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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

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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)


Time value of money

Time value of money

Formulas


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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