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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

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?

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
• (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
• Additional ingredients
• Discount rate: 5%
• Annuity Formula
• PV(Salary Increase) =
• NPV = PV(Salary Increase – Tuition) = \$572,000

CONGRATULATIONS!

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)

### 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

\$100

t = 0

1

2

3

4

\$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
• More generally
• If the bank offers you an interest rate r,
• The PV of C dollars, n years from now, is
TVM: Formulas
• The mother of all finance formulas:
• 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,
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
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
• 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
• 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 is just a growing perpetuity with 0 growth
• Similar interpretation to a growing perpetuity
Deriving the Perpetuity Formula
• It’s just some clever factoring:
• Notice the thing in [] is the PV
• Solve for PV
TVM Formulas: Intuition
• Annuity:
• Intuition:
• This is the difference between two perpetuities
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
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
• 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
• 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 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
• 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 money

Valuing Bonds

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
• 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
• 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
• Other TVM formulas
• Growing Perpetuity
• Perpetuity
• Annuity
• Valuing Bonds