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Course: MATH 380, Fall 2009
School: Lake County
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Word Count: 325

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FUNCTION INVERSE THEOREM IMPLIES IMPLICIT FUNCTION THEOREM EUGENE LERMAN We show that the implicit function theorem can be deduced from the inverse function theorem. We consider only the case of f : R2 R. The proof in higher dimensions is analogous. The only real dierence is a more cumbersome notation. We assume that there is a point (a, b) R2 so that f (a, b) = 0. We want to show y that there is a neighborhood...

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FUNCTION INVERSE THEOREM IMPLIES IMPLICIT FUNCTION THEOREM EUGENE LERMAN We show that the implicit function theorem can be deduced from the inverse function theorem. We consider only the case of f : R2 R. The proof in higher dimensions is analogous. The only real dierence is a more cumbersome notation. We assume that there is a point (a, b) R2 so that f (a, b) = 0. We want to show y that there is a neighborhood U of a in R, V of b in R and a function g : U V so that f (x, g(x)) = c for x U . Here, c = f (a, b). Proof. Consider the map H(x, y) := (x, f (x, y)). It is a map from R to R . We will argue that it is invertible near (a, b). Indeed, its dierential DH(a, b) is 1 0 f f (a, b) y (a, b) x 2 2 Hence det DH(a, b) = 1 f (a, b), is which not zero by assumption. Hence DH(a, b) is y invertible. Hence, by the inverse function theorem, the map H is invertible near (a, b). Denote the inverse by G. It is of the form G(u, v) = (G1 (u, v), G2 (u, v)) for some real-valued functions G2 , G1 dened on a neighborhood of H(a, b) = (a, f (a, b)) = (a, c) in R2 . Since G and H are inverses of each other, (u, v) = H(G(u, v)) = H(G1 (u, v), G2 (u, v)) = (G1 (u, v), f (G1 (u, v), G2 (u, ...

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Lake County - MATH - 380
Lake County - MATH - 380
Lake County - MATH - 423
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Lake County - FIN - 432
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Lake County - FIN - 432
Finance 432 Class Exercise #17 Find a partner for this exercise. List both of your names below: _ _ Refer to the CDO spreadsheets handed out in class today. You are the actuary at a mid sized life insurance company. An investment banker has just made a pr
Lake County - FIN - 432
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Lake County - FIN - 432
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Lake County - FIN - 432
Payment Interest Rate 0.1 0.1 0.11 Year100,000 PV 14.5 15 15.5 26822 23939 21276#1 ED #2 EC #3 Change in r Change in Economic Value23.17366.9 -0.02 53.7%Check for reasonability Interest rate Year of payment PV Change in PV #4 k m n t T f(t)0.08 13 3
Lake County - FIN - 432
Finance 432 Class Exercise 10 Form groups of 2. List your names below: _ _ 1. Write the function for determining the present value of a 10 year zero coupon bond with a maturity value of \$1000, where r is the interest rate. P(r) = 2.Calculate the price o
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Lake County - FIN - 432
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Lake County - FIN - 432
1-a Spot Rates 1 year 2 year 3 year0.02 0.02 0.02Implied Forward Rates 1f1 0.0187 2f1 0.0204 1f2 0.0220 1-b Graph of term structure Time (in years) 0.083 2.15 0.250 2.35 0.5 2.41 1 2.29 2 2.08 3 2.12 5 2.64 7 3.01 10 3.52 20 4.23 30 4.23Term Structure
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Finance 432 Spring 2008 Class Exercise #3 ValueatRisk 1. A financial institution has \$30 million invested in stocks. The company uses a model to simulate the portfolio's annual return. The results of running this simulation for 100 iterations, ranked by t
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Finance432 Spring2008 ClassExercise#3 ValueatRisk Formintogroupsof2.Listbothofyournames: _ 1. _Afinancialinstitutionhas\$30millioninvestedinstocks.Thecompanyusesa modeltosimulatetheportfoliosannualreturn.Theresultsofrunningthis simulationfor100iterations,
Lake County - FIN - 432
1Date1/1/2008 7/1/2008 1/1/2009 7/1/2009 1/1/2010 7/1/2010 1/1/2011 7/1/2011 1/1/2012 7/1/2012LIBOR 3.07 5.62 7.84 8.02 9.89 9.33 7.62 10.19 11.03 6.15 27 25 1 0.04 0.25 0.59 0.34 0.72 0.63 4.29872 4.30026Net Interest Expense 3.07 5.62 7.00 7.00 7.89
Lake County - FIN - 432
1Date 1/1/2008 7/1/2008 1/1/2009 7/1/2009 1/1/2010 7/1/2010 1/1/2011 7/1/2011 1/1/2012 7/1/2012LIBOR 3.07 5.62 7.84 8.02 9.89 9.33 7.62 10.19 11.03 6.15 27 25 1 0.04 0.3 0.54 0.24 0.71 0.59 4.75806 4.76026Net Interest Expense 3.07 5.62 6.00 6.02 7.89 7
Lake County - FIN - 432
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Lake County - FIN - 432
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1)The first thing you need to do is calibrate the interest rates @ T = 1. T=0 T=1 T=2 Principal \$100.00 Coupon \$5.00 Present Value \$98.22 Coupon \$5.00 \$100.00 6.9032% 4.5000% Present Value \$100.78 Coupon \$5.00 4.1870%Principal \$100.00 Coupon \$5.00Princ
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Lake County - FIN - 432
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YearU. S. Treasury Interest Rate Data Month 6 month 2 year 10 year 1952 1 1.72 1.88 2.49 1952 2 1.75 1.97 2.64 1952 3 1.68 1.9 2.52 1952 4 1.77 1.87 2.44 1952 5 1.84 1.95 2.39 1952 6 1.88 2.05 2.42 1952 7 1.94 2.13 2.47 1952 8 2 2.21 2.56 1952 9 1.94 2.1
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Q1Interest Rate 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20% 21% 22% 23% 24% 25% 26% 27% 28% 29% 30%Price \$100,000.00 \$89,869.72 \$81,262.70 \$73,903.56 \$67,571.10 \$62,087.15 \$57,307.68 \$53,115.93 \$49,416.92 \$46,133.12 \$43,201
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