1 Basic Growth Models

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1 UCLA Aderso MGMT37B: Fudametals i Fiace Fall 015) Week #1 rofessor Eduardo Schwartz November 9, 015 Hadout writte by Sheje Hshieh 1 Basic Growth Models 1.1 Cotiuous Compoudig roof: lim 1 + i m = expi) 1) m m) lim 1 + i m = lim 1 + m m) i ) m ) m m )) = exp lim l1 + m im 1 ) im 1 im m 1 = expi) ) The last two lies are equivalet because l1 + im 1 ) 1 + im 1 ) 1 lim = lim im 1 0 im 1 im We use l Hôpital s rule above. As m, im 1 0. = 1 3) 1. Sum of Geometric Series For r 1, r k = 4) lease me at sheje.hshieh.1@aderso.ucla.edu if there are ay errors. 1

2 roof: ) r k = r k 1 = r k 1 1 r k+1 r k = 1 + r k r = 5) From this, we ca get the geeral expressio b r k = k=a b a 1 r k = b+1 = ra r b+1 r k a 6) If r < 1, the ifiite sum coverges to r k = 1 7) 1.3 erpetuity Usually, for the preset value of a perpetuity, we assume there is o paymet i period k = 0. The preset value therefore has the followig expressio: r k = 1 1 = r 8) If r = 1 1+i, the ) k 1 = i i 9)

3 If the paymets are growig each period as well, the, assumig o paymet i period k = 0 ad the growth rate is strictly less tha the iterest rate i.e., g < i), the preset value is 1 + g)k i) k = 1 + g ) k 1 + g = 1+g 1+i 1 + i 1 + g 1 1+g 1+i = i g 10) 1.4 Auity Assumig o paymet i period k = 0, the preset value of a auity edig at k = T is 1 + i) k 1 + i) = k+t i i) T i = i i) T ) 11) If the paymets are growig each period as well, the the preset value of a auity edig at period k = T is 1 + g)k i) k Review of Statistics ) T 1 + g 1 + g)k 1 = i 1 + i) k i g ) ) T 1 + g 1 + i.1 Defiitios ad roperties of Mea, Covariace, ad Correlatio CovX, Y ) = E[X E[X])Y E[Y ])] = E[XY ] E[X]E[Y ] CovX, a) = 0 CovaX, by ) = abcovx, Y ) CovX + a, Y + b) = CovX, Y ) CovaX + by, cw + dv ) = accovx, W ) + adcovx, V ) + bccovy, W ) + bdcovy, V ) ) V ar a i X i = a i V arx i ) + a i a j CovX i, X j ) ρ X,Y = CovX, Y ) V arx)v ary ) 1 i<j 1) 3

4 . Arithmetic Mea-Geometric Mea Iequality l 1 ) 1 + r i ) 1 ) l1 + r i ) = l 1 + r i ) r i ) 1 + r i ) ) This iequality holds due to Jese s Iequality, sice l ) is a strictly icreasig cocave fuctio. It should be clear that the arithmetic mea is a ubiased estimate of the true oe-period mea returs. I cotrast, the geometric average has a dowward bias. However, the arithmetic average igores compoudig. To gauge past returs, the geometric average is probably a better measuremet. To measure future expected returs, the arithmetic average is probably better sice it is a ubiased estimate. 3 ractice Questios 3.1 Sample Exam Questio 01) The U.S. Treasury aouced it will offer a ew bod to the market place ad you have bee asked to estimate its worth. The bod will make a total of 50 paymets of $50, but these paymets occur every three years i.e., this bod makes its last paymet i 150 years). There is o face value or pricipal. These paymets, however, do ot go to the ower of the bod. Rather these paymets go ito a secret govermet cotrolled accout that ears 7% per year. At maturity, the ower of the bod simply turs i the bod to get the balace of the secret accout. Questio: What price would you recommed give that the market will require a 10% aual rate of retur o this bod? Solutio: Note that the paymets arrive every 3 years, while the secret accout is compouded aually. First, we should calculate the preset value of the auity. Suppose each period k is 3 years. The effective 3-year rate is r = ) The preset value is therefore ) 50 ) ) The future value is ) Next, we discout the future value at the required rate: ) ) 150 4

5 3. Chapter 5, Questio 18 Bodie, Kae, ad Marcus 014) Cosider the followig log-term ivestmet data. The price of a 10-year $100 par zero coupo iflatio-idexed bod is $ A real-estate property is expected to yield % per quarter omial) with a stadard deviatio of the effective) quarterly rate of 10% Compute the aual rate o the real bod r) 10 = r = ) 1 16) The cotiuously compouded aual real iterest rate is l ) Compute the cotiuously compouded aual risk premium o the real-estate ivestmet. The expected) effective aual rate for the real estate ivestmet is ) The cotiuously compouded aual retur is r cc = l ) The aual risk free rate is ot give i the textbook, but assume that the aual risk-free rate is The the cotiuously compouded risk-free rate is l ) The aual cotiuously compouded risk premium is = Fid the stadard deviatio of the cotiuously compouded aual excess retur o the real-estate ivestmet. Sice the risk-free rate is o-stochastic, the variace of the aual excess retur is just the variace of the risky real-estate ivestmet. As doe i the textbook, we assume that future returs o the real estate ivestmet has a log-ormal distributio: 1 + r) T = expr cc T ) = expµt + T ɛ) l1 + r)t = r cc T = µt + T ɛ NµT, σ T ) 17) r is the effective retur over a period e.g., 1 year), r cc is the cotiuously compouded retur over the period covered by r, µ is the expected retur over the same period, σ is the retur variace over the same period, ad ɛ is a stadard ormal radom variable i.e., ɛ N0, 1)). From the equatios above, we ca coclude that r cc Nµ, σ T 1 ). Usig the properties of the log-ormal distributio, we have the followig mea ad variace for 5

6 1 + r) T : E[1 + r) T ] = exp T V ar1 + r) T ) = exp T )) µ + σ µ + σ )) expσ T ) 1) 18) For this problem we wat to fid σ i aual terms T = 1). Substitutig all the parameters we prepared earlier, we solve for the followig equatio: )) V arr) = = exp σ expσ ) 1) 19) We assume that that quarterly variace is additive you ca do this if the uderlyig radom variable is cotiuously compouded, but this is ot ecessarily true with just the effective returs). We solve for σ umerically via R: 1 fidme <- fuctios,g,c){ c^ - exp*g + s^) * exps^) - 1) } uirootfidme,g=0.079,c=.1*sqrt4), c0,1))$root #

1 The Power of Compounding

1 The Power of Compounding 1 The Power of Compoudig 1.1 Simple vs Compoud Iterest You deposit $1,000 i a bak that pays 5% iterest each year. At the ed of the year you will have eared $50. The bak seds you a check for $50 dollars.