FINN403
Financial Modelling
Fall 2023
Week 1
Instructor:
Talha Farrukh
•
Representation of behavior/ real world observations through
mathematical approach designed to anticipate range of outcomes
•
Tools used to set and manage a suite of variable assumptions in order to
predict the financial outcomes of an opportunity
•
A numerical story….
•
Forecasting wealth by putting money away now/investing
•
Something to do with money and fashion
•
I really have no idea
•
A complex spreadsheet
What is Financial Modelling?
•
A business case that determines whether to go ahead with a project
•
Investment Analysis of a Joint venture
•
An actual vs budget monthly variance analysis
•
Dashboard report
•
Risk assessment
Some examples
•
Spreadsheet is any type of information stored in Excel including a
financial model. Raw Data, check list
•
Financial Model is more structured
Variable assumptions
– Inputs and outputs
– Calculations
– Scenarios
–
•
Financial Model should be more dynamic
•
Shows a hypothetical outcome
Difference between a Spreadsheet and a FM
•
Risk Management
•
Project Planning
•
KPI and Benchmarking
•
Dashboards
•
Balance Scorecards
Types and purposes of FM
•
Most people have it
•
Faster
•
Knowledge
•
Easy!
•
Add Ons
Why use excel
•
Spreadsheet and technical skills (Vlookup, formulas)
•
Industry knowledge (Regulatory constraints, industry standards)
•
Accounting knowledge (P&L, BS)
•
Business knowledge(cost of capital, KPI)
•
Aesthetic skills
•
Communication and Language skills
•
Numeracy skills
•
Ability to think logically
What skills you need to be a good FM
The ideal Financial Modeller?
•
Cover sheet
•
Input sheet
•
Output, Summary and Scenario Sheets
•
Calculations and workings
•
Error Check sheet
Anatomy of a Model
Excel
Techniques
Corporate
Finance
Option
Pricing
Course Elements
Bonds
Portfolio
Models
•
Attendance and CP -10%
•
Spreadsheet Assignments - 20% (with N-1 and no petitions accepted)
•
Mid Term - 25%
•
Final – 30%
•
Group Project- 15%
•
All assessments will be held in lab
Course Components
•
Net Present Value
•
Internal Rate of Return
•
Payment Schedules and Loan Tables
•
Future Value
•
Pension and Accumulation problems
•
Continuously compounded interest
•
Time Dated cash flows (XNPV & XIPR)
Excel applications
•
You have decided to put away $1000 in your investment account every year for the next 5 years. You will
make the first deposit starting at end of the year. How much will you have in your account by the end of 5
years if annual interest rate is 3%? What if the deposit is made at the start of the year?
•
Your father has promised you $800 every year starting after one year for the next 7 years . How much is
this worth today if Banks are paying interest of 3% per annum? What is the value if he gives you the
money at the end of the year?
•
Suppose your bank qoutes an APR of 12% compounded monthly. What is the Effective Annual Rate?
•
If the banks are paying an annual interest rate of 5%, how long would it take you to double your
investment?
Problems
Present Value and Net Present Value
•
Both concepts are related to the value today of a set of future anticipated cash flows.
•
As an example, suppose we are valuing an investment that promises $100 per year at
the end of this and the next four years.
•
We suppose that these cash flows are risk-free
•
A bank pays an annual interest rate of 4% on a 5-year deposit
A
B
C
D
1
COMPUTING THE PRESENT VALUE
2 Discount rate
4%
Present value
3
Year
Cash flow
4
0
5
1
100
96.15 <--=B5/(1+$B$2)^A5
6
2
100
92.46 <--=B6/(1+$B$2)^A6
7
3
100
88.90 <--=B7/(1+$B$2)^A7
8
4
100
85.48 <--=B8/(1+$B$2)^A8
9
5
100
82.19 <--=B9/(1+$B$2)^A9
10
11 Present value
12
Summing cells C5:C9
445.18 <--=SUM(C5:C9)
13
Using Excel's NPV function
445.18 <--=NPV(B2,B5:B9)
14
Using Excel's PV function
445.18 <--=PV(B2,5,-100)
Financial Modeling 5th ed. Chapter 1
The Difference between Excel’s PV and NPV
Functions
•
NPV can handle any series of cash flows, PV can handle only constant cash flows.
