In troduction to Com putational Finance and Financial Econometrics Chapter 1 Asset Return C alculations Eric Ziv ot Dep artment of E c ono m ics, Univ er sity of Washington December 3 1, 1998 Updated: January 7, 2002 1TheTimeValueofMoney Consider an amount $V invested for n years at a simple interest rate of R per ann um ( where R i s expressed as a decimal). If compounding tak es place only at the end of the year the future value after n years is FV n =$V · (1 + R) n . Example 1 Consider putting $1000 in a n interest checking account that pays a simple annual pe rcentage rate o f 3%. The future value after n =1, 5 and 10 years is, respectively, FV 1 = $1000 · (1.03) 1 = $1030 FV 5 = $1000 · (1.03) 5 = $1159.27 FV 10 = $1000 · (1.03) 10 = $1343.92. If in teres t is paid m time per yea r t h en t he futur e value afte r n years is FV m n =$V · µ 1+ R m ¶ m·n . 1 R m isoftenreferredtoastheperiodic interest rate.Asm, the frequency of compounding, increases the rate becomes continuously compounded and it can be shown that fu tur e value becomes FV c n =lim m→∞ $V · µ 1+ R m ¶ m·n =$V · e R·n , where e (·) is the exponential function and e 1 =2.71828. Example 2 If the simple annual percentage rate is 10% then the value of $1000 at the end of one year (n =1)for different values of m isgiveninthe table below. Compounding Frequency Value of $1000 at end of 1 year (R =10%) Ann ually (m =1) 1100 Quarterly (m =4) 1103.8 Weekly (m =52) 1105.1 Daily (m = 365) 1105 .515 Con tin uously (m = ∞) 1105.517 We now consider the r ela tionship bet ween simple inte r est rat es, periodic rates, effective annual rates and contin uously compounded rates. Suppose an in ve stment pay s a periodic interest rate of 2% eac h qu arter. This g ives rise to a simple annual r ate o f 8 % (2% ×4 quarters). At the e nd of the year, $1000 invested accrues to $1000 · µ 1+ 0.08 4 ¶ 4·1 = $1082.40. The effective annual ra te, R A , on the in vestm ent is determined by the rela- tionship $1000 · (1 + R A ) = $1082.40, wh ich gives R A =8.24%. The effective annual rate is greater than the simple ann u al rate due to the pa yment of in terest on interest. The general relationship bet ween the simple ann u al rate R with pa yments m time per yea r and th e effective annual rate, R A , is (1 + R A )= µ 1+ R m ¶ m·1 . 2 Example 3 To determine the simple annual ra te with quarterly paym en ts that produ ces an e ffective annual rate of 12%,wesolve 1.12 = µ 1+ R 4 ¶ 4 =⇒ R = ³ (1.12) 1/4 − 1 ´ · 4 =0.0287 · 4 =0.1148 Suppose we wish to calculate a value for a contin uously compounded rate, R c , when we know the m−period sim ple rate R. The relationship between suchratesisgivenby e R c = µ 1+ R m ¶ m . (1) Solving ( 1) for R c gives R c = m ln µ 1+ R m ¶ , (2) and solving (1) for R gives R = m ³ e R c /m − 1 ´ . (3) Example 4 Suppose an investment pays a periodic interest r ate o f 5% every six months ( m =2,R/2=0.05). I n the market this would be quoted as having an annu al pe rcentage rate of 10%. An investment of $100 yields $100 · (1.05) 2 = $110.25 after one year. The effec tive annual ra te is then 10.25%. Suppos e we wish to convert the sim ple annual rate of R =10%to an e qu ivalent continuou sly compounded ra te. Using (2) with m =2gives R c =2· ln(1.05) = 0.09758. That is, if interest is compounded continuou sly at an a nnual rate of 9.758% then $100 invested today would gro w to $100 · e 0.09758 = $110.25. 2 A s se t Retu r n Calcula t io n s 3 2.1 Simple Returns Let P t denote the price in mon th t of an asset that pay s no dividends and let P t−1 deno te the price in month t − 1 1 . Then the one mon th simple net return on an investment in the asset between months t − 1 and t is defined as R t = P t − P t−1 P t−1 =%∆P t . (4) Writing P t −P t−1 P t−1 = P t P t−1 − 1,wecandefine the sim ple gross return as 1+R t = P t P t−1 . (5) Notice that the one mon th gross return has the interp retation of the future value of $1 invested in the asset for one month. Unless otherwise stated, when we refer to retu rns we mean net ret urns. (mention that simple return s cannot be less t han 1 (100%) since pr ices cannot be negativ e) Example 5 Consider a one mo nth investm ent in Microsoft stock. Suppose you buy the stoc k in month t − 1 at P t−1 = $85 and sell the sto ck the next month for P t = $90. Further a ss ume that M icrosoft does not pay a divi dend between months t −1 and t. The one month simple net and gross returns are then R t = $90 − $85 $85 = $90 $85 − 1=1.0588 − 1=0.0588, 1+R t =1.0588. The o n e month investment i n Micro so ft yielded a 5.88% per month return. Alternatively, $1 invested in Microsoft stock in m onth t − 1 grew to $1.0588 in month t. 2.2 Multi-period returns The simple two-month return on an in ve stment in an asset betwe en mont hs t − 2 and t is defined as R t (2) = P t − P t−2 P t−2 = P t P t−2 − 1. 