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Review: Time Value of MoneyPowerPoint Presentation

Review: Time Value of Money

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### Time value of money

### Time value of money

This session:

- The mother of all finance formulas

- Other TVM formulas
- Growing Perpetuity
- Perpetuity
- Annuity

- Valuing Bonds

This should be a review

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

- (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

- Additional ingredients
- Discount rate: 5%
- Annuity Formula

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

CONGRATULATIONS!

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)

Formulas

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 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 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

- 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

- 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

- 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

- 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 Formulas

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

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

- 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

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

Deriving the Perpetuity Formula

- It’s just some clever factoring:
- Notice the thing in [] is the PV
- Solve for PV

TVM Formulas: Intuition

- Annuity:
- Intuition:
- This is the difference between two perpetuities

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

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

- 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

- 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

- 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

- 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

- 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

Valuing Bonds

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

- 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

- 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 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

- 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”

- 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

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