A
B
C
D
COMPUTING THE PRESENT VALUE
In this example the cash flows are not equal
Either discount each cash flow separately or use Excel's NPV function
Excel's PV doesn't work for this case
1
2 Discount rate
3
Year
4
0
5
1
6
2
7
3
8
4
9
5
10
11 Present value
12
Summing cells C5:C9
13
Using Excel's NPV function
4%
Cash
100
200
300
400
500
Present value
96.15
184.91
266.70
341.92
410.96
1,300.65 <--=SUM(C5:C9)
1,300.65 <--=NPV(B2,B5:B9)
<--=B5/(1+$B$2)^A5
<--=B6/(1+$B$2)^A6
<--=B7/(1+$B$2)^A7
<--=B8/(1+$B$2)^A8
<--=B9/(1+$B$2)^A9
Excel’s NPV Function Is Misnamed!
•
In standard finance terminology, the present value of a series of cash flows is the
value today of the future cash flows:
–
•
𝐶𝐹𝑡
1+𝑟 𝑡
The net present value is the present value minus the cost of acquiring the asset
(the cash flow at time zero):
–
•
𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 = σ𝑁
𝑡=1
𝑁𝑒𝑡 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 = σ𝑁
𝑡=0
𝐶𝐹𝑡
1+𝑟 𝑡
=
𝐶
𝐹0
ต
↑
In many cases
CF0 <0
meaning that it represents the
price paid for the asset.
+
σ𝑁
𝑡=1
𝐶𝐹𝑡
1+𝑟 𝑡
↑
This is the present
value,given by Excel’s
NPV function.
Excel’s differs somewhat from the standard finance nomenclature.
The Net Present Value (NPV)
•
The NPV is calculated as: 𝑁𝑃𝑉 = 𝐶𝐹0 + σ𝑁
𝑡=1
–
•
𝐶𝐹𝑡
1+𝑟 𝑡
where 𝐶𝐹𝑡 is the investment’s cash flow at time 𝑡 and 𝐶𝐹0 is today’s cash flow.
Suppose, for example, that the series of five cash flows of $100 is sold for $250. Then, as shown below, the NPV = 195.18, r=4%.
A
B
C
D
1
COMPUTING THE NET PRESENT VALUE
2 Discount rate
4%
Present value
3
Year
Cash flow
4
0
-250
-250.00 <--=B4/(1+$B$2)^A4
5
1
100
96.15 <--=B5/(1+$B$2)^A5
6
2
100
92.46 <--=B6/(1+$B$2)^A6
7
3
100
88.90 <--=B7/(1+$B$2)^A7
8
4
100
85.48 <--=B8/(1+$B$2)^A8
9
5
100
82.19 <--=B9/(1+$B$2)^A9
10
11 Net present value
12
Summing cells C4:C9
195.18 <--=SUM(C4:C9)
13
Using Excel's NPV function
195.18 <--=B4+NPV(B2,B5:B9)
•
The NPV represents the wealth increment that accrues to the purchaser of the cash flows.
•
The market price of our five cash flows of 100 ought to be 445.18.
The Present Value of an Annuity and a Perpetuity
•
An annuity is a security that pays a constant cash flow in each period in the future.
•
When the periodic cash flow is “C” and the appropriate discount rate is r, then the value
today of the annuity, paying n periods its present value, is:
–
𝑃𝑉 𝑜𝑓 𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =
𝐶
1+𝑟 1
+
𝐶
1+𝑟 2
+⋯ +
A
1
2
3
4
5
6
7
8
𝐶
1+𝑟 𝑛
=
𝐶
𝑟
1−
1
1+𝑟 𝑛
B
C
COMPUTING THE VALUE OF AN ANNUITY
Periodic payment, C
Number of future periods paid, n
Discount rate, r
Present value of annuity
Using formula
Using Excel's PV function
1,000
5
6%
4,212.36 <--=B2*(1-1/(1+B4)^B3)/B4
4,212.36 <--=PV(B4,B3,-B2)
Annuity payment
9
Period
10
0
11
1
1,000.00 <--=B2
12
2
1,000.00
13
3
1,000.00
14
4
1,000.00
15
5
1,000.00
16
NPV function
17 Present value using Excel'sFinancial
4,212.36
Modeling 5th ed.
Chapter<--=NPV(B4,B11:B15)
1
The Present Value of a Perpetuity
•
When the series is infinite, it is called perpetuity.
•
Computing the value of a perpetuity (an infinite annuity) is even simpler, as
shown in the spreadsheet below.
