Econ173A_topic2

Econ173A_topic2 - m)mt = V0(1+rear)t (1 + r/m)mt =...

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m) mt = V 0 (1+r ear ) t (1 + r/m) mt = (1+r ear ) t r/m) = t Ln(1+r ear ) Ln(1+r ear ) = mLn(1 + r/m) 1 + r/m) m r ear = (1 + r/m) m − 1 Ec 173A TIME VALUE OF MONEY p. 1 Ec 173A – FINANCIAL MARKETS LECTURE NOTES Foster, UCSD 13:14:47 A. Interest Compounding, Growth and Discounting 1. Compound Growth: a) Stocks and flows. 1) Stock -- a quantity measured at an instant of time. 2) Flow -- quantity measured over a period of time; the rate of change in some stock. 3) Stock may change level discretely (interest paid at end of quarter) or continuously (H 2 O evaporating from lake). NOTATION Symbol Definition X t Level of stock X at end of period t X t 1-period change in X during period t: X t ≡ X t − X t−1 X(t) Level of stock X at instant of time t or X’(t) Instantaneous rate of change of X at time t: X’(t) ≡ dX(t)/dt r Interest or discount rate g Constant compound growth rate of stock X b) Discrete exponential growth model. c) Continuous exponential growth model. 2. Future Values and Compound Interest: a) An interest rate (r) can be viewed 3 ways: 1) A cost of borrowing. 2) A growth rate of value of money invested in interest-bearing debt securities. 3) A discount factor for finding the present equivalent value of payments to be made or received in the future. b) Interest compounded annually. 1) If principal V 0 earns interest at annual rate r, compounded at the end of each year, its value rises according to discrete exponential growth equation V t = V 0 (1+r) t . 2) Example -- V 0 = $500, r = 0.04 (4%): V 1 = V 0 (1+r) = 500 (1.04) = $520.00 V 2 = V 1 (1+r) = V 0 (1+r) 2 = 500 (1.04) 2 = $540.80
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Ec 173A TIME VALUE OF MONEY p. 2 V 3 = V 0 (1+r) 3 = 500 (1.04) 3 = 500(1.1249) = $562.43 c) Interest compounded more frequently. 1) If V 0 earns interest at periodic rate r/m, compounded m times per year, its value at the end of t years is given by V t = V 0 (1 + r/m) mt . 2) Example -- V 0 = $500, r= 4%, m = 4, so r/m=1% (quarterly compounding): In 3 months: V 1/4 = V 0 (1 + r/m) 4*0.25 = 500 (1.01) = $505.00 You won’t really need a financial calculator for any of these, but this is the sequence: N=mt=1, I/Y=r/m=1, PV=500, FV=? In 6 months: V 1/2 = V 0 (1 + r/m) 4*0.5 = 500 (1.0201) = $510.05 In 9 months: V 3/4 = V 0 (1 + r/m) 4*0.75 = 500 (1.0303) = $515.15 In 2.5 years V 5/2 = V 0 (1 + r/m) 4*2.5 = 500 (1.1046) = $552.31 3) Example -- V 0 = $500, r= 4%, m = 2, so r/m=2% (semiannual compounding): In 3 months: V 1/4 = V 0 (1 + r/m) 2*0.25 = 500 (1.0995) = $504.9752 In 6 months: V 1/2 = V 0 (1 + r/m) 2*0.5 = 500 (1.02) = $510 In 9 months: V 3/4 = V 0 (1 + r/m) 2*0.75 = 500 (1.0301) = $515.07 In 2.5 years: V 5/2 = V 0 (1 + r/m) 2*2.5 = 500 (1.1041) = $552.05 4) Example -- V 0 = $500, r = 4%, m = 1 (annually), or 2 (semiannually), or 4 (quarterly), or 12 (monthly), or 365 (daily): In 1 year, m=1: V 1 = V 0 (1 + r/m) 1*1 = 500 (1.04) = $520 In 1 year, m=2: V 1 = V 0 (1 + r/m)
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Econ173A_topic2 - m)mt = V0(1+rear)t (1 + r/m)mt =...

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