NTop-2 - Ec 173A FINANCIAL MARKETS LECTURE NOTES Foster,...

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Ec 173A – FINANCIAL MARKETS LECTURE NOTES Foster, UCSD October 21, 2010 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 X 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. 1 1 1 0 ) 1 ( - - - - = = + = t t t t t t t X X X X X g and g X X c) Continuous exponential growth model. ) ( ) ( ) 0 ( ) ( t X t X g and e X t X gt = = 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 V 3 = V 0 (1+r) 3 = 500 (1.04) 3 = 500(1.1249) = $562.43
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Ec 173A PERSONAL FINANCE p. 2 of 9 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 m = 1%, m = 4: In 3 months: V 1/4 = V 0 (1 + r m ) = 500 (1.01) = $505.00 In 6 months: V 1/2 = V 0 (1 + r m ) 2 = 500 (1.0201) = $510.05 In 9 months: V 3/4 = V 0 (1 + r m ) 3 = 500 (1.0303) = $515.15 In 1 year: V 1 = V 0 (1 + r m ) 4 = 500 (1.0406) = $520.30 d) Continuous compounding. 1) Some investments and bank accounts report annual rate of interest r, compounded con-tinuously (and deposited daily or weekly). For continuous compounding, m and initial value V(0) increases as in a continuous exponential growth model. The value at the end of t years is given by V(t) = V(0)e rt . 2) Example -- V(0) = $500, r = 4%: In 2.0 years: V(2.0) = V(0)e 2.0r = 500 e .080 = $541.64 In 3.7 years: V(3.7) = V(0)e 3.7r = 500 e .148 = $579.76 e) Annuities -- A stream of equal cash flows every year for T years. There are two kinds: Ordinary (“deferred”) annuity -- cash flows occur at end of each year Annuity due - cash flows occur at beginning of each year f) Future value of an annuity due (FVAD). 1)
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NTop-2 - Ec 173A FINANCIAL MARKETS LECTURE NOTES Foster,...

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