# Chapter 2 Present Value

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1 Chapter 2 Present Value Road Map Part A Introduction to finance. Financial decisions and financial markets. Present value. Part B Valuation of assets, given discount rates. Part C Determination of risk-adjusted discount rates. Part D Introduction to derivatives. Main Issues Present Value Compound Interest Rates Nominal versus Real Cash Flows and Discount Rates Shortcuts to Special Cash Flows

2 Chapter 2 Present Value Valuing Cash Flows Visualizing cash flows. CF 1 CF T t =0 t =1 t = T time CF 0 Example. Drug company develops a flu vaccine. Strategy A: To bring to market in 1 year, invest \$1 B (billion) now and returns \$500 M (million), \$400 M and \$300 M in years 1, 2 and 3 respectively. Strategy B: To bring to market in 2 years, invest \$200 M in years 0 and 1. Returns \$300 M in years 2 and 3. Which strategy creates more value? Problem. How to value/compare CF streams. Fall 2006 c J. Wang Lecture Notes

3 2-2 Present Value Chapter Future Value (FV) How much will \$1 today be worth in one year? Current interest rate is r, say,4%. \$1 investable at a rate of return r =4%. FV in 1 year is FV =1+r =\$1.04. FV in t years is FV = \$1 (1+r) (1+r) = (1+r) t. Example. Bank pays an annual interest of 4% on 2-year CDs and you deposit \$10,000. What is your balance two years later? FV =10, 000 ( ) 2 = \$10, Lecture Notes c J. Wang Fall 2006

4 Chapter 2 Present Value Present Value (PV) We can ask the question in reverse (interest rate r =4%). What is the PV of \$1 received a year from now? Consider putting away 1/1.04 today. A year later receive: 1 (1+0.04) = The PV of \$1 received a year from now is: 1 1+r = The present value of \$1 received t years from now is: PV = 1 (1+r) t. Example. (A) \$10 M in 5 years or (B) \$15 M in 15 years. Which is better if r =5%? PV A = =7.84. PV B = 15 = Fall 2006 c J. Wang Lecture Notes

5 2-4 Present Value Chapter 2 Solution to Example. Flu Vaccine. Assume that r =5%. Strategy A: Time Cash Flow -1, Present Value -1, Total PV 98.2 Strategy B: Time Cash Flow Present Value Total PV Firm should choose strategy B, and its value would increase by \$140.8 M Lecture Notes c J. Wang Fall 2006

6 Chapter 2 Present Value Compound Interest Rates 2.1 APR and EAR Sometimes, interest rate is quoted as an annual percentage rate (APR) with an associated compounding interval. Example. Bank of America s one-year CD offers 5% APR, with semi-annual compounding. If you invest \$10,000, how much money do you have at the end of one year? What is the actual annual rate of interest you earn? Quoted APR of r APR =5%is not the actual annual rate. It is only used to compute the 6-month interest rate as follows: (5%)(1/2) = 2.5%. Investing \$10,000, at the end of one year you have: 10, 000( )( ) = 10, In the second 6-month period, you earn interest on interest. The actual annual rate, the effective annual rate (EAR), is r EAR =( ) 2 1=5.0625%. Annual rates typically refer to EARs. Fall 2006 c J. Wang Lecture Notes

7 2-6 Present Value Chapter Compounding Let r APR be the annual percentage rate and k be the number of compounding intervals per year. One dollar invested today yields: ( 1+ r APR k ) k dollars in one year. Effective annual rate, r EAR is given by: (1 + r EAR )= ( 1+ r APR k ) k or r EAR = ( 1+ r APR k ) k 1. Example. Suppose r APR =5%: k Value of \$1 in a year r EAR % % % % 8, %... e 0.05 = % Here, e Lecture Notes c J. Wang Fall 2006

8 Chapter 2 Present Value Real vs. Nominal CFs and Rates Nominal vs. Real CFs Inflation is 4% per year. You expect to receive \$1.04 in one year, what is this CF really worth next year? The real or inflation adjusted value of \$1.04 in a year is Real CF = Nominal CF 1+inflation = =\$1.00. In general, at annual inflation rate of i we have (Real CF) t = (Nominal CF) t (1 + i) t. Nominal vs. Real Rates Nominal interest rates - typical market rates. Real interest rates - interest rates adjusted for inflation. Example. \$1.00 invested at a 6% interest rate grows to \$1.06 next year. If inflation is 4% per year, then the real value is \$1.06/1.04 = The real return is 1.9%. 1+r real = 1+r nominal. 1+i Fall 2006 c J. Wang Lecture Notes

