FINA1310 CORPORATE FINANCE
Lecture 2. The Time Value of Money
Instructor: Dr. Mingzhu TAI
HKU Business School
The University of Hong Kong
Review of Lecture 1
• Three key questions in corporate finance
• Cash flow from assets
• Forms of business organizations
• Corporation and Financial markets
• The goal of financial management and business ethics
• The agency problem
• Textbook Reading: Chapter 1, Chapter 2.4
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Lecture 2: Time Value of Money
2
The Time Value of Money
$1 today > $1 in the future
• Why?
‒ Inflation makes money worth less in term of purchasing power
‒ Real return: you can use today’s money to create more value over time
• You can earn interest from the money in hand
• The following questions remain:
‒ How much of today’s money is equivalent to $1 next year?
‒ How much next year’s money is worth $1 today?
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Lecture 2: Time Value of Money
3
Warm-up Problems
1. A financial advisor offers you an investment contract under which you get
$12,000 back in 3 years. You need to invest $10,000 today into this contract;
‒ If the interest rate is 6.5%, would you make this investment?
2. You have $2,000 now and want to save it to buy an Apple Watch which costs
$3,000
‒ If the interest rate is 5% per month, how long do you need to wait?
3. You need to borrow some money. You can borrow $1,000 from a friend and pay
back $1,250 in 3 years.
‒ What is your borrowing cost?
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Lecture 2: Time Value of Money
4
Today’s Roadmap
• Basic concepts and common terminology
‒ Timeline of cash flows
‒ Basic terms related to the timing of cash flows
• Calculating the future value by compounding
• Calculating the present value by discounting
• Connecting future and present values
‒ How to find the interest rate
‒ How to determine the number of periods
• Textbook Reading: Chapter 5
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Lecture 2: Time Value of Money
5
Today’s Roadmap
• Basic concepts and common terminology
‒ Timeline of cash flows
‒ Basic terms related to the timing of cash flows
• Calculating the future value by compounding
• Calculating the present value by discounting
• Connecting future and present values
‒ How to find the interest rate
‒ How to determine the number of periods
• Textbook Reading: Chapter 5
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Lecture 2: Time Value of Money
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Timeline
t=1
t=0
End of the
first period
Beginning of the
fourth period
End of the
third period
Today
t=2
t=3
t=4
Beginning of the
second period
• Unless otherwise stated, t=0 represents today (the decision point)
• Periods can be defined as any time intervals (days, months, quarters, years, …)
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Lecture 2: Time Value of Money
7
Timeline of Cash Flows
CF0
CF1
CF2
CF3
CF4
t=0
t=1
t=2
t=3
t=4
• Unless otherwise stated, cash flows occur at the end of the period
• Cash inflow: positive amount
• Cash outflow: negative amount
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Lecture 2: Time Value of Money
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Problem with Timing
Option 1:
CF1=+$100
t=0
t=1
Option 1:
CF3=+$110
t=2
t=3
t=4
• Different cash flows occur at different periods:
‒ Problem 1: How to compare CF1 with CF3?
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Lecture 2: Time Value of Money
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Problem with Timing
CF0=-$100
t=0
CF4=+$120
t=1
t=2
t=3
t=4
• Different cash flows occur at different periods:
‒ Problem 2: What is the net value of cash outflow CF0<0 and inflow CF4>0?
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Lecture 2: Time Value of Money
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Today’s Roadmap
• Basic concepts and common terminology
‒ Timeline of cash flows
‒ Basic terms related to the timing of cash flows
• Calculating the future value by compounding
• Calculating the present value by discounting
• Connecting future and present values
‒ How to find the interest rate
‒ How to determine the number of periods
• Textbook Reading: Chapter 5
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Solution for the Timing Issue
Future
What is the value of
CF4 at the present?
t=0
CF4
t=1
t=2
CF0
t=4
What is the value of
CF0 in the future?
Present
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t=3
Lecture 2: Time Value of Money
12
Basic Concepts
1. Present Value (PV)
‒ The value of a cash flow at an earlier period (Time 0)
‒ Discounting: translating a future value to a present value
2. Future Value (FV)
‒ The value of a cash flow at some time in the future
‒ Compounding: converting a present value to a future value
Discounting at rate r
PV(CF4)
t=0
CF0
Present
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t=1
t=2
Compounding at rate r
Lecture 2: Time Value of Money
t=3
Future
CF4
t=4
FV(CF0 )
13
Basic Concepts
3. Interest Rate
‒ The rate used to discount a FV to a PV or to compound a PV to a FV
‒ An “exchange rate” between cash flows in different periods
§ Discount rate
§ Required return
§ Cost of capital
§ …
PV(CF4)
t=0
CF0
Present
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Future
CF4
Discounting at rate r
t=1
t=2
Compounding at rate r
Lecture 2: Time Value of Money
t=3
t=4
FV(CF0 )
14
Basic Concepts
4. Principal
‒ The original amount you borrow or invest
‒ The face value of a bond
PV(CF4)
t=0
CF0
Present
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Future
CF4
Discounting at rate r
t=1
t=2
Compounding at rate r
Lecture 2: Time Value of Money
t=3
t=4
FV(CF0 )
15
Today’s Roadmap
• Basic concepts and common terminology
‒ Timeline of cash flows
‒ Basic terms related to the timing of cash flows
• Calculating the future value by compounding
• Calculating the present value by discounting
• Connecting future and present values
‒ How to find the interest rate
‒ How to determine the number of periods
• Textbook Reading: Chapter 5
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Lecture 2: Time Value of Money
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Future Value and Compounding
• Investing for one single period
FV1 ?
