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REVIEW: TIME VALUE OF MONEY SMF Prep Workshop Andrew Chen - OSU

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This session: The mother of all finance formulas Other TVM formulas Growing Perpetuity Perpetuity Annuity Valuing Bonds This should be a review

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$53,000 Thank you. Is it worth it? (yes) How much is it worth?

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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

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NPV of the SMF $35,000 in 2050 is not the same thing as $35,000 today. (Change in Salary) x (Working Years) = $35,000 x 45 = $1.575 million (Benefits) – (Costs) = $1.575 million - $50,500 = $1.525 million

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NPV of the SMF: the right way CONGRATULATIONS!

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NPV of the SMF: tweaking A few problems: 1. Forgot to include lost salary while in school 2. Screwed up salary timing: your salary increase should be delayed by a year 3. Why a 5% discount rate? (The interested student should calculate a better NPV)

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TIME VALUE OF MONEY Formulas

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TVM: the basic idea $100 today is not the same as $100 four years from now t = $100

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TVM: the basic idea Suppose your bank offers you 3% interest t = $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

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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

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TVM: Formulas The mother of all finance formulas: In principle, this is all you need to know.

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TVM: Formulas The key: Present values add up If the bank offers you interest rate r And you receive C 1, C 2, C 3,…, C n at the end of years 1, 2, 3, …, n,

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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?

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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?

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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?

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TVM Formulas Growing Perpetuity Perpetuity Annuity Note: for all formulas, the first cash flow C is at time 1

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TVM Formulas No need to memorize In exams, youll get a formula sheet In real life, youll use Excel or Matlab But its useful to memorize them Back-of-the-envelope calculations Intuition *First impressions

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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)

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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?

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Perpetuity Formula Perpetuity: Intuition: This is just a growing perpetuity with 0 growth Similar interpretation to a growing perpetuity

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Deriving the Perpetuity Formula

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TVM Formulas: Intuition Annuity: Intuition: This is the difference between two perpetuities

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Annuity Example Youve 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? *Whats wrong with this analysis?

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Timing Details Growing Perpetuity Perpetuity Annuity Note: for all formulas, the first cash flow C is at time 1

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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?

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Timing Example 2 Your aunt gave you a loan to buy the food truck and understood that itd 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?

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Future Values

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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?

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Solving for interest rates

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Solving for interest rates numerically

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TIME VALUE OF MONEY Valuing Bonds

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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

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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?

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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?

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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

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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?

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Why its 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?

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TVM Wrapup: We covered… The mother of all finance formulas Other TVM formulas Growing Perpetuity Perpetuity Annuity Valuing Bonds

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