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bonds - Spring 2011 Duke University Econ 172 Lecture notes...

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) Emma Rasiel, 2009 1 Econ 172 Lecture notes Spring 2011 © Emma Rasiel, 2011 Duke University Present Value = Future Cash Flow (1 + R) t Greed Fear Impatience 2
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) Emma Rasiel, 2009 3 Discounted Cash Flows 4 Percentage Changes Yesterday’s price on an asset was P t-1 = 110, today’s price is P t = 121. What is the percentage price increase? Percentage change = = 10% Suppose instead the asset price fell from 110 to 99. What is the percentage price decrease? Percentage change = = - 10% 121 - 110 110 99 - 110 110
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) Emma Rasiel, 2009 5 Annual Rates of Return Suppose we observe last year’s stock price P t-1 , and today’s price P t What is the annual rate of return on the stock? It is the percentage change… R t = - 1 = P t P t-1 P t – P t-1 P t-1 Now suppose we observe monthly prices P t-1 , P t The monthly rate of return R month t = How do we convert this to an annualized rate of return? R annualized = (R 1month + 1) 12 - 1 What if we have 12 monthly returns: R 1 , R 2 , … , R 12 How do we obtain the annualized rate of return: R annualized = (Average[R 1 , R 2 , … , R 12 ] + 1) 12 – 1 P t – P t-1 P t-1 Why might we want to convert to an annualized return? 6 Rate of Return Examples Example 1: Last year’s stock price was $101. Today’s price is $106.5. What is the rate of return on the stock? Gross Return = = 1.054 Net return = 5.4% Example 2: Seven weekly closing levels on Gold in 2006: Feb 1, 06: 572.50 Feb 8, 06: 558.70 Feb 15, 06: 547.50 Feb 22, 06: 561.70 Mar 1, 06: 571.10 Mar 8, 06: 549.50 Mar 15, 06: 559.50 Weekly returns: r 1 = –2.41% r 2 = 2.00% r 3 = 2.59% r 4 = 1.67% r 5 = -3.78% r 6 = 1.82% Average weekly return: –0.35% Annualized return: (1 – 0.35%) 52 -1 = –16.74% 106.5 101
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) Emma Rasiel, 2009 7 Compounded Rates of Return You have $100 to invest. You are offered the following two rates: (i) 5.00% per annum (ii) 4.96% per annum with semi-annual compounding What does (ii) mean? By convention : given a quoted (annual) rate with semi-annual compounding…divide the rate in half, and receive that amount of interest every half-year. (i) 100(1.05) = $105 (ii) 100(1.0248)(1.0248) = $105.02 8 More Compounding Examples Bank offers 4% per annum savings rate for your 1 year, $100 investment… with no compounding: 100 (1.04) = $104.00 with semi-annual compounding: 100(1 + 0.04/2) 2 = 100 (1.02) 2 = $104.04 with monthly compounding: 100(1 + 0.04/12) 12 = 100(1 + 4%/12) 12 = 104.07 Year-end cash = K (1 + R/m) m General Formula : Invest K dollars for one year at R% per annum, m -period compounding: Where is the extra money coming from?
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) Emma Rasiel, 2009 9 Continuous Compounding Bank offers 4% per annum with continuous compounding ; you invest $100 as m ⇒ ∝ 100 (1 + 0.04/m) m = 100 e 0.04 = 104.08 Year-end cash = K e Rt General Formula : Invest K dollars for t years at a rate of R% per annum, continuous compounding 10 Example: $100 today. Invest at 6% per annum. Calculate Future Value: 100 + 6% x 100 = 100 (1 + 0.06) = 106 And in two years' time? 106 + (6% x 106) = 112.36 Generalize : Present Value PV dollars, t years, R is discount rate Future Value FV is: FV = PV (l+R) t Future Value
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) Emma Rasiel, 2009 11 Future Value Example If you have $100 today, and invest it at 6% per year, what will be its future value in three years?
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