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Review: Time Value of Money. SMF Prep Workshop. Andrew Chen - OSU. This session:. The mother of all finance formulas. 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?.

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review time value of money

Review: Time Value of Money

SMF Prep Workshop

Andrew Chen - OSU

this session
This session:
  • The mother of all finance formulas
  • Other TVM formulas
    • Growing Perpetuity
    • Perpetuity
    • Annuity
  • Valuing Bonds

This should be a review

53 000
$53,000
  • Thank you.
  • Is it worth it?
    • (yes)

How much is it worth?

npv of the smf ingredients
NPV of the SMF: Ingredients
  • Tuition / Fees: $53,000
  • New Salary: $85,000
    • (Median Fisher MBA)
  • Old Salary: $50,000
    • (Nice round number)
  • Years ‘till retirement: 40
npv of the smf
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 million
  • $35,000 in 2050 is not the same thing as $35,000 today.
npv of the smf the right way
NPV of the SMF: the right way
  • Additional ingredients
    • Discount rate: 5%
    • Annuity Formula
  • PV(Salary Increase) =
  • NPV = PV(Salary Increase – Tuition) = $572,000

CONGRATULATIONS!

npv of the smf tweaking
NPV of the SMF: tweaking
  • A few problems:
    • 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)
tvm the basic idea
TVM: the basic idea
  • $100 today is not the same as $100 four years from now

t = 0

1

2

3

4

$100

t = 0

1

2

3

4

$100

tvm the basic idea1
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 idea2
TVM: the basic idea
  • Flip that around:
    • $113 four years from now is worth
  • More generally
    • If the bank offers you an interest rate r,
    • The PV of C dollars, n years from now, is
tvm formulas
TVM: Formulas
  • The mother of all finance formulas:
  • In “principle,” this is all you need to know.
tvm formulas1
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,
basic tvm formula example 1
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
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
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 formulas2
TVM Formulas
  • Growing Perpetuity
  • Perpetuity
  • Annuity
  • Note: for all formulas, the first cash flow C is at time 1
tvm formulas3
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 calculations
    • Intuition
    • *First impressions
tvm formulas intuition
TVM Formulas: Intuition
  • Growing Perpetuity:
  • Intuition:
    • 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
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 Formula
  • Perpetuity:
  • Intuition:
    • This is just a growing perpetuity with 0 growth
    • Similar interpretation to a growing perpetuity
deriving the perpetuity formula
Deriving the Perpetuity Formula
  • It’s just some clever factoring:
  • Notice the thing in [] is the PV
  • Solve for PV
tvm formulas intuition1
TVM Formulas: Intuition
  • Annuity:
  • Intuition:
    • This is the difference between two perpetuities
annuity example
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
Timing Details
  • Growing Perpetuity
  • Perpetuity
  • Annuity
  • Note: for all formulas, the first cash flow C is at time 1
timing example 1
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
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
Future Values
  • Any of the formulas can be used to find future values by rearranging the basic equation
    • is the same as or
  • 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
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
Solving for interest rates
  • Sometimes you can solve for the interest rate:
    • Growing Perpetuity: can re-arranged to be
  • Other times, you can’t
    • Annuity: cannot be solved for r by using algebra
solving for interest rates numerically
Solving for interest rates numerically
  • But you can solve for r in 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 money1

Time value of money

Valuing Bonds

valuing bonds jargon
Valuing Bonds: Jargon
  • Face value: the amount used to calculate the coupon
    • Usually repaid at maturity
  • Coupon: a regular payment paid until the maturity
  • APR: “annualized” interest rate computed by simple multiplication
    • Does not take into account compounding interest
  • Yield-to-Maturity (YTM): the interest rate
valuing bonds example 1
Valuing Bonds: Example 1
  • You are thinking of buying a 5-year, $1000 face-value 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
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: 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
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
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
TVM Wrapup: We covered…
  • The mother of all finance formulas
  • Other TVM formulas
    • Growing Perpetuity
    • Perpetuity
    • Annuity
  • Valuing Bonds
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