1. Internal Rate of Return 內部報酬率
張智星 (Roger Jang)
台大資工系 多媒體檢索實驗室

2. 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
Quiz:
Definition of IRR in plain texts!
The term “internal” indicates it does not take
environmental factors (e.g., inflation) into consideration.

3. IRR Example Is this investment plan good? Profit=250?
Bank interest rate=2%
Bank interest rate=5%
What is the equivalent interest rate?
NPV: Net present value
 NPV=121.07
 NPV=-51.73

4. Definition of IRR in an equation!
IRR Computation
IRR satisfies the following equation:
Or equivalently:
Quiz:
Definition of IRR in an equation!
Net
Present
value
Cash flow table:
NPV
NFV
Net
Future
value

5. IRR Example An simple example of IRR Cash flow table Equation Year (i)
Cash flow (ci)
-1234 1
362 2
548 3
481

6. IRR Computed by MATLAB Case 1: Simplified formula
Case 2: Formula used by the banks
fcn=' /(1+x)+548/(1+x)^2+481/(1+x)^3';
x0=0;
r=fzero(fcn, x0)
fcn=' /(1+x/12)^12+548/(1+x/12)^24+481/(1+x/12)^36';
x0=0;
r=fzero(fcn, x0)

7. Display Intermediate Results
fcn=' /(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
initial interval
search
search
search
search
Search for a zero in the interval [-0.08, 0.08]:
Func-count x f(x) Procedure
initial
interpolation
interpolation
interpolation
e interpolation
e interpolation
.....
e interpolation
Zero found in the interval [-0.08, 0.08]

8. Plot NPV as a function of IRR
Function handle to an anonymous function
/(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
Note that npv is a function handle
Quiz: Can you write a function to find all roots
of a given equation in a given interval?

9. 儲蓄險比較
郵局六年期吉利保險
遠雄好鑽養老保險

10. 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
-16320 1
-16157 2 3 4 5 6
100000
IRR
0.839%
 Yearly compounding

11. Element-wise operation
郵局六年期吉利保險
irrFind.m
Main script
function r=irrFind(cashFlowVec, x0)
x0);
function npv=npvCompute(x)
n=length(cashFlowVec);
npv=sum(cashFlowVec./((1+x).^(0:n-1)));
end
Note that:
npvCompute is a function
@npvCompute is a function handle
Function of function
Element-wise operation
Required!
cashFlowVec=[-16320, , , , , , ];
x0=0;
r=irrFind(cashFlowVec, x0)
 r=0.839% (Yearly compounding)

12. 年化報酬率 年化報酬率 應用：投資比較 說明一、說明二 方案一： 2年賺20%  r=9.54%

13. References
References for IRR
綠角財經筆記  A very good example of IRR
Wiki