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# Financial Markets and Economic Fluctuations ECON 423

UNC

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This 4 page Class Notes was uploaded by Erna Gislason on Sunday October 25, 2015. The Class Notes belongs to ECON 423 at University of North Carolina - Chapel Hill taught by Michael Aguilar in Fall. Since its upload, it has received 27 views. For similar materials see /class/228687/econ-423-university-of-north-carolina-chapel-hill in Economcs at University of North Carolina - Chapel Hill.

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Date Created: 10/25/15

Economics 423 Spring 2014 Prof Aguilar Financial Markets UNC at Chapel Hill Student Name Honor Code Signature HW 1 Due 020614 beginning of Class Instructions 0 Students may work together but each must turn in their own assignment 0 Answers should be typed whenever possible and submitted with this page as a cover sheet 0 Please submit any computer code used to complete the assignment If you are using EXCEL then you can print out formuals by hitting Ctrl N Specialzied functions for time value of money bond valuation and the like are NOT permitted 0 Please explain all answers thoroughly There may be more than one correct solution 0 Point Allocation 1a1b1c212pts 2a2b2c122pts 35pts 4114172r310t5 1 You are purchasing a 25000 car The nancing of cer at the dealership quotes you a 5yr loan with an APR of 5 Assume payments are made monthly and that the monthly interest rate is 112 of the annual interest rate a What is the xed payment associated with this loan Provide an analytical as well as a numerical expression Solution We know that the PDV of a xed payment loan is PDV FP ELI We 71 can solve easily for the xed payment as FP PDV ELI In our case FP 47178 b What is the interest paid principal paid and amount due in month 20 Explain your numerical solution Solution In any given month the xed payment is split between interest and principal The interest amount is the monthly interest rate times the amount of the loan outstanding The portion paid to principal is the difference between the xed payment and the interest paid For month 20 the interest paid is 7395 which comes from 17747 due on the loan times the monthly interest rate The difference between the xed payment and the interest paid is 39783 which is the principal paid The amount due can then be found as 17747 39783 1734928 c Suppose you had 5000 available to put toward the purchase of the car You could either i use that money to lower your loan amount and invest the monthly savings in a series of discount bonds all maturing at the end of month 60 note the monthly savings are accrued beginning in the rst period of the loan not today or ii you could use the 5 000 to purchase a single asset that compounds at a set interest rate each month note the rst compounding period occurs in the rst month of the loan not today In either case the monthly interest rates are identical to that used for the car loan Which scenario do you prefer Provide analytical expressions and numerical justi cation State all necessary assumptions Solution By putting 5000 towards a downpayment the loan amount drops by 5000 which lowers the monthly payments to 37742 This is a monthly savings of 9436 per month If each month s savings were invested in an OD maturing at the end of the 60th month the future value of this stream would be 641679 Of course this assumes that the discount rate remains unchanged for the entire period Alterna tively I could invest the 5000 in an asset that grows by the monthly APR each period which would have a future value of 641679 Therefore I should be indif ferent between these two options if the earnings described are the only factors in my decision making process 2 You are a nancial engineer tasked with creating a new security called a 77Jobless Bond The asset is designed as follows The owner of this 10000 face value coupon bond is gauranteed an annual coupon of no less than 50 In those years where the unemployment rate is greater than 6 the coupon is adjusted For each tenth of a percentage point of Page 2 unemployment greater than 6 the coupon is increased by 1 For consistency let us assume that the present date is December 2003 a Explain the rationale behind the 77Jobless Bond Solution The purpose of the security is to provide an income stream that varies with the probability that the investor will become jobless As the unemployment rate rises the chances that the investor is out of work rises and so that coupons increase accordingly b Provide an analytical expression for the PDV of this asset Assume a generic 71 period horizon Solution T 50 UR 760 10 10000 PDV 21 t c Now assume this asset has a 10yr maturity Find the PDV of this asset Use annual unemployment data from 2004 to 2013 You may use the December values of the unemployment rate FRED Code UNRATE for adjusting the coupon Moreover you may assume that the discount rate is the December monthly value of the 10yr US Treasury Bond FRED Code G810 Solution The unemployment rate breaches the 6 mark during the crisis of 2008 through 2011 In those periods the coupon is increased The PDV is around 7 220 A key assumption is what we are expecting for the discount rate over the life of the asset Perhaps we use the last point available Or maybe we assume there was a forecasting mechansim Without speci cs on that mechanism we might say that the 77best77 one could do is if they had perfect foresight Many possibilities are acceptable See excel for further details 3 You are watching CNBC and you hear a bond investor discussing a particular security It s a 3yr 1000 face value bond with a 10 coupon rate selling at par The coupon is paid annually The discounting frequency is annual The 1yr Treasury rate which we will assume is the correct discount rate currently is 1 The investor says that if he sells the bond early at the end of year two prior to the year three coupon being paid he expects a 4 CAGR The investor says that he expects interest rates to change at a xed rate over the next three years Find an analytical expression for the rate of increase in the discount rate he must be expecting Solution 1 i i i 20 1 14 The CAGR associated w1th selling at the end of year two is 7 1 04 The sales price P3 Since rates are expected to change at a constant rate we can write is 1 z3i0 where i0 is today s discount rate and z is the object of interest 20 C F3 13 We can now write the CAGR as w 104 Solving for z yields z Page 3 13 7 1 31 7 1 Plugging in the known values provides us with z m 167 4 Consider an investor with exponential utility U ezpW facing a gamble that costs 10 to enter With probability 7139 the 10 will turn into 15 and with probability 1 7 7139 the 10 will turn into 5 a Assume 7r 05 Characterize this investor as either risk averse neutral or loving Will this investor partake in the gamble How much must you pay the investor NOT to partake in this gamble Solution The exponential utility function and its convex shape suggest a risk loving in vestor We can illustrate this in several ways The 05U5 05U15 163458289 and UW 2202647 The expected utility generated from gambling far outweighs that of not gambling In order to determine the amount needed to sway the investor from gambling we need to compute 7 WCE To calculate WCE we set UWCE 7 empWCE 163458289 7 WCE log1 634 58289 1431 This can be interpreted as how much the investor would hav to get paid in order to make her just as happy as the gamble Analogously we can compute the risk premium as 7 WCE 10 7 1431 7431 which can be interpreted as the investor must be given 431 in order to be convinced not to gamble b Now suppose the probability of heads is not necessarily 05 Find the level of 7139 that would cause the the investor to behave as if they were risk neutral Solution You can t turn this investor into a risk a neutral agent Their utility function belies that notion However you can alter the associated probabilities of the gamble such that they behave AS IF they were risk neutral We know that a risk neutral investor is indifferent between gambling and not This can be represented as zero risk premium or in other words WOE First we can nd the WCE as UWCE 7 eWCE 7r515 1 7 7r55 7 WCE ln7r515 17 7r55 Next set this certainty equivalent equal to expected wealth ln7r515 1 77055 10 7 7139515 17 7r55 510 7 7139 m 007 We need a very small weight on the high outcome to make us just as happy as not gambling Page 4

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