8 question
1. Time value of money
Chapters 3-7
Outline
What is/how to understand time value of money (TVM)?
How to understand discount rates?
How to calculate present value and future value?
Applications in asset valuation
Annuities, perpetuities, bonds, stocks, etc.
Using Excel formula to solve values
Tools
Regular / financial calculator (not required but may come handy)
Excel (required and crucial)
The value of an asset
Suppose an asset paid you $1 at the end of yesterday (t=-1), today, and three days in the future
What is the price of such an asset, suppose the value of money stays constant over time?
…as of when?
As of the beginning of yesterday, P = $1 * 5 = $5 since you will receive 5 payments
As of the beginning of today, P = $4 since it entitles you only 4 payments
…
As of t = 4, P = 0
t = -1
0
1
2
3
$1
$1
$1
$1
$1
The value of an asset
Depends on
The time spot
Future entitled payoffs
Alternative thinking: it is worth how much one can sell for / is willing to pay for
If one buys the asset from you at the beginning of today, she will receive 4 payments of $4 so P=$4
Not related with payoff in the past or its total payoff
Not determined by the total payoff
Always look forward when pricing an asset
When time introduces uncertainty
There is always some degree of uncertainty (i.e., risk) for money outside pocket
Derivatives > stocks > bonds > savings > cash
To induce investors from keeping cash,
Banks offer interest rate
Bonds offer coupons/interest payments
Stocks and derivatives offer upside for value increase
Uncertainty = Time value of money
Compensation for bearing the risk of the asset’s future payments
Types of Uncertainty
From underlying asset itself: higher rating corporate bonds offer lower yield
From market: stock prices tanked when Covid hit in March
The value of an asset
What is the price of such an asset as of the beginning of today, assuming a benchmark discount rate of 2%?
“Discount rate” is the rate to adjust future payments by its uncertainty
“Benchmark” means assets with similar uncertainty offer the same rate
(assuming the rate is constant over time)
Each part estimates the present value (PV) of $1, discounted from each period
The sum of all PV is called net present value (NPV)
(How we come up with this 2% will be explained in later lectures but let’s assume we know it already.)
t = -1
0
1
2
3
$1
$1
$1
$1
$1
Example
Tom saved $100 in a bank savings account Jan 1, 2014. The bank offered a 3% annual interest rate and Tom watched his savings grow as follows up till Jan 1, 2018:
t = 0
1
2
3
4
Q1: On Jan 1, 2014, what was Tom’s expectation on his savings after 4 years, i.e., on Jan 1, 2018?
$113 = the future value of $100 in four years
Q2: On Jan 1, 2014, if Tom expected to collect $113 on Jan 1, 2018, how much should he save then?
$100 = the present value of $113 in four years
t = 0
1
2
3
4
Q3: Why would banks offer interest on savings?
TVM is unlikely the only answer. Very little uncertainty on bank savings.
Banks
Make money by lending deposits at a higher rate so willing to pay a bit to attract deposits
Depositors
Money safe and easy to mange in bank
Rates set by Federal Reserve
Two interest rates: one between banks, and one between bank and depositors
Fed sets the first rate, controlling how costly banks borrow from each other
Banks set the second rate, resulting from their demand for capital
E.g., when Fed increases inter-banks interest rate, it becomes more costly for banks to borrow from other banks, so the demand to borrow from deposits increases hence higher savings interest rate, and vice versa.
Competition from other banks to attract savings
Interest rate vs. discount rate vs. cost of capital
Interest rate
Associated with specific type of asset
Often assigned by the issuer (bank savings, bonds, loans, etc.)
Loan rate >> deposit rate
Discount rate
Specifically refers to the rate used to discount future cash flows in cash flow analysis
Depends on the nature (degree of uncertainty) of the asset
Cost of capital
A corporate finance term that specifically refers to a firm’s overall cost of raising money from investors
Combination of cost of debt and cost of equity
Commonly used as discount rate in valuing firm cash flows
In a more general case…
Timeline t
Cash flows Ct at each t
If we know the constant discount rate
The present value of is , hence
Asset’s Net present value or NPV =
(1. What is the NPV at t=1? t=3? 2. What if r is time-variant?)
t = 0
1
2
3…
n…
C1
C2
C3…
Cn…
C0
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12
Example
What is the NPV of a project that earns $1,000, $2000, $3000 at the end of the first three years and liquidates at $20,000 at the end of the fourth year? The discount rate is assumed to be 10%. If you need to invest $15,000 to initiate the project, is it a good project you should take? What if it costs $20,000?
NPV of 4 years
Worth taking if initial investment < NPV
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Valuing different assets using NPV formula
NPV =
Perpetuity
Infinite same/growing payments
Annuity
Finite same payments over n
Bond
Coupon bond: Finite same payment before n, ending payment at n
Stocks
If held forever: Infinite same/growing dividends (in a simple way)
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Perpetuities
Perpetuity: infinite number of payments
Same payments each period
If cash flows grow at constant rate (growing perpetuity),
t = 0
1
2
3
C
……
PV
C
C
C
…
t = 0
1
2
3
C(1+g)3
……
PV
C(1+g)2
C(1+g)
C
…
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Perpetuities
Perpetuity without growth
Growing Perpetuity
What if ?
