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Financial Markets and Economic Fluctuations

by: Ferne Miller

Financial Markets and Economic Fluctuations ECON 423

Marketplace > University of North Carolina - Chapel Hill > Economcs > ECON 423 > Financial Markets and Economic Fluctuations
Ferne Miller
GPA 3.96


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This 7 page Class Notes was uploaded by Ferne Miller on Sunday October 25, 2015. The Class Notes belongs to ECON 423 at University of North Carolina - Chapel Hill taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/228696/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 185 Michael Salemi Notes on Present Value and Internal Rate of Return 1 Assets such as bonds and stocks are held because they give the owner the right to receive a stream of payments Examples 0 Streams of P ments BOND Ct CH1 C Hm Maturity Value 100 100 100 1000 STOCK Dt DH 1 DHm Sale Price at Date tm 50 60 40 PHm 2 Present Value Because a dollar received in the future is less valuable than a dollar in hand today we need some procedure to evaluate the current value of future payments In these notes we assume that all payments will occur without any risk of default The concept we will use to compute the value of future payments is present value Suppose the interest rate is eXpected to be constant over the Myear life of the asset and suppose the asset promises to pay Sm dollars in period tj Then present value is de ned to be pV L Squot SVS St39M 39 1 R 1 R2 1 R3 1 RM How to ThinkAbout Present Value The present value of a stream of payments is an amount that would permit you to replicate the stream provided you can borrow and lend at the same rate of interest used in the PV computation Example Let R 08 Consider a three year bond with 11 coupon and par value of 1000 The stream of payments is 110 at end of years 1 and 2 and 1110 at the end of year 3 Applying the above formula we can compute the present value of this stream as follows PV 110 H 110 1110 107731 108 1082 1083 But what does it mean for the present value to be 107731 The following table shows that with 107731 in hand one has just enough to replicate the stream of payments that the bond promises to pay Year Balance Earned Interest Yearend Payout 1 107731 8618 110 2 105349 8428 110 3 102777 8222 110 1000 4 000 Generalization ofPresent Value Formula The previous de nition of present value assumed a constant rate of interest NeXt we de ne present value for a situation where the interest rate is not eXpected to remain constant in the future Suppose the interest rate is eXpected to be R1 in the rst year t to t1 R2 in the second year t1 to t2 and so forth The stream of payments is SH SH SHM In the present value formula payment Sm is discounted by dividing by a product of interest rate terms If R1 R for all j the following formula simpli es to the one on the previous page PV L St Sl 3 t 1 39RI 1 39Rllll 39RZ 1 39Rllll R21 R3 StM 1 R1 1 HRM Example M3 S11S12110andS131110 R1 08 R2 09 R3 10 Pv1 110 o o 110 1110 105249 108 108109 108109110 Continuous Time Discounting 2 3 Theseries 1 R 1 R 1 R u n is a discretetime discount function Discrete time is 1 1 1 when it is to hypothesize that time is a sequence of years months weeks or days in which case the right R to use is a yearly monthly weekly or daily interest rate respectively In some settings it is reasonable to hypothesize that time evolves continuously When time evolves continuously the right discount function is 5quot for s 39t where R is a continuously compounded rate If we measure time in years the sequence 3R corresponding to that above is e39R e39 7 Example Let R 10 The following table compares the discrete and continuous time discount factors for horizons of between one and ve years 1 K 909 826 751 683 621 l R e39KR 905 819 741 670 607 3 Internal Rate of Return Internal rate of return IR is another concept used to evaluate a stream of payments When one knows the price of the asset and the stream of payments that it promises it is standard to ask how much the asset yields Internal rate of return provides the answer to this question If the asset is a bond internal rate of return is called the yield to maturity of the bond Consider the stream SH 1 S H2 SHM with market price PS The internal rate of return IRR is that number such that the market price is equal to the present value using IRR as a discount rate Note that the de nition of IR is implicit rather than eXplicit S S 1 IRR 1IRR2 lIRRM St39 1 t39 2 t39 M PS t Example Pst 900 M 3 s st2 100 st3 1100 100 100 o o 1100 900 1 IRR 1 IRR2 1 IRR3 The formula may be rewritten as a cubic equation 1 IRR3 900 1 IRR2100 1 IRR 100 1100 Typically computer algorithms are used to solve for internal rate of return However there is an IR approximation formula Let a bond with M years to maturity have an annual coupon of C and a market price of PS C H1000 PS 100 H1000 900 IRR H 3 13333 140 1000 PS 1000 900 950 2 2 4 Present Value with Constant Periodic Payments There is a simpli cation of the present value formula that holds if the annual coupon is constant If the coupons is C per year and par value is F and maturity in M years present value is 1 R 1R2 1 RM 1 RM FCR 1 M 1 R Let F 1000 and C 100 The following table gives present value for various values of R and M 5 Return ofa Financial Asset The return of a nancial asset is a very different concept from the present value of an asset although the two concepts are related Return on an asset can be de ned retrospectively or prospectively Retrospective Return C pt ooPti Retumt 1t tquot1 Prospective Return C pt 1 pt Retumtt 1 Pt Questions Explain how the above present value table shows that longmaturity bonds are riskier than short maturity bonds 2 How accurate is the approximation