REVIEW: TIME VALUE OF
MONEY
Andrew Chen - OSU
SMF Prep Workshop
This session:

The mother of all finance formulas
$𝐶
𝑃𝑉 =
(1 + 𝑟)𝑛

Other TVM formulas
 Growing
Perpetuity
 Perpetuity
 Annuity

Valuing Bonds
This should be a review
$53,000


Thank you.
Is it worth it?
 (yes)
How much is it worth?
NPV of the SMF: Ingredients


Tuition / Fees: $53,000
New Salary: $85,000
 (Median

Old Salary: $50,000
 (Nice

Fisher MBA)
round number)
Years ‘till retirement: 40
NPV of the SMF


(Change in Salary) x (Working Years) =
$35,000 x 45 = $1.575 million
(Benefits) – (Costs) = $1.575 million - $50,500
= $1.525
 $35,000
million
in 2050 is not the same thing as $35,000 today.
NPV of the SMF: the right way

Additional ingredients
 Discount
rate: 5%
 Annuity Formula


1  (1  r )  n 
PV  C 

r


1−(1.05)−45
0.05
PV(Salary Increase) = $35,000
= $601,000
NPV = PV(Salary Increase – Tuition)
= $572,000
CONGRATULATIONS!
NPV of the SMF: tweaking

A few problems:
1.
2.
3.

Forgot to include lost salary while in school
Screwed up salary timing: your salary increase should
be delayed by a year
Why a 5% discount rate?
(The interested student should calculate a better
NPV)
TIME VALUE OF MONEY
Formulas
TVM: the basic idea

$100 today is not the same as $100 four years from now
t=0
1
2
3
4
1
2
3
4
$100
t=0
$100
TVM: the basic idea

Suppose your bank offers you 3% interest
t=0
1
2
3
4
$100
$100 x (1.03)
$100 x (1.03)^2
$100 x (1.03)^3
$100 x (1.03)^4
= $113

$100 today is worth $113 four years from now
TVM: the basic idea

Flip that around:

$113 four years from now is worth
$113
$100 =
(1 + 0.03)4

More generally

If the bank offers you an interest rate r,

The PV of C dollars, n years from now, is
$𝐶
𝑃𝑉 =
(1 + 𝑟)𝑛
TVM: Formulas

The mother of all finance formulas:
$𝐶
𝑃𝑉 =
𝑛
(1 + 𝑟)

In “principle,” this is all you need to know.
TVM: Formulas

The key: Present values add up

If the bank offers you interest rate r
And you receive C1, C2, C3 ,… , Cn
 at the end of years 1, 2, 3, …, n,

$𝐶1
$𝐶2
$𝐶3
$𝐶𝑛
𝑃𝑉 =
+
+
…+
1
2
3
(1 + 𝑟)
(1 + 𝑟)
(1 + 𝑟)
(1 + 𝑟)𝑛
Basic TVM Formula: Example 1

A zero-coupon bond will pay $15,000 in 10 years.
Similar bonds have an interest rate of 6% per year
 What
is the bond worth today?
Basic TVM Formula: Example 2

You need to buy a car. Your rich uncle will lend you
money as long as you pay him back with interest (at
6% per year) within 4 years. You think you can pay
him $5,000 next year and $8,000 each year after
that.
 How
much can you borrow from your uncle?
Basic TVM Formula: Example 3

Your crazy uncle has a business plan that will
generate $100 every year forever. He claims that
an appropriate discount rate is 5%.
 How
much does he think his business plan is worth?
TVM Formulas

Growing Perpetuity

Perpetuity

Annuity

Note: for all formulas, the first cash flow C is at time 1
C
PV 
rg
C
PV 
r
1  (1  r )  n 
PV  C 

r


TVM Formulas

No need to memorize
 In
exams, you’ll get a formula sheet
 In real life, you’ll use Excel or Matlab

But it’s useful to memorize them
 Back-of-the-envelope
 Intuition
 *First
impressions
calculations
TVM Formulas: Intuition

Growing Perpetuity:

Intuition:
C
PV 
rg
 As
the discount rate goes up, PV goes down
 As the growth rate goes up, PV goes up

(This is a nice one to memorize)
Growing Perpetuity 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?
Perpetuity Formula

Perpetuity:

Intuition:
 This
C
PV 
r
is just a growing perpetuity with 0 growth
 Similar interpretation to a growing perpetuity
Deriving the Perpetuity Formula


It’s just some clever factoring:
 𝑃𝑉
=
 𝑃𝑉
=
+
+
1
1
+
…
(1+𝑟)2
(1+𝑟)3
1
1
1
+
…
(1+𝑟) (1+𝑟)
(1+𝑟)2
Notice the thing in [] is the PV
 𝑃𝑉