1 We make the c onv ention that the default inv e stment horizon is one month and that the price is the closing price at the end of the month. This is completely arbitrary and is used only to simplify calculations. 4 Since P t P t−2 = P t P t−1 · P t−1 P t−2 thetwo-monthreturncanberewrittenas R t (2) = P t P t−1 · P t−1 P t−2 − 1 =(1+R t )(1 + R t−1 ) −1. Then the simple t wo-month gross return becomes 1+R t (2) = (1 + R t )(1 + R t−1 )=1+R t−1 + R t + R t−1 R t , which is a geometric (m ultiplicative) sum of the two simple one-month gross returns and not the simple sum of the one month returns. If, ho wever, R t−1 and R t are small then R t−1 R t ≈ 0 and 1+R t (2) ≈ 1+R t−1 + R t so that R t (2) ≈ R t−1 + R t . In general, the k-mon th gross return is defined as the geometric average of k one month gross ret urns 1+R t (k)=(1+R t )(1 + R t−1 ) ···(1 + R t−k+1 ) = k−1 Y j=0 (1 + R t−j ). Example 6 Continuing with the previ ous example, suppose that t he price of Microsoft stock in mo nth t−2 is $80 andnodividendispaidbetweenmonths t − 2 and t. The two month net return is R t (2) = $90 −$80 $80 = $90 $80 − 1=1.1250 − 1=0.1250, or 1 2.50% pe r two months. The t wo one month return s ar e R t−1 = $85 − $80 $80 =1.0625 − 1=0.0625 R t = $90 − 85 $85 =1.0588 − 1=0.0588, and the geometric average of the two one m on th gross returns is 1+R t (2) = 1.0625 × 1.0588 = 1.1250. 5 2.3 Annualizing returns Very often returns ov er different horizons are annualized, i.e. con verted to an annual retu rn, to facilitate com paris on s with other investments. The an- n u alization p rocess depends on the holding period of the in vestment and an implicit assumption about compounding. We illustrate with several exam- ples. To start, if our i nvestment horizon is o ne year th en th e a nnu al gr oss an d net returns are just 1+R A =1+R t (12) = P t P t−12 =(1+R t )(1 + R t−1 ) ···(1 + R t−11 ), R A = P t P t−12 − 1=(1+R t )(1 + R t−1 ) ···(1 + R t−11 ) −1. In this case, no c ompounding is required to create an annu al return. Next, con sider a one month inve stme nt in an a sset with r etu rn R t . What is the annualized return on th is investment? If we assum e that we receiv e thesamereturnR = R t every month for th e year then t he gross 12 month or gross annual return is 1+R A =1+R t (12) = (1 + R) 12 . Notice th at the annu al gross retu rn is defined as the monthly return c om- pounded for 12 mon ths. The net ann ual return is then R A =(1+R) 12 − 1. Example 7 In the first example, the one month re turn, R t , on Microsoft stoc k was 5.88%. If we assume that we can get this return for 12 months then the a nnualized return i s R A =(1.0588) 12 − 1=1.9850 − 1=0.9850 or 98.50% per year. Pretty good! Now, consider a t wo month investm ent with r et urn R t (2). If we assume that we receiv e the same two mo nth r eturn R(2) = R t (2) for the next 6 t wo month periods then the gross and net annual returns are 1+R A =(1+R(2)) 6 , R A =(1+R(2)) 6 − 1. 6 Here the ann ual gross return is defined as the two mon th return compounded for 6 months. Example 8 In the se cond example, the two month r e turn, R t (2), on Mi- crosoft stock was 12.5%. If we assume that we can get this two m onth return for the next 6 two month periods then the annualized re turn is R A =(1.1250) 6 − 1=2.0273 − 1=1.0273 or 1 02.73 % per year. To complicate matters, now suppose that our investmen t horizon is t wo y ears. That is we start our investment at time t − 24 and cash out at time t. The two ye ar gross return is then 1+R t (24) = P t P t−24 . What is the annual return on this two y ear inves tm ent? To determine the annu al return w e solve the follow in g relatio ns h ip for R A : (1 + R A ) 2 =1+R t (24) =⇒ R A =(1+R t (24)) 1/2 − 1. In this case, the annual return is compounded tw ice to get the two year return and t he relationsh ip is then so lved for the annu al re turn. Example 9 Supp ose that the price of Microsoft sto ck 24 months ago is P t−24 = $50 and the price tod ay is P t =$90. The two year gross return is 1+R t (24) = $90 $50 =1.8000 which yields a two yea r net return of R t (24) = 80%. The annual return f or this investm ent is defined as R A =(1.800) 1/2 − 1=1.3416 − 1=0.3416 or 3 4.16% pe r year. 2.4 Adjusting for dividends If an asset pays a dividend, D t , sometime between month s t − 1 and t,the return calculation becomes R t = P t + D t − P t−1 P t−1 = P t − P t−1 P t−1 + D t P t−1 where P t −P t−1 P t−1 is referred as the capital gain and D t P t−1 is referred to as the dividend yield. 