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 ingredientsDiscount rate: 5%Annuity FormulaPV(Salary Increase) = $35,000 1− (1.05) − =$601,000NPV = PV(Salary Increase – Tuition) = $572,000CONGRATULATIONS!
7 NPV of the SMF: tweaking A few problems:Forgot to include lost salary while in schoolScrewed up salary timing: your salary increase should be delayed by a yearWhy a 5% discount rate?(The interested student should calculate a better NPV)
9 TVM: the basic idea$100 today is not the same as $100 four years from nowt = 01234$100t = 01234$100
10 TVM: the basic idea Suppose your bank offers you 3% interest 1234$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) 4More generallyIf the bank offers you an interest rate r,The PV of C dollars, n years from now, is𝑃𝑉= $𝐶 (1+𝑟) 𝑛
12 𝑃𝑉= $𝐶 (1+𝑟) 𝑛 TVM: Formulas The mother of all finance formulas: 𝑃𝑉= $𝐶 (1+𝑟) 𝑛In “principle,” this is all you need to know.
13 TVM: Formulas The key: Present values add up If the bank offers you interest rate rAnd you receive C1, C2, C3 ,… , Cnat 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 yearWhat 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 sheetIn real life, you’ll use Excel or MatlabBut it’s useful to memorize themBack-of-the-envelope calculationsIntuition*First impressions
19 TVM Formulas: Intuition Growing Perpetuity:Intuition:As the discount rate goes up, PV goes downAs 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 growthSimilar 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 PV𝑃𝑉= 1 (1+𝑟) + 1 (1+𝑟) 𝑃𝑉Solve for PV𝑃𝑉= 1 𝑟
23 TVM Formulas: Intuition Annuity:Intuition:This is the difference between two perpetuities
24 Annuity ExampleYou’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 DetailsGrowing PerpetuityPerpetuityAnnuityNote: for all formulas, the first cash flow C is at time 1
26 Timing Example 1Your 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 rateHow much is your food truck worth?
27 Timing Example 2Your 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 ValuesAny of the formulas can be used to find future values by rearranging the basic equation𝑃𝑉= 𝐶 (1+𝑟) 𝑛 is the same as 𝐶= 1+𝑟 𝑛 𝑃𝑉 or𝐹𝑉= 1+𝑟 𝑛 (𝑃𝑉)Then do a two-step1) Use PV formulas to take cash flows to the present2) 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 +𝑔Other times, you can’tAnnuity: 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 rExcel has similar functions for finding the PV and nPV(r,n,-C) gives you PVNper(r,-C,PV) gives you n
33 Valuing Bonds: JargonFace value: the amount used to calculate the couponUsually repaid at maturityCoupon: a regular payment paid until the maturityAPR: “annualized” interest rate computed by simple multiplicationDoes not take into account compounding interestYield-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 JargonBonds are typically issued at par: Price is equal to the face valueHere, the coupon rate = interest rateAfter issuance, prices fluctuate. The price may beAt a premium: price > parAt 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 formulas𝑃𝑉= $𝐶 (1+𝑟) 𝑛Other TVM formulasGrowing PerpetuityPerpetuityAnnuityValuing Bonds
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