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Internal Rate of Return 內部報酬率. 張智星 (Roge Jang) [email protected] http://mirlab.org/jang 台大資工系 多媒體檢索實驗室. 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 … - PowerPoint PPT Presentation
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Scientific Computing: Compounding
Intro to IRR
Internal rate of return (IRR) Definition: The rate at which an investment
plan breaks even Also known as
Effective interest rate Annualized effective compounded return rate …
Application: Comparison of 2 investment plans
The term “internal” indicates it does not takeenvironmental factors (e.g, inflation) into consideration.
Scientific Computing: Compounding
IRR Example Is this investment plan good? Profit=250?
Bank interest rate=2% NPV=121.07
Bank interest rate=5% NPV=-51.73
What is the equivalent interest rate?
NPV: Net present value
Scientific Computing: Compounding
IRR Computation IRR satisfies the following equation:
Or equivalently:
RRr
n
ic
r
cNPV
i
n
ii
i
I:
periods ofnumber total:
periodat flowcash :
010
010
n
i
ini rcNFV
NetPresentvalue
NetFuturevalue
0c 1c 2c 3c nc
Cash flow table:
NPV NFV
Scientific Computing: Compounding
IRR Example An simple example of IRR
Cash flow table
Equation
06.00
1
481
1
548
1
3621234 321
r
rrrNPV
1234 362 548Year (i)Cash flow
(ci)
0 -1234
1 362
2 548
3 481
481
Scientific Computing: Compounding
IRR Computed by MATLAB Case 1
Case 2
%96.50
1
481
1
548
1
3621234 321
r
rrrNPV
%80.50
12/1
481
12/1
548
12/1
3621234 362412
r
rrrNPV
fcn='-1234+362/(1+x)+548/(1+x)^2+481/(1+x)^3';x0=0;r=fzero(fcn, x0)
fcn='-1234+362/(1+x/12)^12+548/(1+x/12)^24+481/(1+x/12)^36';x0=0;r=fzero(fcn, x0)
Scientific Computing: CompoundingDisplay Intermediate
Resultsfcn='-1234+362/(1+x/12)^12+548/(1+x/12)^24+481/(1+x/12)^36';x0=0;opt=optimset('Display', 'iter');r=fzero(fcn, 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] .....
Scientific Computing: CompoundingPlot NPV as a function of
IRR
npv=@(x) -1234+362/(1+x/12)^12+548/(1+x/12)^24+481/(1+x/12)^36;x0=0;opt=optimset('Display', 'iter');r=fzero(npv, x0, opt);ezplot(npv, [0, 0.08]);line(r, npv(r), 'marker', 'o', 'color', 'r');grid on
Anonymous functions
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
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 -16320
1 -16157 -16157
2 -16157 -16157
3 -16157 -16157
4 -16157 -16157
5 -16157 -16157
6 100000
100000
IRR0.839
%
Yearly compounding
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))); endend
cashFlowVec=[-16320, -16157, -16157, -16157, -16157, -16157, 100000];x0=0;r=irrFind(cashFlowVec, x0) r=0.839% (Yearly compounding)
Scientific Computing: Compounding
年化報酬率 年化報酬率
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%
Scientific Computing: Compounding
References References for IRR
綠角財經筆記 A very good example of IRR
Wiki
