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REVIEW: TIME VALUE OF MONEY Andrew Chen - OSU SMF Prep Workshop This session: ο± The mother of all finance formulas $πΆ ππ = (1 + π)π ο± Other TVM formulas ο± Growing Perpetuity ο± Perpetuity ο± Annuity ο± Valuing Bonds This should be a review $53,000 ο± ο± Thank you. Is it worth it? ο± (yes) How much is it worth? NPV of the SMF: Ingredients ο± ο± Tuition / Fees: $53,000 New Salary: $85,000 ο± (Median ο± Old Salary: $50,000 ο± (Nice ο± Fisher MBA) round number) Years ‘till retirement: 40 NPV of the SMF ο± ο± (Change in Salary) x (Working Years) = $35,000 x 45 = $1.575 million (Benefits) – (Costs) = $1.575 million - $50,500 = $1.525 ο± $35,000 million in 2050 is not the same thing as $35,000 today. NPV of the SMF: the right way ο± Additional ingredients ο± Discount rate: 5% ο± Annuity Formula ο± ο± ο©1 ο (1 ο« r ) ο n οΉ PV ο½ C οͺ οΊ r ο« ο» 1−(1.05)−45 0.05 PV(Salary Increase) = $35,000 = $601,000 NPV = PV(Salary Increase – Tuition) = $572,000 CONGRATULATIONS! NPV of the SMF: tweaking ο± A few problems: 1. 2. 3. ο± 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 ο± $100 today is not the same as $100 four years from now t=0 1 2 3 4 1 2 3 4 $100 t=0 $100 TVM: the basic idea ο± 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 ο± $100 today is worth $113 four years from now TVM: the basic idea ο± Flip that around: ο± $113 four years from now is worth $113 $100 = (1 + 0.03)4 ο± More generally ο± If the bank offers you an interest rate r, ο± The PV of C dollars, n years from now, is $πΆ ππ = (1 + π)π TVM: Formulas ο± The mother of all finance formulas: $πΆ ππ = π (1 + π) ο± In “principle,” this is all you need to know. TVM: Formulas ο± The key: Present values add up ο± If the bank offers you interest rate r And you receive C1, C2, C3 ,… , Cn ο± at the end of years 1, 2, 3, …, n, ο± $πΆ1 $πΆ2 $πΆ3 $πΆπ ππ = + + …+ 1 2 3 (1 + π) (1 + π) (1 + π) (1 + π)π Basic TVM Formula: Example 1 ο± A zero-coupon bond will pay $15,000 in 10 years. Similar bonds have an interest rate of 6% per year ο± What is the bond worth today? Basic TVM Formula: Example 2 ο± 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. ο± How much can you borrow from your uncle? Basic TVM Formula: Example 3 ο± Your crazy uncle has a business plan that will generate $100 every year forever. He claims that an appropriate discount rate is 5%. ο± How much does he think his business plan is worth? TVM Formulas ο± Growing Perpetuity ο± Perpetuity ο± Annuity ο± Note: for all formulas, the first cash flow C is at time 1 C PV ο½ rοg C PV ο½ r ο©1 ο (1 ο« r ) ο n οΉ PV ο½ C οͺ οΊ r ο« ο» TVM Formulas ο± No need to memorize ο± In exams, you’ll get a formula sheet ο± In real life, you’ll use Excel or Matlab ο± But it’s useful to memorize them ο± Back-of-the-envelope ο± Intuition ο± *First impressions calculations TVM Formulas: Intuition ο± Growing Perpetuity: ο± Intuition: C PV ο½ rοg ο± As the discount rate goes up, PV goes down ο± As the growth rate goes up, PV goes up ο± (This is a nice one to memorize) Growing Perpetuity Example ο± A stock pays out a $2 dividend every year. The dividend grows at 1% per year, and the discount rate is 6%. ο± How much is the stock worth? Perpetuity Formula ο± Perpetuity: ο± Intuition: ο± This C PV ο½ r is just a growing perpetuity with 0 growth ο± Similar interpretation to a growing perpetuity Deriving the Perpetuity Formula ο± ο± It’s just some clever factoring: