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# Review: Time Value of Money - PowerPoint PPT Presentation

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

• 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?

• Tuition / Fees: \$53,000

• New Salary: \$85,000

• (Median Fisher MBA)

• Old Salary: \$50,000

• (Nice round number)

• Years ‘till retirement: 40

• (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.

• Discount rate: 5%

• Annuity Formula

• PV(Salary Increase) =

• NPV = PV(Salary Increase – Tuition) = \$572,000

CONGRATULATIONS!

• 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

• \$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

• 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

• 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

• The mother of all finance formulas:

• In “principle,” this is all you need to know.

• 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,

• 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?

• 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?

• 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?

• Growing Perpetuity

• Perpetuity

• Annuity

• Note: for all formulas, the first cash flow C is at time 1

• 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

• 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)

• 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:

• Intuition:

• This is just a growing perpetuity with 0 growth

• Similar interpretation to a growing perpetuity

• It’s just some clever factoring:

• Notice the thing in [] is the PV

• Solve for PV

• Annuity:

• Intuition:

• This is the difference between two perpetuities

• 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?

• Growing Perpetuity

• Perpetuity

• Annuity

• Note: for all formulas, the first cash flow C is at time 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?

• 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?

• 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

• 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?

• 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

• 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

• 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

• 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?

• 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 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

• 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).

• What is the new bond price?

• 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