formula for IRR for a ve year note with a 100 coupon and a price of 122516 3 How can one use present value to compute an prospective return for a ten year bond that you buy in 2004 and eXpect to seel in 2005 4 You buy a ten year bond in 2004 and sell it in 2005 Under what conditions will the retrospective return you compute in 2005 turn out to be lower than the prospective return you compute in 04 Economics 423 Michael Salemi Modeling the Price ofA Share of Common Stock 1 Market Fundamentals Model The mainstream model of share prices is the fundamentals model The model says that shares are valued because and only because they are a claim to expected future dividends Assume 1 Agents are risk neutral so that they care only about expected returns 11 Agents have common beliefs about what determines the value of a share 111 Agents forecasts are rational The agents know as much as the economist IV Agents are price takers Let PZYt be the price ofa share of company Z at time t and let D Zytbe the dividend paid by Z at time t Then the fundamentals model says that the price of a share is the present value of expected future dividends P E Dzt1 Et Dzt2 a El Dzts 1 R5 zquot llR 1R2 One version of the fundamentals model adds hypotheses about how rms earn pro ts and pay dividends Given those additional hypotheses the fundamentals model predicts how the price of a share will related to the parameters of the dividend model For simplicity of notation we drop the Zsubscript so that Pt P 27 t Suppose the rm pays dividends equal to a xed proportion of pro ts and pro ts are equal to a standard return to capital plus a random component The rm uses retained pro ts to increase its capital stock Let Qt be pro t per share at time t Kt capital per share at time t and let U t be an unforecastable random shock to pro ts D quotQ i U Qt K t Km K1 Q Plugging the equations into the present value expression for stock prices permits derivation of the following 5p115p 39 7 PquotWKquotfK p R 5 If R 10 c 25 and c 12 5 2512 1 7512 0327 Kt 10 75 12 01 2 The quotBubblesquot Model Equation 1 implies P z Em EP1 1 R 1 R Suppose we modify assumptions II and III to permit diversity of opinion The fundamentals model says that the eXpected value of neXt period s stock price Et PH 1 must be quotjusti edquot by forecasts of dividends The bubbles model weakens that requirement and says that if anything happens that makes people think the stock will be more valuable in the future then the current price will rise as a result The eXpectations of the traders are selfful lling 3 The Random Walk Model Reinstate II and III Suppose we measure time in days rather than months or years Then R the per period rate of interest is a small number Because it is unlikely that a dividend will be paid in the neXt period EtDH 1 will be zero for most stocks and time periods Then the above equation reduces to This property is called the Martingale property One example of a martingale is a random walk such as the following equation for stock prices where Vt is a quotwhite noisequot random variable 4 The Capital Asset Pricing Model CAPM Drop assumption I and assume that people require compensation for bearing risk Rearrange the bubbles model equation to derive R the eXpected oneperiod return from holding a share of Z zt7 How large should RZYt be In a risk neutral world the required return on stocks would be approximately the required return on longmaturity treasury bonds CAPM suggests that accounting for risk aversion provides and an alternative answer Rm Rt I32 RMt Rt where Rt is the quotrisk freequot rate Treasury Bond yield RMt is the average return on an optimally diversi ed portfolio and 392 39 RMt 39R is the risk premium built by the market into RZYt The parameter 392 measures the correlation between returns on the Z share and the overall market return Econ 423 Michael Salemi Stock Pricing Exercise In this exercise students will work with the fundamental value hypothesis of stock market prices to determine the appropriate price for the stock of a ctional company They will also determine what sort of changes in the economic environment would change the price of the ctional stock and by how much that price would change The XYZ company is a biotechnical products company that is about to make an initial public offering of shares The current earnings of the company are zero The consensus View of investment analysts is that the prospects for future earnings of XYZ are well described by the following table of scenarios Scenario A B C D E F Probability of 16 16 16 16 16 16 Scenario Permanent Level 000 200 200 500 500 1500 of XYZ Earnings per share In scenario A the rm never has earnings while in scenario C earnings are eXpected to be permanently 200 per share Suppose the yield to maturity of 20 year treasury bonds is 500 Questions 1 What price will the fundamental theory of stock prices predict for the shares of XYZ How did you arrive at your answer 2 What would be the effect on the IPO price of XYZ shares if on the day before the IPO XYZ surprisingly announced that it had sold a patent for a new product and thereby earned 500 per share Your answer should depend on whether this announcement affects the table of scenarios a Suppose the table of scenarios remains unchanged How should the share price change b Suppose analysts interpret the announcement to mean that the probability of scenario D has risen to 13 while the probability of scenario A has fallen to zero How should the share price change 3 What would be the effect on the current price of XYZ shares if the treasury bond rate falls to 450 EXplain your reasoning 4 Why might the actual price of shares of XYZ be greater than the share price you concluded in part A


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