# Review: Time Value of Money

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

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

\$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, = \$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) = \$35,000 1− (1.05) − =\$601,000 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)

Time value of money 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
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: \$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

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.

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

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 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

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: 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

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 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

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: 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

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

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