REVIEW: TIME VALUE OF MONEY

SMF Prep Workshop Andrew Chen - OSU

This session:

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

$53,000 Thank you. Is it worth it?

(yes)

How much is it worth?

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

$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

NPV of the SMF: the right way

Additional ingredients Discount rate: 5% Annuity Formula

PV(Salary Increase) =

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

r

rCPV

n)1(1

CONGRATULATIONS!

NPV of the SMF: tweaking

A few problems:1. Forgot to include lost salary while in school2. Screwed up salary timing: your salary

increase should be delayed by a year3. Why a 5% discount rate?

(The interested student should calculate a better NPV)

TIME VALUE OF MONEY

Formulas

TVM: the basic idea

$100 today is not the same as $100 four years from nowt = 0 1 2 3 4

t = 0 1 2 3 4

$100

$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

$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

TVM: Formulas

𝑃𝑉=$𝐶

(1+𝑟)𝑛

The mother of all finance formulas:

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

TVM: Formulas

𝑃𝑉=$𝐶1

(1+𝑟)1+$𝐶2

(1+𝑟 )2+$𝐶3

(1+𝑟 )3…+

$𝐶𝑛

(1+𝑟 )𝑛

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

r

rCPV

n)1(1

r

CPV

gr

CPV

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)

gr

CPV

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

r

CPV

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

r

rCPV

n)1(1

r

C

rr

C

r

rC

nn

1

1)1(1

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

r

rCPV

n)1(1

r

CPV

gr

CPV

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

TIME VALUE OF MONEY

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

𝑃𝑉=$𝐶

(1+𝑟)𝑛

Other TVM formulas Growing Perpetuity Perpetuity Annuity

Valuing Bonds