0580469 -4.44271e-09 interpolation 14 0.0580469 -2.27374e-13 interpolation..... 23 0.0580469 -2.27374e-13 interpolation Zero found in the interval [-0.08, 0.08].....

7. Scientific Computing: Compounding 儲蓄險比較 郵局六年期吉利保險 遠雄好鑽養老保險

8. Scientific Computing: Compounding 郵局六年期吉利保險 詳細列表 Cash flow table: age of 30, yearly payment 6 年期儲蓄險 - 郵局六年期吉利保險 年度保費領回小計 0-16320 1-16157 2 3 4 5 6 100000 IRR0.839%  Yearly compounding

9. Scientific Computing: Compounding 郵局六年期吉利保險 irrFind.m Main script function r=irrFind(cashFlowVec, x0) r=fzero(@npvCompute, x0); function npv=npvCompute(x) n=length(cashFlowVec); npv=sum(cashFlowVec./((1+x).^(0:n-1))); end cashFlowVec=[-16320, -16157, -16157, -16157, -16157, -16157, 100000]; x0=0; r=irrFind(cashFlowVec, x0)  r=0.839% (Yearly compounding)

10. Scientific Computing: Compounding 年化報酬率 說明一、說明二 說明一說明二 應用：投資比較 方案一： 2 年賺 20%  r=9.54% 方案二： 5 年賺 50%  r=8.45% 方案三： 10 年賺 100%  r=7.18%