$100
r=10%
t=0
t=1
t=2
t=3
t=4
• This is equivalent to ask:
If you invest $100 in a bank account with an interest rate of 10% per year, how much
will you get in a year as your total investment proceeds?
𝐹𝑉! = $100 + $100×10% = $100× 1 + 10% = $100×1.1 = $110
• General Formula:
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𝐹𝑉! = 𝑃𝑉× 1 + 𝑟
Lecture 2: Time Value of Money
17
Future Value and Compounding
• Investing for multiple periods
FV2 ?
$100
r=10%
r=10%
t=0
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t=1
t=2
Lecture 2: Time Value of Money
t=3
t=4
18
Future Value and Compounding
• Investing for multiple periods
FV2 ?
$100
r=10%
r=10%
t=0
t=1
t=2
t=3
t=4
• Simple Interest
𝐹𝑉! = $100 + $100×0.1 + $100×0.1 = $120
Principal
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Interest
of Year 1
Lecture 2: Time Value of Money
Interest
of Year 2
19
Future Value and Compounding
• Investing for multiple periods
FVt ?
$100
r=10%
r=10%
1
0
……
2
r=10%
……
t
• Simple Interest
General formula:
𝐹𝑉" = 𝑃𝑉 + 𝑃𝑉×𝑟 + ⋯ + 𝑃𝑉×𝑟 = 𝑃𝑉×(1 + 𝑡×𝑟)
t periods
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Lecture 2: Time Value of Money
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Future Value and Compounding
• Investing for multiple periods
$100
r=10%
r=10%
t=1
t=0
FV2 ?
t=2
t=3
t=4
• Compound Interest
‒ Year 1:
‒ Year 2:
𝐹𝑉! = $100× 1 + 𝑟 = $100× 1 + 0.1 = $110
‒ Together:
𝐹𝑉" = 𝐹𝑉!× 1 + 𝑟 = $110× 1 + 0.1 = $121
𝐹𝑉" = 𝐹𝑉!× 1 + 𝑟 = $100× 1 + 0.1 × 1 + 0.1
= $100× 1 + 0.1 " = $121
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Lecture 2: Time Value of Money
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Future Value and Compounding
• Investing for multiple periods
FVt ?
$100
r=10%
r=10%
0
• Compound Interest
1
……
2
r=10%
……
t
General formula:
𝐹𝑉" = 𝑃𝑉 × 1 + 𝑟 …× 1 + 𝑟 = 𝑃𝑉× 1 + 𝑟 "
t periods
• The Future Value Interest Factor: 1 + 𝑟 "
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Simple VS Compound Interest
• Simple Interest: 𝐹𝑉# = 𝑃𝑉×(1 + 𝑡×𝑟)
Take out the $10 interest
Reinvest the $100 principal
$100
$110
r=10%
FV2=$120
r=10%
t=1
t=0
t=2
t=3
t=4
• Compound Interest: 𝐹𝑉# = 𝑃𝑉× 1 + 𝑟 #
Reinvest the $100 principal + the $10 interest
$100
$110
r=10%
t=0
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FV2=$121
r=10%
t=1
t=2
Lecture 2: Time Value of Money
t=3
t=4
23
The Effect of Compounding
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Lecture 2: Time Value of Money
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The Effect of Compounding
• The effect of compound could be huge as the investment horizon
grows:
Period
Year 1
Year 2
Year 3
Year 4
Year 5
…
Year 10
…
Year 100
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Principal
$100
$100
$100
$100
$100
Simple
Interest
$10
$20
$30
$40
$50
Compound
Interest
$0.00
$1.00
$3.10
$6.41
$11.05
Total Amount
(Simple)
$110.00
$120.00
$130.00
$140.00
$150.00
Total Amount
(Compound)
$110.00
$121.00
$133.10
$146.41
$161.05
Compound - Simple
$0.00
$1.00
$3.10
$6.41
$11.05
$100
$100
$59.37
$200.00
$259.37
$59.37
$100
$1,000
$1,376,961.23
$1,100.00
$1,378,061.23
$1,376,961.23
Lecture 2: Time Value of Money
25
Example of FV: Usury Loans
• An online-lending platform offers to lend you $1,000 in January, and a repayment
of $1,100 next month.