1
(1+𝑟)
1
(1+𝑟)
=
1
(1+𝑟)
Solve for PV
 𝑃𝑉
=
1
𝑟
+
1
(1+𝑟)
𝑃𝑉
TVM Formulas: Intuition

Annuity:

Intuition:
 This
1  (1  r )  n 
PV  C 

r


is the difference between two perpetuities
1  (1  r )  C  1  C
C

  
r

 r 1 r  r
n
n
Annuity Example

You’ve won a $30 million lottery. You can either
take the money as (a) 30 payments of $1 million
per year (starting one year from today) or (b) as
$15 million paid today. Use an 8% discount rate.
 Which
option should you take?
 *What’s wrong with this analysis?
Timing Details

Growing Perpetuity

Perpetuity

Annuity

Note: for all formulas, the first cash flow C is at time 1
C
PV 
rg
C
PV 
r
1  (1  r )  n 
PV  C 

r


Timing Example 1

Your food truck has earned $1,000 each year (at
the end of the year). You expect this to continue for
4 years, and for the earnings to grow after that at
7% forever. Use a 10% discount rate
 How
much is your food truck worth?
Timing Example 2

Your aunt gave you a loan to buy the food truck
and understood that it’d take time for the profits to
come in. She said you can pay her $1000 at the
end of each year for 10 years with the first
payment coming in exactly 4 years from now. Use
a 10% discount rate.
 How
much did she lend you?
Future Values

Any of the formulas can be used to find future
values by rearranging the basic equation
 𝑃𝑉
=
𝐶
(1+𝑟)𝑛
is the same as 𝐶 = 1 + 𝑟
𝑛
𝑃𝑉
or
𝑛
 𝐹𝑉 = 1 + 𝑟 (𝑃𝑉)

Then do a two-step
 1)
Use PV formulas to take cash flows to the present
 2) Use FV formula to move to the future
Future Values: Example

You want expand your food truck business by
getting a second truck. You figure you can save
$500 each year and your bank pays you 3%
interest.
 How
much can you spend on your truck in 10 years?
Solving for interest rates

Sometimes you can solve for the interest rate:
 Growing
𝑟=

𝐶
𝑃𝑉
Perpetuity: 𝑃𝑉 =
𝐶
𝑟−𝑔
can re-arranged to be
+𝑔
Other times, you can’t
 Annuity:
𝑃𝑉 =
using algebra
𝐶
1
(1 −
)
𝑟
1+𝑟 𝑛
cannot be solved for r by
Solving for interest rates numerically

But you can solve for r in 𝑃𝑉 =
𝐶
1
(1 −
)
𝑟
1+𝑟 𝑛
by
using Excel.
 Rate(n,-C,PV)

gives you r
Excel has similar functions for finding the PV and n
 PV(r,n,-C)
gives you PV
 Nper(r,-C,PV) gives you n
TIME VALUE OF MONEY
Valuing Bonds
Valuing Bonds: Jargon

Face value: the amount used to calculate the
coupon
 Usually


Coupon: a regular payment paid until the maturity
APR: “annualized” interest rate computed by
simple multiplication
 Does

repaid at maturity
not take into account compounding interest
Yield-to-Maturity (YTM): the interest rate
Valuing Bonds: Example 1

You are thinking of buying a 5-year, $1000 facevalue bond with a 5% coupon rate and semiannual
coupons. Suppose the YTM on comparable bonds is
6.3% (APR with seminannual compounding).
 How
much is the bond worth?
Valuing Bonds: Example 2

A $1000 face value bond pays a 8% semiannual
coupon and matures in 10 years. Similar bonds
trade at a YTM of 8% (semiannual APR)
 How
much is the bond worth?
Bonds: More Jargon

Bonds are typically issued at par: Price is equal to
the face value
 Here,

the coupon rate = interest rate
After issuance, prices fluctuate. The price may be
 At
a premium: price > par
 At a discount: price < par
Valuing Bonds: Example 3

A software firm issues a 10 year $1000 bond at
par. The bond pays a 12% annual coupon. Two
years later, there is good news about the industry,
and interests rates for similar firms fall to 8%
(annual).
 Does
the bond trade at a premium or discount?
 What is the new bond price?
Why it’s called “Yield to Maturity”

A software firm issues a 10 year $1000 bond at
par. The bond pays a 12% annual coupon. Two
years later, there is good news about the industry,
and interests rates for similar firms fall to 8%
(annual).
 If
you bought the bond at issue and held it to maturity,
what “effective interest rate” did you get?
 If you bought it at issue and sold it two years later,
what “effective interest rate” did you get?
TVM Wrapup: We covered…

The mother of all finance formulas
$𝐶
𝑃𝑉 =
(1 + 𝑟)𝑛

Other TVM formulas
 Growing
Perpetuity
 Perpetuity
 Annuity

Valuing Bonds