TVM - Andrew Y. Chen

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```REVIEW: TIME VALUE OF
MONEY
Andrew Chen - OSU
SMF Prep Workshop
This session:
ο±
The mother of all finance formulas
\$πΆ
ππ =
(1 + π)π
ο±
Other TVM formulas
ο± Growing
Perpetuity
ο± Perpetuity
ο± Annuity
ο±
Valuing Bonds
This should be a review
\$53,000
ο±
ο±
Thank you.
Is it worth it?
ο± (yes)
How much is it worth?
NPV of the SMF: Ingredients
ο±
ο±
Tuition / Fees: \$53,000
New Salary: \$85,000
ο± (Median
ο±
Old Salary: \$50,000
ο± (Nice
ο±
Fisher MBA)
round number)
Years ‘till retirement: 40
NPV of the SMF
ο±
ο±
(Change in Salary) x (Working Years) =
\$35,000 x 45 = \$1.575 million
(Benefits) – (Costs) = \$1.575 million - \$50,500
= \$1.525
ο± \$35,000
million
in 2050 is not the same thing as \$35,000 today.
NPV of the SMF: the right way
ο±
Additional ingredients
ο± Discount
rate: 5%
ο± Annuity Formula
ο±
ο±
ο©1 ο­ (1 ο« r ) ο­ n οΉ
PV ο½ C οͺ
οΊ
r
ο«
ο»
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
ο±
A few problems:
1.
2.
3.
ο±
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
ο±
\$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
ο±
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
ο±
\$100 today is worth \$113 four years from now
TVM: the basic idea
ο±
Flip that around:
ο±
\$113 four years from now is worth
\$113
\$100 =
(1 + 0.03)4
ο±
More generally
ο±
If the bank offers you an interest rate r,
ο±
The PV of C dollars, n years from now, is
\$πΆ
ππ =
(1 + π)π
TVM: Formulas
ο±
The mother of all finance formulas:
\$πΆ
ππ =
π
(1 + π)
ο±
In “principle,” this is all you need to know.
TVM: Formulas
ο±
The key: Present values add up
ο±
If the bank offers you interest rate r
And you receive C1, C2, C3 ,… , Cn
ο± at the end of years 1, 2, 3, …, n,
ο±
\$πΆ1
\$πΆ2
\$πΆ3
\$πΆπ
ππ =
+
+
…+
1
2
3
(1 + π)
(1 + π)
(1 + π)
(1 + π)π
Basic TVM Formula: Example 1
ο±
A zero-coupon bond will pay \$15,000 in 10 years.
Similar bonds have an interest rate of 6% per year
ο± What
is the bond worth today?
Basic TVM Formula: Example 2
ο±
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.
ο± How
much can you borrow from your uncle?
Basic TVM Formula: Example 3
ο±
Your crazy uncle has a business plan that will
generate \$100 every year forever. He claims that
an appropriate discount rate is 5%.
ο± How
much does he think his business plan is worth?
TVM Formulas
ο±
Growing Perpetuity
ο±
Perpetuity
ο±
Annuity
ο±
Note: for all formulas, the first cash flow C is at time 1
C
PV ο½
rο­g
C
PV ο½
r
ο©1 ο­ (1 ο« r ) ο­ n οΉ
PV ο½ C οͺ
οΊ
r
ο«
ο»
TVM Formulas
ο±
No need to memorize
ο± In
exams, you’ll get a formula sheet
ο± In real life, you’ll use Excel or Matlab
ο±
But it’s useful to memorize them
ο± Back-of-the-envelope
ο± Intuition
ο± *First
impressions
calculations
TVM Formulas: Intuition
ο±
Growing Perpetuity:
ο±
Intuition:
C
PV ο½
rο­g
ο± As
the discount rate goes up, PV goes down
ο± As the growth rate goes up, PV goes up
ο±
(This is a nice one to memorize)
Growing Perpetuity Example
ο±
A stock pays out a \$2 dividend every year. The
dividend grows at 1% per year, and the discount
rate is 6%.
ο± How
much is the stock worth?