A
1
THE VALUE OF
2 Periodic payment, C
3 Discount rate, r
4 Present value of annuity
B
C
D
E
A PERPETUITY (AN INFINITE ANNUITY)
1,000
6%
16,666.67 <--=B2/B3
The Value of a Growing Annuity
•
A growing annuity pays out a sum C, which grows at a periodic growth rate g. If
the annuity is finite, its value today is given by:
–
𝑃𝑉 𝑜𝑓 𝑔𝑟𝑜𝑤𝑖𝑛𝑔 𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =
𝐶
1+𝑟 1
𝐶 1+𝑔
+
1+𝑟 2
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
𝐶 1+𝑔 2
+
1+𝑟 3
B
𝐶 1+𝑔 𝑛−1
+⋯+
1+𝑟 𝑛
C
COMPUTING THE VALUE OF A FINITE GROWING ANNUITY
First payment, C
Growth rate of payments, g
Number of future periods paid, n
Discount rate, r
Present value of annuity
Using formula
Period
0
1
2
3
4
5
Present value using Excel's NPV function
1,000
3%
5
6%
4,457.43 <--=B2*(1-((1+B3)/(1+B5))^B4)/(B5-B3)
Annuity payment
1,000.00
1,030.00
1,060.90
1,092.73
1,125.51
<--=B2
<--=$B$2*(1+$B$3)^(A12-$A$11)
<--=$B$2*(1+$B$3)^(A13-$A$11)
<--=$B$2*(1+$B$3)^(A14-$A$11)
<--=$B$2*(1+$B$3)^(A15-$A$11)
4,457.43 <--=NPV(B5,B11:B15)
=
𝐶
𝑟−𝑔
1−
1+𝑔 𝑛
1+𝑟
The Value of a Growing Perpetuity
•
Taking this formula and letting n → , we can compute the value of a growing
perpetuity (infinite growing annuity) as follows:
–
𝑃𝑉 𝑜𝑓 𝑔𝑟𝑜𝑤𝑖𝑛𝑔 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦 =
𝐶
1+𝑟 1
𝐶 1+𝑔
+
1+𝑟 2
A
1
2
3
4
5
B
𝐶 1+𝑔 2
𝐶
+
+.
.
.
=
, 𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑
1+𝑟 3
𝑟−𝑔
𝑔<𝑟
C
COMPUTING THE VALUE OF A GROWING
PERPETUITY (AN INFINITE GROWING ANNUITY)
Periodic payment, C
Growth rate of payments, g
Discount rate, r
Present value of annuity
1,000 <-- Starting at date 1
3%
6%
33,333.33 <--=IF(B3<B4,B2/(B4-B3),"NA")
21
The Gordon Formula
•
The Gordon formula values a stock by discounting its future anticipated dividends
at the cost of equity 𝑟𝐸 . Letting 𝑃0 be the current stock price, 𝐷𝑖𝑣0 the current
dividend, and 𝑔 the growth rate of future dividends, then:
–
•
𝑃0 =
𝐷𝑖𝑣0 1+𝑔 𝑡
∞
σ𝑡=1
.
1+𝑟 𝑡
𝐸
Using the formula for an infinite growing annuity, we can write this as
–
𝑃0 =
𝐷𝑖𝑣0 1+𝑔
𝑟𝐸 −𝑔
, 𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝑔 < 𝑟𝐸 .
The Internal Rate of Return (IRR)
•
The internal rate of return (IRR) is defined as the compound rate of return r,
which makes the NPV equal to zero:
–
𝐼𝑅𝑅 = 𝑟ȁ𝑁𝑃𝑉(𝑟) = 0
–
→ 𝐶𝐹0 + σ𝑁
𝑡=1
𝐶𝐹𝑡
1+𝐼𝑅𝑅 𝑡
=0
A
1 COMPUTING THE
2
Year
3
0
4
1
5
2
6
3
7
4
8
5
9
10 Internal rate or return (IRR)
•
B
C
INTERNAL
RATE OF RETURN
Cash
flow
-800
100
200
300
400
500
19.54% <-- =IRR(B3:B8)
Note that the Excel IRR function includes as arguments all of the cash flows of the
investment, including the first (in this case negative) cash flow of –800.
Investment Tables and the Internal Rate of
Return
•
The IRR is the compound rate of return paid by the investment.
•
To understand this fully, it helps to make an investment table, which shows
annual investment value after accumulation of interest rate and the cash
payment.
