Review: Time Value of Money

1. Review: Time Value of Money SMF Prep Workshop Andrew Chen - OSU

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

3. $53,000 • Thank you. • Is it worth it? • (yes) How much is it worth?

4. 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

5. 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.

6. 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!

7. 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)

8. Time value of money Formulas

9. 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

10. 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

11. 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

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

13. 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,

14. 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?

15. 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?

16. 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?

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

18. 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

19. 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)

20. 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?

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

22. Deriving the Perpetuity Formula • It’s just some clever factoring: • Notice the thing in [] is the PV • Solve for PV

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

24. 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?

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

26. 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?

27. 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?

28. 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

29. 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?

30. 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

31. 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

32. Time value of money Valuing Bonds

33. 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

34. 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?

35. 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?

36. 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

37. 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?

38. 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?

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