7 3 Continu ously Compounded R eturns 3.1 One Pe riod Returns Let R t denote the simple monthly return on an in ve stment . The co ntinuously compounde d monthly r eturn, r t , is defined as r t =ln(1+R t )=ln à P t P t−1 ! (6) where ln(·) is the natural log function 2 .Toseewhyr t is called the con- tinuously compounded return, take the exponen tial of both sides of (6) to give e r t =1+R t = P t P t−1 . Rearranging w e get P t = P t−1 e r t , so that r t is the con tinuou sly compounded grow th rate in prices between months t − 1 and t. ThisistobecontrastedwithR t which i s t h e simple growth rate in prices bet ween months t −1 an d t w ithou t any compounding. Furtherm ore, since ln ³ x y ´ =ln(x) − ln(y) it follows that r t =ln à P t P t−1 ! =ln(P t ) −ln(P t−1 ) = p t − p t−1 where p t =ln(P t ). Hence, the contin uously compounded mon thly return, r t , can be computed simply b y taking the first difference of the natural loga- rithms of mon thly prices. Example 10 Using the price and re tu rn data from e xam ple 1, the contin u- ously compou nd ed monthly return on Mi crosoft stock can be com p uted in two ways: r t =ln(1.0588) = 0.0571 2 The contin uously compounded return is always defined since asset prices, P t , are alw ays non-negativ e. Properties of logarithms and exponentials are discussed in the ap- pendix to this chapter. 8 or r t =ln(90)− ln(85) = 4.4998 − 4.4427 = 0.0571. Notice that r t is slightly smaller than R t . Why? Giv en a monthly continuously compounded return r t , is straightfo rward to solv e back for the cor responding simple net return R t : R t = e r t − 1 Hence, nothing is lost by considering contin uously com pounded returns in- steadofsimplereturns. Example 11 In the previous example, the co ntinuous ly compo un ded monthly re tu rn on Microsoft stock is r t =5.71%. The implied simple net return is then R t = e .0571 − 1=0.0588. Con tin uously compounded returns are ve ry similar to s imple returns as long as the retur n is relatively small, wh ich it generally will be for mont hly or daily returns. For modeling and statistical purposes, however, it is mu ch mo re conve nie nt to use con tinu o usly compounded returns due to the additivity property of mu ltiperiod contin uously com pounded returns and unless noted otherwise from h ere on we will work w ith con tinuously compounded retu rn s. 3.2 Mu lti-Period Returns The computation of multi-period con tin uously compounded returns is con- siderably easier than the computation of multi-period simple returns. To illustrate, consider the two month continuously compounded return defined as r t (2) = ln(1 + R t (2)) = ln à P t P t−2 ! = p t − p t−2 . Ta k ing e xponent ials of both sides shows that P t = P t−2 e r t (2) 9 so t hat r t (2) is the con tin uously com pounded gro w th rate o f prices between months t − 2 and t. Using P t P t−2 = P t P t−1 · P t−1 P t−2 and t h e fact th at ln( x · y)= ln(x)+ln(y) it follows that r t (2) = ln à P t P t−1 · P t−1 P t−2 ! =ln à P t P t−1 ! +ln à P t−1 P t−2 ! = r t + r t−1 . Hence t he contin uously compounded two m onth return is just the sum of the t w o contin uously compounded one mon th returns. Recall that with simple returns the two month return is of a multiplica tive form (geometric averag e). Example 12 Using the data from example 2, the continuously compounded two month re turn on Microsoft stock can be compu ted in two equiv alent ways. The first w a y u ses the differ ence in the lo gs of P t and P t−2 : r t (2) = ln(90) − ln(80) = 4.4998 − 4.3820 = 0.1178. The s econd way use s the sum of the two continuously compounded one month returns. Here r t =ln(90)− ln(85) = 0.0571 and r t−1 = ln(85) − ln(80) = 0.0607 so that r t (2) = 0.0571 + 0.0607 = 0.1178. Notice that r t (2) = 0.1178 <R t (2) = 0.1250. The continuou sly com pounded k−monthreturnisdefine d by r t (k)=ln(1+R t (k)) = ln à P t P t−k ! = p t − p t−k . Using similar manipu lations to the ones used for the continuously com- pounded tw o month return we may express the con tin uously compounded k−month return a s the sum of k contin uously compounded monthly returns: r t (k)= k−1 X j=0 r t−j . The additivitity of continuously compounded return s to form multiperiod returns is an importan t property for statistical modeling purposes. 10 [...]