Perpetuity Formula
ο±
Perpetuity:
ο±
Intuition:
ο± This
C
PV ο½
r
is just a growing perpetuity with 0 growth
ο± Similar interpretation to a growing perpetuity
Deriving the Perpetuity Formula
ο±
ο±
It’s just some clever factoring:
ο± ππ
=
ο± ππ
=
+
+
1
1
+
…
(1+π)2
(1+π)3
1
1
1
+
…
(1+π) (1+π)
(1+π)2
Notice the thing in [] is the PV
ο± ππ
ο±
1
(1+π)
1
(1+π)
=
1
(1+π)
Solve for PV
ο± ππ
=
1
π
+
1
(1+π)
ππ
TVM Formulas: Intuition
ο±
Annuity:
ο±
Intuition:
ο± This
ο©1 ο­ (1 ο« r ) ο­ n οΉ
PV ο½ C οͺ
οΊ
r
ο«
ο»
is the difference between two perpetuities
ο©1 ο­ (1 ο« r ) οΉ C ο¦ 1 οΆ C
Cοͺ
ο·
οΊ ο½ ο­ο§
r
ο«
ο» r ο¨1ο« r οΈ r
ο­n
n
Annuity Example
ο±
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.
ο± Which
option should you take?
ο± *What’s wrong with this analysis?
Timing Details
ο±
Growing Perpetuity
ο±
Perpetuity
ο±
Annuity
ο±
Note: for all formulas, the first cash flow C is at time 1
C
PV ο½
rο­g
C
PV ο½
r
ο©1 ο­ (1 ο« r ) ο­ n οΉ
PV ο½ C οͺ
οΊ
r
ο«
ο»
Timing Example 1
ο±
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
ο± How
much is your food truck worth?
Timing Example 2
ο±
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.
ο± How
much did she lend you?
Future Values
ο±
Any of the formulas can be used to find future
values by rearranging the basic equation
ο± ππ
=
πΆ
(1+π)π
is the same as πΆ = 1 + π
π
ππ
or
π
ο± πΉπ = 1 + π (ππ)
ο±
Then do a two-step
ο± 1)
Use PV formulas to take cash flows to the present
ο± 2) Use FV formula to move to the future
Future Values: Example
ο±
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.
ο± How
much can you spend on your truck in 10 years?
Solving for interest rates
ο±
Sometimes you can solve for the interest rate:
ο± Growing
π=
ο±
πΆ
ππ
Perpetuity: ππ =
πΆ
π−π
can re-arranged to be
+π
Other times, you can’t
ο± Annuity:
ππ =
using algebra
πΆ
1
(1 −
)
π
1+π π
cannot be solved for r by
Solving for interest rates numerically
ο±
But you can solve for r in ππ =
πΆ
1
(1 −
)
π
1+π π
by
using Excel.
ο± Rate(n,-C,PV)
ο±
gives you r
Excel has similar functions for finding the PV and n
ο± PV(r,n,-C)
gives you PV
ο± Nper(r,-C,PV) gives you n
TIME VALUE OF MONEY
Valuing Bonds
Valuing Bonds: Jargon
ο±
Face value: the amount used to calculate the
coupon
ο± Usually
ο±
ο±
Coupon: a regular payment paid until the maturity
APR: “annualized” interest rate computed by
simple multiplication
ο± Does
ο±
repaid at maturity
not take into account compounding interest
Yield-to-Maturity (YTM): the interest rate
Valuing Bonds: Example 1
ο±
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).
ο± How
much is the bond worth?
Valuing Bonds: Example 2
ο±
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)
ο± How
much is the bond worth?
Bonds: More Jargon
ο±
Bonds are typically issued at par: Price is equal to
the face value
ο± Here,
ο±
the coupon rate = interest rate
After issuance, prices fluctuate. The price may be
ο± At
a premium: price &gt; par
ο± At a discount: price &lt; par
Valuing Bonds: Example 3
ο±
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).
ο± Does
the bond trade at a premium or discount?
ο± What is the new bond price?
Why it’s called “Yield to Maturity”
ο±
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).
ο± If
you bought the bond at issue and held it to maturity,
what “effective interest rate” did you get?
ο± If you bought it at issue and sold it two years later,
what “effective interest rate” did you get?
TVM Wrapup: We covered…
ο±
The mother of all finance formulas
\$πΆ
ππ =
(1 + π)π
ο±
Other TVM formulas
ο± Growing
Perpetuity
ο± Perpetuity
ο± Annuity
ο±
Valuing Bonds
```