A
B
C
D
INTERNAL RATE OF RETURN
1
Cash flow
2
Year
3
0
-800
4
1
100
5
2
200
6
3
300
7
4
400
8
5
500
9
10 Internal rate or return (IRR)
19.54% <-- =IRR(B3:B8)
11
12
USING THE IRR IN AN INVESTMENT
13
=B17*$B$10
14
=-B3
E
F
TABLE
=B4
15
16
17
18
19
20
21
22
23
24
25
Year
1
2
3
4
5
=E17
Investment at
beginning of year
800.00
856.31
823.61
684.53
418.28
Income
156.31
167.31
160.92
133.75
81.72
Cash flow
Investment at
at end of year end of year
100.00
856.31
<-- =B17+C17-D17
200.00
823.61
300.00
684.53
400.00
418.28
500.00
0.00
The remaining investment principal in the year
after the last cash flow is zero, indicating that all
the principal has been repaid.
Multiple Internal Rates of Return
•
Sometimes a series of cash flows has more than one IRR.
•
In the example below we can tell that the cash flows in cells B6:B11 have two IRRs, since the NPV graph
crosses the x-axis twice.
A
1
2 Discount rate
3 NPV
B
C
D
E
6%
-3.99 <-- =NPV(B2,B7:B11)+B6
4
5
Year
Cash flow
6
0
-145
7
1
100
8
2
100
9
3
100
10
4
100
11
5
-275
12
13
Two IRRs
14
5
15
16
0
17
0%
10%
20%
30%
18
-5
19
20
-10
21
Discount rate
22
-15
23
-20
24
25
-25
26
27
28
29 Identifying the two IRRs
30 First IRR
8.78% <-- =IRR(B6:B11,0)
31 Second IRR
26.65% <-- =IRR(B6:B11,0.3)
Net present value
F
G
H
I
MULTIPLE INTERNAL RATES OF RETURN
40%
DATA TABLE
Discount
rate
NPV
-3.99 Table header, <-- =B3
0%
-20.00
3%
-10.51
6%
-3.99
9%
0.24
12%
2.69
15%
3.77
18%
3.80
21%
3.02
24%
1.62
27%
-0.24
30%
-2.44
33%
-4.90
36%
-7.53
39%
-10.27
Note: For a discussion of how to create data
tables in Excel, see Chapter 28.
Excel’s IRR function allows us to add an extra argument that
will help us find both IRRs.
Instead of writing =IRR(B6:B11), we write =IRR(B6:B11,guess).
The argument guess is a starting point for the algorithm that
Excel uses to find the IRR; by adjusting the guess, we can
identify both the IRRs.
If you don’t put in a guess, Excel defaults to a guess of 10%.
Flat Payment Schedules
•
A “flat” repayment for a loan is also called “a mortgage”.
•
For example: You take a loan for $10,000 at an interest rate of 7% per year. You
make a series of payments that will pay off the loan and the interest over 10
years.
•
We can use Excel’s PMT function to determine how much each annual payment
should be.
Flat Payment
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
B
C
D
E
F
FLAT PAYMENT SCHEDULES
Loan principal
Interest rate
Loan term
Annual payment
10,000
7%
6 <-- Number of years over which loan is repaid
2,097.96 <-- =PMT(B3,B4,-B2)
=B2
Year
1
2
3
4
5
6
7
=B10-E10
=$B$3*B10
Principal at
beginning of year
10,000.00
8,602.04
7,106.23
5,505.70
3,793.15
1,960.71
0.00
Payment at
end of year
2,097.96
2,097.96
2,097.96
2,097.96
2,097.96
2,097.96
Split payment into:
Return of
principal
Interest
700.00
1,397.96 <-- =C10-D10
602.14
1,495.82
497.44
1,600.52
385.40
1,712.56
265.52
1,832.44
137.25
1,960.71
Future Values – Intro
•
Suppose you deposit $1,000 in an account today, leaving it there for 10 years.
Suppose the account accumulates an annual interest of 10%. How much will you
have at the end of 10 years?
Future Value
A
1
2 Interest
3
B
C
D
E
SIMPLE FUTURE VALUE
10%
Account balance, Interest earned Total in account,
Year
4
beginning of year
during year
end of year
5
1
1,000.00
100.00
1,100.00 <-- =C5+B5
6
2
1,100.00
110.00
1,210.00 <-- =C6+B6
7
3
1,210.00
121.00
1,331.00
8
4
1,331.00
133.10
1,464.10
9
5
1,464.10
146.41
1,610.51 =$B$2*B5
10
6
1,610.51
161.05
1,771.56
11
7
1,771.56
177.16
1,948.72
12
8
1,948.72
194.87
2,143.59
13
9
2,143.59
214.36
2,357.95
=D5
14
10
2,357.95
235.79
2,593.74
15
16 A simpler way
2,593.74 <-- =B5*(1+B2)^10
Future Value (2)
•
Now consider the following problem:
–
–
–
You intend to open a savings account.