9 2-8 Present Value Chapter 2 Example. Sales is \$1M this year and is expected to have real growth of 2% next year. Inflation is expected to be 4%. The appropriate nominal discount rate is 5%. What is the present value of next year s sales revenue? Next year s nominal sales forecast: = PV = = Next year s real sales forecast: = Real discount rate: r real = 1+r n 1+i 1= =0.9615%. PV = = Important Rules: Discount nominal CFs by nominal discount rates. Discount real CFs by real discount rates Lecture Notes c J. Wang Fall 2006

10 Chapter 2 Present Value Shortcuts to Special Cash Flows 4.1 Annuity A A A t =0 1 2 T time Today is t =0and cash flow starts at t =1. PV (Annuity) = A 1+r + A (1+r) = A 1 [ ] 1 1. r (1+r) T A 1+r) T Example. An insurance company sells an annuity of \$10,000 per year for 20 years. Suppose r =5%. What should the company sell it for? PV = 10, ( 1 = 124, ) =10, FV (Annuity) = PV (Annuity) (1+r) T. Fall 2006 c J. Wang Lecture Notes

11 2-10 Present Value Chapter Annuity with Growth AA(1+g) A(1+g) T t =0 1 2 T time PV (growing annuity) = A = A [ 1 1+r + 1+g ] (1+r) + + (1+g)T 1 2 (1+r) T [ 1 r g 1 ( ) ] T 1+g 1+r if r g T 1+r if r = g. Example. Saving for retirement - Suppose that you are now 30 and would like \$2 million at age 65 for your retirement. You would like to save each year an amount that grows by 5% each year. How much should you start saving now, assuming that r=8%? A = 2, 000, =6, Lecture Notes c J. Wang Fall 2006

12 Chapter 2 Present Value Perpetuity A A A t = time A perpetuity is an annuity with infinite maturity. Example. You just won the lottery and it pays \$100,000 a year for 20 years. Are you a millionaire? Suppose that r = 10%. PV = 100, ( = 851, 356. ) = 100, What if the payments last for 50 years? PV = 100, ( = 991, 481. ) = 100, How about forever - a perpetuity? PV = 100, 000/0.10 = 1, 000, 000. PV (Perpetuity) = A r. Fall 2006 c J. Wang Lecture Notes

13 2-12 Present Value Chapter Perpetuity with Growth AA(1+g) A(1+g) 2 t = time PV (Perpetuity with growth) = A r g where A is the first CF one period from now. Example. Super Growth Inc. will pay an annual dividend next year of \$3. After analyzing the company, you expect the dividend after that to grow at the rate of 5% per year forever. Also for companies of this risk class, the expected return is 10%. What should be Super Growth s price per share? PV = 3 3( ) 3( ) = = Lecture Notes c J. Wang Fall 2006

14 Chapter 2 Present Value 2-13 Example. Mortgage calculation in the U.S. Pay 20% down payment, and borrow the rest from the bank using the property as collateral. Pay a fixed monthly payment for the life of the mortgage. Have the option to prepay the mortgage anytime before the maturity date of the mortgage. Suppose that you bought a house for \$500,000 with \$100,000 down payment and financed the rest with a thirty-year fixed rate mortgage at 8.5% APR compounded monthly. The monthly payment M is determined by 400, 000 = = 360 M [1 + (0.085/12)] t t=1 M (0.085/12) { 1 } 1 [1 + (0.085/12)] 360 = M (0.9212) (0.085/12). M =\$3, Your effective annual interest rate (EAR): [1 + (0.085/12)] 12 1= =8.839%. Fall 2006 c J. Wang Lecture Notes

15 2-14 Present Value Chapter 2 The monthly payments are as follows: t (month) Principal Interest Sum Remaining P , , , , , , , , Total monthly payment is the same for each month. The percentage of principal payment increases over time. The percentage of interest payment decreases over time Lecture Notes c J. Wang Fall 2006

16 Chapter 2 Present Value Homework Readings: BMA Chapter 3. Assignment: Problem Set 1. Fall 2006 c J. Wang Lecture Notes

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