• However, you cannot payback the money next month, and the lender offers a
“rollover”, lending you $1,100 new debt to pay the old debt:
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Date
Borrow
Repayment
Jan
$1,000
- $1,100
Feb
$1,100
- $1,210
Mar
$1,210
- $1,331
…
…
…
Dec
$2,853
- $3,138
Lecture 2: Time Value of Money
26
The Effect of Compounding
• The effect of Interest Rate:
r ↑ ⟹ FV ↑
• The effect of Time:
t ↑ ⟹ FV ↑
• Interest Rate & Time:
20%
$6
Future Value of $1
‒ FV grows with t
§ at an increasing speed
§ by more when r is larger
$7
$5
15%
$4
$3
10%
$2
5%
0%
$1
$0
1
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2
3
Lecture 2: Time Value of Money
4
5
6
Time
(years)
7
8
9
10
27
Compound Growth
• Compounding in savings
‒ Your money grows compoundingly when you deposit it into an interest-baring
account
• Compounding in corporate finance
‒ When we talk about the growth rate of sales/ profits/ dividends, it means that
they have compound growth
• Compounding in the macroeconomy
‒ GDP growth, CPI growth (the inflation rate), …
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Lecture 2: Time Value of Money
28
Today’s Roadmap
• Basic concepts and common terminology
‒ Timeline of cash flows
‒ Basic terms related to the timing of cash flows
• Calculating the future value by compounding
• Calculating the present value by discounting
• Connecting future and present values
‒ How to find the interest rate
‒ How to determine the number of periods
• Textbook Reading: Chapter 5
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Lecture 2: Time Value of Money
29
Present Value and Discounting
• Investing for one single period
$100
PV ?
r=10%
t=0
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t=1
t=2
Lecture 2: Time Value of Money
t=3
t=4
30
Present Value and Discounting
• Investing for one single period
$100
PV ?
r=10%
t=0
• General Formula:
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t=1
t=2
t=3
t=4
𝑃𝑉× 1 + 0.1 = 𝐹𝑉$ = $100
$100
𝑃𝑉 =
= $90.9
1.1
𝐹𝑉$
𝑃𝑉 =
(1 + 𝑟)
Lecture 2: Time Value of Money
31
Present Value and Discounting
• Investing for multiple periods
$100
PV ?
r=10%
r=10%
t=1
t=0
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t=3
t=4
𝑃𝑉× 1 + 0.1 " = 𝐹𝑉" = $100
$100
𝑃𝑉 =
= $87.34
"
1.1
• General Formula:
• Discount factor:
t=2
!
𝐹𝑉"
𝑃𝑉 =
1+𝑟 "
!"# "
Lecture 2: Time Value of Money
32
The Effect of Discounting
• The effect of Interest Rate:
r ↑ ⟹ PV ↓
• The effect of Time:
t ↑ ⟹ PV ↓
t ➞ ∞ ⟹ PV ➞ 0
1.2
‒ PV reduces with t
at a decreasing rate
1
Present Value of $1
• Interest Rate & Time:
0%
0.8
0.6
5%
0.4
10%
0.2
15%
20%
0
1
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2
3
Lecture 2: Time Value of Money
4
5
6
Time (Years)
7
8
9
10
33
Connecting FV and PV
• Basic Rule:
To get FV, you compound: 𝐹𝑉# = 𝑃𝑉× 1 + 𝑟 #
$%!
To get PV, you discount: 𝑃𝑉 =
!
!&'
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Lecture 2: Time Value of Money
34
An Example of PV & FV
A financial advisor offers you an investment contract under which you get $12,000
back in 3 years.
• You need to invest $10,000 today into this contract;
• Instead, you can deposit your money at an interest rate of 6.5% per year ;
Will you sign the investment contract?
• Present Value of the payoff:
𝑃𝑉 =
$"#,%%%
$"#,%%%
=
= $9,934
"&%.%() !
".#%*
• Future value of the investment:
𝐹𝑉+ = $10,000× 1 + 0.065 + = $12079.5
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Today’s Roadmap
• Basic concepts and common terminology
‒ Timeline of cash flows
‒ Basic terms related to the timing of cash flows
• Calculating the future value by compounding
• Calculating the present value by discounting
• Connecting future and present values
‒ How to find the interest rate
‒ How to determine the number of periods
• Textbook Reading: Chapter 5
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Connecting FV and PV: Interest Rate
• Basic Rule:
To get FV, you compound: 𝐹𝑉, = 𝑃𝑉× 1 + 𝑟 ,
-."