ο± ππ = ο± ππ = + + 1 1 + … (1+π)2 (1+π)3 1 1 1 + … (1+π) (1+π) (1+π)2 Notice the thing in [] is the PV ο± ππ ο± 1 (1+π) 1 (1+π) = 1 (1+π) Solve for PV ο± ππ = 1 π + 1 (1+π) ππ TVM Formulas: Intuition ο± Annuity: ο± Intuition: ο± This ο©1 ο (1 ο« r ) ο n οΉ PV ο½ C οͺ οΊ r ο« ο» is the difference between two perpetuities ο©1 ο (1 ο« r ) οΉ C ο¦ 1 οΆ C Cοͺ ο· οΊ ο½ οο§ r ο« ο» r ο¨1ο« r οΈ r οn n Annuity Example ο± 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. ο± Which option should you take? ο± *What’s wrong with this analysis? Timing Details ο± Growing Perpetuity ο± Perpetuity ο± Annuity ο± Note: for all formulas, the first cash flow C is at time 1 C PV ο½ rοg C PV ο½ r ο©1 ο (1 ο« r ) ο n οΉ PV ο½ C οͺ οΊ r ο« ο» Timing Example 1 ο± 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 ο± How much is your food truck worth? Timing Example 2 ο± 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. ο± How much did she lend you? Future Values ο± Any of the formulas can be used to find future values by rearranging the basic equation ο± ππ = πΆ (1+π)π is the same as πΆ = 1 + π π ππ or π ο± πΉπ = 1 + π (ππ) ο± Then do a two-step ο± 1) Use PV formulas to take cash flows to the present ο± 2) Use FV formula to move to the future Future Values: Example ο± 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. ο± How much can you spend on your truck in 10 years? Solving for interest rates ο± Sometimes you can solve for the interest rate: ο± Growing π= ο± πΆ ππ Perpetuity: ππ = πΆ π−π can re-arranged to be +π Other times, you can’t ο± Annuity: ππ = using algebra πΆ 1 (1 − ) π 1+π π cannot be solved for r by Solving for interest rates numerically ο± But you can solve for r in ππ = πΆ 1 (1 − ) π 1+π π by using Excel. ο± Rate(n,-C,PV) ο± gives you r Excel has similar functions for finding the PV and n ο± PV(r,n,-C) gives you PV ο± Nper(r,-C,PV) gives you n TIME VALUE OF MONEY Valuing Bonds Valuing Bonds: Jargon ο± Face value: the amount used to calculate the coupon ο± Usually ο± ο± Coupon: a regular payment paid until the maturity APR: “annualized” interest rate computed by simple multiplication ο± Does ο± repaid at maturity not take into account compounding interest Yield-to-Maturity (YTM): the interest rate Valuing Bonds: Example 1 ο± You are thinking of buying a 5-year, $1000 facevalue bond with a 5% coupon rate and semiannual coupons. Suppose the YTM on comparable bonds is 6.3% (APR with seminannual compounding). ο± How much is the bond worth? Valuing Bonds: Example 2 ο± 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) ο± How much is the bond worth? Bonds: More Jargon ο± Bonds are typically issued at par: Price is equal to the face value ο± Here, ο± the coupon rate = interest rate After issuance, prices fluctuate. The price may be ο± At a premium: price > par ο± At a discount: price < par Valuing Bonds: Example 3 ο± 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). ο± Does the bond trade at a premium or discount? ο± What is the new bond price? Why it’s called “Yield to Maturity” ο± 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). ο± If you bought the bond at issue and held it to maturity, what “effective interest rate” did you get? ο± If you bought it at issue and sold it two years later, what “effective interest rate” did you get? TVM Wrapup: We covered… ο± The mother of all finance formulas $πΆ ππ = (1 + π)π ο± Other TVM formulas ο± Growing Perpetuity ο± Perpetuity ο± Annuity ο± Valuing Bonds