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Econ173A_topic4

Econ173A_topic4 - Ec 173A-FINANCIAL MARKETS Foster UCSD...

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Ec 173A—FINANCIAL MARKETS LECTURE NOTES Foster, UCSD 12-Jul-08 TOPIC 4 – RISK & RETURN A. Risk Factors in Interest Rates 1. Interest Compounding and Growth: [Optional Review from Topic 2] a) 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 b) 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 c) Continuous compounding. 1) Some investments and bank accounts report annual rate of interest r, compounded continuously (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 1.0 years: V(1.0) = V(0)e 1.0r = 500 e .040 = $520.41 In 3.7 years: V(3.7) = V(0)e 3.7r = 500 e .148 = $579.76 d) Finding annual interest rates from periodic rates. 1) Annual percentage rate (APR). ( 29 m m r r apr × = ≡ the periodic rate × no. of compounding periods/year. For a loan offered at 1%/month, r apr = 1% × 12 = 12%/year. 2) Effective (equivalent) annual rate (EAR). r ear ≡ annual rate resulting in same growth of principal as quoted periodic rate. If r/m = quoted rate compounded m times/year: For a loan offered at 1%/month, r ear = (1.01) 12 − 1 = 12.68%/year. ( 29 1 1 - + = m ear m r r
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Ec 173A RISK & RETURN p. 2 2. Building an Interest Rate: a) I am willing to give up 100 loaves of bread this year if I can have at least 102 loaves of bread 1 year from now. My brother in law wants to borrow $100 for a year. What interest rate should I charge? 1) If bread costs $1/loaf and I think this price will remain constant, I might lend $100 for one year at r = 2%, having $102 when loan is paid off, which will buy me 102 loaves of bread. (Of course, if I could get a higher rate on other loans, then I will charge my brother in law the higher rate.) 2) If I expect bread price to rise 4% to $1.04, I will charge r = (1.02)(1.04) − 1 = 6.08%. I get $106.08 next year, with real purchasing power of 106.08/1.04 = 102 loaves. 3) If I estimate a 5% risk default, I charge r = (1.02)(1.04)/0.95 − 1 = 11.66%. I get back $111.66 if loan is repaid, and $0 if loan defaults, for an expected payoff of 0.95(111.66) + 0.05(0) = $106.08, with purchasing power of 102 loaves. b) Generalization. 1) In the above 3 cases, we found the needed interest rate r by the following formula, where: r = nominal or actual interest rate r p = minimum compensatory (pure) interest rate i = predicted rate of price inflation d = risk factor 2) So (1 + r)(1 − d) = (1 + r p )(1 + i), or r = d + r×d + r p + i + i×r p . If we drop small terms like r×d and i×r p , we get a linear approximation: d i r r p + + .
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