Your initial deposit of $1,000 today (the beginning of year 1) will be followed by a similar
deposit at the beginning of years 2, … , 10 (total of 10 deposits at the beginning of each year).
If the account earns 10% per year, how much will you have in the account at the end of year
10?
A
1
2 Interest
3 Annual deposit
4 Number of deposits
5
Year
6
7
1
8
2
9
3
10
4
11
5
12
6
13
7
14
8
15
9
16
10
17
18 Future value
B
C
D
E
F
FUTURE VALUE WITH ANNUAL DEPOSITS
10%
1,000 <-- Made today and at beginning of each of next 9 years
10
Account balance,
Deposit at
Interest earned Total in account,
beginning of year beginning of year
during year
end year
0.00
1,000
100.00
1,100.00 <-- =D7+C7+B7
1,100.00
1,000
210.00
2,310.00 <-- =D8+C8+B8
2,310.00
1,000
331.00
3,641.00
3,641.00
1,000
464.10
5,105.10
5,105.10
1,000
610.51
6,715.61 =$B$2*(C7+B7)
6,715.61
1,000
771.56
8,487.17
8,487.17
1,000
948.72
10,435.89
10,435.89
1,000
1,143.59
12,579.48
12,579.48
1,000
1,357.95
14,937.42
14,937.42
1,000
1,593.74
17,531.17
17,531.17 <-- =FV(B2,B4,-B3,,1)
=E7
A Pension Problem—Complicating the FV
Problem
•
You are currently 55 years old and intend to retire at age 60. To make your
retirement easier, you intend to start a retirement account:
–
–
•
At the beginning of each of years 1, 2, 3, 4, 5 (that is, starting today and at the
beginning of each of the next four years), you intend to make a deposit into
the retirement account. You think that the account will earn 7% per year.
After retirement at age 60, you anticipate living eight more years. At the
beginning of each of these years you want to withdraw $30,000 from your
retirement account. Your account balances will continue to earn 7%.
How much should you deposit annually in the account?
31
Retirement Problem
A
1
2 Interest
3 Annual deposit
4 Annual retirement withdrawal
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Year
1
2
3
4
5
6
7
8
9
10
11
12
13
B
C
D
E
F
A RETIREMENT PROBLEM
7%
31,150.60
30,000.00
=$B$2*(C7+B7)
Account balance,
Deposit at
Interest earned Total in account,
beginning of year beginning of year
during year
end year
0.00
31,150.60
2,180.54
33,331.14 <-- =SUM(B7:D7)
33,331.14
31,150.60
4,513.72
68,995.46
68,995.46
31,150.60
7,010.22
107,156.28
107,156.28
31,150.60
9,681.48
147,988.36
147,988.36
31,150.60
12,539.73
191,678.68
191,678.68
-30,000.00
11,317.51
172,996.19
172,996.19
-30,000.00
10,009.73
153,005.92
153,005.92
-30,000.00
8,610.41
131,616.34
131,616.34
-30,000.00
7,113.14
108,729.48
108,729.48
-30,000.00
5,511.06
84,240.55
84,240.55
-30,000.00
3,796.84
58,037.38
58,037.38
-30,000.00
1,962.62
30,000.00
30,000.00
-30,000.00
0.00
0.00
Would You Prefer to Have $1 Million Cash Right Now or a Cent that
Doubles Every Day for 30 Days?
Cent
Y= 2^(x-1)
Continuous Compounding (I)
•
Suppose you deposit $1,000 in a bank account that pays 5% per year.
•
Suppose that the bank interprets “5% per year” to mean that it pays you 2.5% interest
twice a year.
–
–
•
After six months you’ll have $1,025, and after one year you will have
$1,000 ∗ 1 +
0.05 2
2
= $1,050.625.
By this logic, if you get paid interest
n times per year, your accretion at the end of the
𝑛
0.05
year will be $1,000 ∗ 1 +
.
𝑛
•
As n increases this amount converges to 𝑒 0.05 as continuous compounding.
•
As you can see in the next display, $1,000 continuously compounded for one year at 5%
grows to $1,000 ∗ 𝑒 0.05 = $1,051.271 at the end of the year.