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Perpetuities
The British government has issued the closest thing to a perpetuity in WWI to finance its military actions. They are completely paid off in 2015 by the government, almost 100 years later. Suppose when the government redeemed these outstanding perpetuities, it follows our pricing rule. Suppose one contract of such perpetuity pays 5 pounds at the end of every year. Under a annual discount rate of 5%, how much should the government pay Jack, who owned 100 contracts?
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Annuity
Annuity (“annual payments”)
Fixed amount of money paid for the next n periods starting at t=1
Can be thought of as the difference between two perpetuities
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Annuity
Example: You have won the lottery, and you have two options as to how to take the winnings. You can either receive 30 annual payments of $1 million each starting one year from today (think of it as an annuity), or $15 million paid today. Using an 8% discount rate, which option should you take?
Using Excel, cell =PV(0.08, 30, -1, 0), default assumption is payment paid at period end.
Take it today!
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Comments on formula
Sometimes you can solve by hand (infinite dividends)
Most of the times you can’t (finite dividends)
But excel (or financial calculators) can do wonders
RATE(), NPER(), PMT() each solves r, n, and C
PV() solves PV of fixed payments, FV() similar
NPV() solves NPV of cash flow stream
E.g. What is the monthly interest rate of an annuity that pays $10 each month for 10 years that sells for $1000 as of today?
n = 120, PV=$1,000, FV = 0, PMT=$10, r=?
In Excel, RATE(120,10,0,0) = 0.31%
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Bond
Form of debt
Issuer raises capital by selling the bond and promise to pay back at its maturity
Owner receives
Coupon each period
Principal/Face value at maturity
= an annuity + extra payment!
t = 0
1
2
3
C
n
PV
C
C
C
F
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Bond
Bond terminology
Face value: Principal amount to be paid back at maturity; also “par” in short
Coupon: Regular payment paid until maturity (% of face value), conventionally paid semiannually
Coupon = Face value X Coupon rate
Yield to Maturity (YTM) : Discount rate that equates all future cash flows from the bond to the price of the bond
Issued “at par”: Price is equal to face value.
Coupon rate = Interest rate
After issuance, bonds can be traded between investors in the bond market
Bond prices can change over time
Priced at a premium: Price Face value (par); coupon rate > interest rate
Priced at a discount: Price Face value (par); coupon rate < interest rate
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Bond
A bond with face value of $1,000 pays annual coupons at 4% and matures in 10 years. If it has a yield to maturity (discount rate) of 3%, what is its price right now? Is it traded at premium or discount?
, at premium
Using a financial calculator, PMT=40, FV=1000, I/Y = 3%, N=10, solve for PV.
Using Excel, cell =PV(0.03,10, -40, -1000), default assumption is payment paid at period end.
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Stock
Ownership of a corporation
Carries 2 types of right
Voting right: vote on firm decisions
Residual cash flow right: share firm profit after debt holders are paid
Dividend: cash or shares paid to shareholders as percentage of its earnings
Example: a 5% payout ratio means firm pays 5% of its earnings to investors as dividends
Common vs. preferred shareholders
Common: both rights
Preferred: superior cash flow right over common shares, but typically carrying no voting right
Pecking order: debtholder > preferred shareholders > common shareholders
Theoretically, stock price should be NPV of all future dividends
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Stock
If you don’t sell the stock and hold on to it forever
What is it?
Perpetuity!
If a firm maintains a constant “payout rate”, and its earnings internally grow at a constant rate, the stock is a
Growing perpetuity
t = 0
1
2
3
Div…
……
Stock Price
Div3
Div2
Div1
…
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Stock
The Dividend Discount Model (no growth)
Gordon Growth Model (constant growth )
As the discount rate goes up, goes down
As the growth rate goes up, goes up
How do we get ?
Economists often assume that the total return of a stock should equal its equity cost of capital, but more on that later…
Example
A stock pays out a $2 dividend every year. The dividend grows at 1% per year, and the discount rate is 6%. How much is the stock worth?
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To sum up
Time value of money and how to calculate different types of cash flow streams:
C n principal pmt PV asset name
Constant finite no annuity
Constant infinite no perpetuity / DDM stock
Growing infinite no growing perpetuity / DDM stock
Constant finite yes bond
What are the issues with our formula?
Critical: Discount rates are time-variant
Quick review
Principle of valuing assets: look for future cash flows
Concept of TVM: time brings risk
NPV model used to price assets
Is this the only way of valuation? (think about that)
Does it always hold?
Applications of NPV model
Next…
See how NPV method applies to firm valuation: capital budgeting