... distribution and if a distribution has kurtosis less than 3 then the distribution has thinner tails than the normal Sometimes the kurtosis of a random variable is described relative to the kurtosis of a normal random variable This relative value of kurtosis is referred to as excess kurtosis and is de& ned as excess kurt(X) = kurt(X) 3 If excess the excess kurtosis of a random variable is equal to zero... Quantitative Methods in Finance, International Thomson Business Press, London, UK 16 Introduction to Financial Econometrics Chapter 2 Review of Random Variables and Probability Distributions Eric Zivot Department of Economics, University of Washington January 18, 2000 This version: February 21, 2001 1 Random Variables We start with a basic de& nition of a random variable De& nition 1 A Random variable X is... J., A Lo, and C MacKinlay (1997), The Econometrics of Financial Markets, Princeton University Press [2] Handbook of U.W Government and Federal Agency Securities and Related Money Market Instruments, The Pink Book, 34th ed (1990), The First Boston Corporation, Boston, MA [3] Stigum, M (1981), Money Market Calculations: Yields, Break Evens and Arbitrage, Dow Jones Irwin 15 [4] Watsham, T.J and Parramore,... zero then the random variable has the same kurtosis as a normal random variable If excess kurtosis is greater than zero, then kurtosis is larger than that for a normal; if excess kurtosis is less than zero, then kurtosis is less than that for a normal 13 1.6 Linear Functions of a Random Variable Let X be a random variable either discrete or continuous with E[X] = àX , var(X) = 2 and let a and b be known... is special to the normal distribution and may or may not hold for a random variable with a distribution that is not normal 1.6.1 Standardizing a Random Variable Let X be a random variable with E[X] = àX and var(X) = 2 De& a new random ne X variable Z as X àX 1 à Z= = X X X X X which is a linear function aX + b where a = 1 and b = àX This transformation is X X called standardizing the random variable... var(Z) = var(X) = X = 1 X 2 X E[Z] = Hence, standardization creates a new random variable with mean zero and variance 1 In addition, if X is normally distributed then so is Z Example 26 Let X N(2, 4) and suppose we want to & Pr(X > 5) Since X is nd not standard normal we can t use the standard normal tables to evaluate Pr(X > 5) directly We solve the problem by standardizing X as follows: à ả X 2 52 >... situations we want to be able to characterize the probabilistic behavior of two or more random variables simultaneously 2.1 Discrete Random Variables For example, let X denote the monthly return on Microsoft Stock and let Y denote the monthly return on Apple computer For simplicity suppose that the sample spaces for X and Y are SX = {0, 1, 2, 3} and SY = {0, 1} so that the random variables X and Y are discrete... in two non-dividend paying stocks A and B over the next month Let RA denote monthly return on stock A and RB denote the monthly return on stock B These returns are to be treated as random variables since the returns will not be realized until the end of the month We assume that RA N (àA , 2 ) and RB N (àB , 2 ) Hence, ài gives the expected return, E[Ri ], A B on asset i and i gives the typical size... bounded above by some large number It is an open question as to what is the best characterization of the probability distribution of stock prices The log-normal distribution is one possibility1 As another example, consider a one month investment in Microsoft stock That is, we buy 1 share of Microsoft stock today and plan to sell it next month Then the return on this investment is a random variable... standardized value of X Pr Z > 3 can be found directly 2 from the standard normal tables Standardizing a random variable is often done in the construction of test statistics For example, the so-called t-statistic or t-ratio used for testing simple hypotheses on coecients in the linear regression model is constructed by the above standardization process A non-standard random variable X with mean àX and . London, UK. 16 Introduction to Financial Econometrics Chapter 2 Review of Random Variables and Probability Distributions Eric Zivot Department of Economics, University of Washington January 18,. In troduction to Com putational Finance and Financial Econometrics Chapter 1 Asset Return C alculations Eric Ziv ot Dep artment of E c ono m ics, Univ er sity of Washington December 3 1,. Microsoft stock. That is, we buy 1 share of Microsoft stock today and plan to sell it next month. Then the return on this investment is a random variable since we do not know its value today with