•
Continuously compounded for t years, it will grow to $1,000 ∗ 𝑒 0.05∗𝑡
The conclusion: more
compounding periods increase
the future value
–
•
Though there is a clear
asymptotic value.
As we will see below, for
accretion over t years,
–
This value is
𝑒 𝑟𝑡
Continuous Compounding
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
B
C
MULTIPLE COMPOUNDING PERIODS
Initial deposit
Interest rate
Number of compounding periods per year
Interest per compounding period
Accretion in one year
Continuous compounding with Exp
1,000
5%
2
2.500% <--=B3/B4
1,050.625 <--=B2*(1+B5)^B4
1,051.271 <--=B2*EXP(B3)
Effect of Multiple Compounding Periods
1,051.40
1,051.20
End-year accretion
•
1,051.00
1,050.80
1,050.60
1,050.40
1,050.20
1,050.00
Number of compounding intervals
1,049.80
1
10
Compounding periods per year
1
2
4
12
52
100
150
300
365
730
8760
100
End-year
accretion
1,050.000
1,050.625
1,050.945
1,051.162
1,051.246
1,051.258
1,051.262
1,051.267
1,051.267
1,051.269
1,051.271
1000
10000
<--=$B$2*(1+$B$3/A25)^A25
<--=$B$2*(1+$B$3/A26)^A26
Continuous Discounting
•
The accretion factor for continuous compounding at interest r over t years is 𝑒 𝑟𝑡
•
The discount factor for the same period is 𝑒 −𝑟𝑡 .
•
Thus, a cash flow Ct occurring in year t and discounted at continuously
compounded rate r will be worth 𝐶𝑡 × 𝑒 −𝑟𝑡 today.
A
1
2 Interest
3
B
C
D
CONTINUOUS DISCOUNTING
8%
4
Year
Cash flow
5
1
100
6
2
200
7
3
300
8
4
400
9
5
500
10
11 Present value
Continously
discounted PV
92.31 <-- =B5*EXP(-$B$2*A5)
170.43 <-- =B6*EXP(-$B$2*A6)
235.99
290.46
335.16
1,124.35 <-- =SUM(C5:C9)
Continuously Compounded Return from Price
Data
•
Suppose at time 0 you had $1,000 in the bank and suppose that one year later
you had $1,200.
•
If there are n compounding periods per year, you have to solve
–
𝑟=𝑛∗
1,200 1/𝑛
1,000
−1 .
A
•
If n is very large, this converges to
•
𝑟 = 𝑙𝑛
1,200
1,000
= 18.2322%.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
B
C
CALCULATING RETURNS FROM PRICES
Initial deposit
End-of-year value
Number of compounding periods
Implied annual interest rate
1,000
1,200
2
19.09% <-- =((B3/B2)^(1/B4)-1)*B4
Continuous return
18.23% <-- =LN(B3/B2)
Implied annual interest rate with n compounding periods
Number of compounding periods
1
2
4
12
24
365
1000
Rate
19.09% <-- =B5, data table header
20.00%
19.09%
18.65%
18.37%
18.30%
18.24%
18.23%
Discounting Using Dated Cash Flows
•
The function XIRR outputs an
annualized return. It works by
computing the daily IRR and
annualizing it:
–
𝑋𝐼𝑅𝑅 = 1 + 𝑑𝑎𝑖𝑙𝑦𝐼𝑅𝑅
365
− 1.
•
XNPV computes the net present
value of a series of cash flows
occurring on specific dates:
A
B
C
USING XNPV TO COMPUTE THE NET
PRESENT VALUE
A
B
C
USING XIRR TO COMPUTE THE ANNUALIZED
INTERNAL RATE OF RETURN
1
2
Date
3
01-Jan-19
4
03-Mar-19
5
04-Jul-20
6
12-Oct-20
7
25-Dec-20
8
9 IRR
Cash flow
-1,000
150
100
50
1,000
16.67% <-- =XIRR(B3:B7,A3:A7)
1
2 Annual discount rate
3
4
Date
5
01-Jan-19
6
03-Mar-20
7
04-Jul-20
8
12-Oct-21
9
25-Dec-22
10
11 Net present value
12
8%
Cash flow
-1,000
100
195
350
800
136.37 <-- =XNPV(B2,B5:B9,A5:A9)
Note that XNPV has a different syntax from NPV !
XNPV requires all the cash flows, including the initial cash flow,
13 whereas NPV assumes that the first cash flow occurs one period hence.