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

2. This session: This should be a review
The mother of all finance formulas
𝑃𝑉= $𝐶 (1+𝑟) 𝑛
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 $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, = $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) = $35,000 1− (1.05) − =$601,000
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% interest1 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: $100= $113 (1+0.03) 4
$113 four years from now is worth
$100= $113 (1+0.03) 4
More generally
If the bank offers you an interest rate r,
The PV of C dollars, n years from now, is
𝑃𝑉= $𝐶 (1+𝑟) 𝑛

12. 𝑃𝑉= $𝐶 (1+𝑟) 𝑛 TVM: Formulas The mother of all finance formulas:
𝑃𝑉= $𝐶 (1+𝑟) 𝑛
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,
𝑃𝑉= $𝐶 1 (1+𝑟) $𝐶 2 (1+𝑟) $𝐶 3 (1+𝑟) 3 …+ $𝐶 𝑛 (1+𝑟) 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 But it’s useful to memorize them
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:
𝑃𝑉= 1 (1+𝑟) + 1 (1+𝑟) (1+𝑟) 3 …
𝑃𝑉= 1 (1+𝑟) + 1 (1+𝑟) 1 (1+𝑟) + 1 (1+𝑟) 2 …
Notice the thing in [] is the PV
𝑃𝑉= 1 (1+𝑟) + 1 (1+𝑟) 𝑃𝑉
Solve for PV
𝑃𝑉= 1 𝑟

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
𝑃𝑉= 𝐶 (1+𝑟) 𝑛 is the same as 𝐶= 1+𝑟 𝑛 𝑃𝑉 or
𝐹𝑉= 1+𝑟 𝑛 (𝑃𝑉)
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: 𝑃𝑉= 𝐶 𝑟−𝑔 can re-arranged to be 𝑟= 𝐶 𝑃𝑉 +𝑔
Other times, you can’t
Annuity: 𝑃𝑉= 𝐶 𝑟 (1− 𝑟 𝑛 ) cannot be solved for r by using algebra

31. Solving for interest rates numerically
But you can solve for r in 𝑃𝑉= 𝐶 𝑟 (1− 𝑟 𝑛 ) 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
𝑃𝑉= $𝐶 (1+𝑟) 𝑛
Other TVM formulas
Growing Perpetuity
Perpetuity
Annuity
Valuing Bonds