To get PV, you discount: 𝑃𝑉 =
"
"&/
• Find the Interest Rate:
1)
Rearrange:
2) Take the tth root:
𝐹𝑉,
= 1+𝑟 ,
𝑃𝑉
"
𝐹𝑉,
=1+𝑟
𝑃𝑉
3) Find the interest rate:
𝑟=
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"
𝐹𝑉,
−1
𝑃𝑉
Lecture 2: Time Value of Money
37
An Example of Interest Rate
• You need to borrow some money. You can borrow $1,000 from a friend and pay
back $1,250 in 3 years; or you can borrow from a bank which charges an annual
interest rate of 8%. Which one is better?
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Lecture 2: Time Value of Money
38
An Example of Interest Rate
• You need to borrow some money. You can borrow $1,000 from a friend and pay
back $1,250 in 3 years; or you can borrow from a bank which charges an annual
interest rate of 8%. Which one is better?
‒ PV=$1,000
‒ FV3=$1,250
‒ t=3
‒ r=?
‒ Interest rate by your friend:
𝑟=
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!
$1,250
− 1 = 7.72%
$1,000
Lecture 2: Time Value of Money
39
Today’s Roadmap
• Basic concepts and common terminology
‒ Timeline of cash flows
‒ Basic terms related to the timing of cash flows
• Calculating the future value by compounding
• Calculating the present value by discounting
• Connecting future and present values
‒ How to find the interest rate
‒ How to determine the number of periods
• Textbook Reading: Chapter 5
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Connecting FV and PV: Number of Periods
• Basic Rule:
To get FV, you compound: 𝐹𝑉, = 𝑃𝑉× 1 + 𝑟 ,
-."
To get PV, you discount: 𝑃𝑉 =
"
"&/
• Find the number of periods:
1)
Rearrange:
2) Take log:
𝐹𝑉!
= 1+𝑟 !
𝑃𝑉
𝐹𝑉!
log
= 𝑡 log(1 + 𝑟)
𝑃𝑉
3) Find the number of periods:
log(𝐹𝑉! /𝑃𝑉)
𝑡=
log(1 + 𝑟)
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Lecture 2: Time Value of Money
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An Example of Time
• You want to buy an Apple Watch which costs $3,000. You have $2,000 now and
can invest it at a monthly interest rate of 5%. How long do you have to wait to get
enough money for your Apple Watch?
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An Example of Time
• You want to buy an Apple Watch which costs $3,000. You have $2,000 now and
can invest it at a monthly interest rate of 5%. How long do you have to wait to get
enough money for your Apple Watch?
‒ PV=$2,000
‒ FVt=$3,000
‒ r=5%
‒ t=?
‒ The number of periods (months) is:
𝑡=
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log($3,000/$2,000)
= 8.31
log(1 + 0.05)
Lecture 2: Time Value of Money
43
Calculation with Excel
• Check Excel file:
‒ “Excel_ Lecture_2”
• In Spreadsheet:
‒ Calculating Future Value: “FV”
‒ Calculating Present Value: “PV”
‒ Calculating Number of Periods: “No.Periods”
‒ Calculating Interest Rate: “Interest Rate”
• Excel Function:
‒ Future Value = FV(rate,nper,pmt,[pv],[type])
‒ Present Value = PV(rate, nper, pmt, [fv], [type])
‒ Number of Periods = NPER(rate,pmt,pv,[fv],[type])
‒ Interest Rate = RATE(nper, pmt, pv, [fv], [type], [guess])
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A Useful Rule of Thumb
• In many cases we need to know:
‒ How long it takes to double the amount of money with a given interest rate
‒ How high the interest rate should be to double the amount of money within a given
number of periods
• The Rule of 72 to double the money:
72
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 =
𝐼𝑛𝑡𝑒𝑟𝑠𝑡 𝑅𝑎𝑡𝑒 (𝑖𝑛 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒)
‒ Given an interest rate of 9% per year, it takes 8 years to double the money
72
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑟𝑖𝑜𝑑 =
9
‒ To double the money within 12 years, the interest rate should be 6% per year
72
I𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑅𝑎𝑡𝑒 =
12 𝑦𝑒𝑎𝑟𝑠
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45
Summary: What You Should Know
• Basic terms
‒ ①Present value (PV), ②Future value (FV), ③Interest rate, ④ principal
‒ How to draw the time line of cash flows for a given context
• Calculating FV
‒ Formula
‒ The effect of compounding
• Calculating PV
‒ Formula
‒ The effect of discounting
• The link between FV, PV, number of periods, and interest rate
‒ Be able to find the value of the fourth item given the other three
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Lecture 2: Time Value of Money
46
Next Week
• DCF Valuation
‒ Textbook Reading: Chapter 6
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