Asset returns and R applications

Transcription

1 I. Time value of money: Asset returns and R applications Consider an investor looking for potential investments in equity and fixed income markets. Why would one want to put money in these investments instead of spending it now? Apparently, the answer is the return expected to receive in the future. If the expectation were that the return would be equal or less than the initial money invested, a rational investor would not make any investment yet rather spend it right away. A better example often cited is the case of savings account. Why would people deposit money with the bank if the amount they get in the future would be the same as that of the beginning? In that case, it would probably be optimal to use it sooner than later since there is no point putting money in banks. In fact, banks often pay interests to depositors to compensate for the waiting time; hence, the term time value of money arises. In other words, time value of money reflects the concept that the money today is worth more than that in the future. Let PV be the amount invested today; i, the annual interest rate; n, number of periods into the future; FV, future value after n periods. If interest is compounded once per year, future value after n years is: FV = PV*(1i) n (1) The above formula reflects the time value of money: the initial amount, called present value (PV), grows according to interest rate and number of years. For example, consider investing $1000 in money market account that pays 6% annual interest rate. Future values after first, second, and third year are: FV(1) = $1000*(10.06) = $1060 FV(2) = $1000*(1.06) 2 = $1060*(1.06) = $ FV(3) = $1000*(1.06) 3 = $1123.6*(1.06) = $ From the future value formula, one can derive present value: " PV = () (2) Suppose a 3-year note with face value $100 pays 5% annual coupon and annual interest rate is 6%, the price (present value) of this note: Price = PV = ""."." ""."." ""."." = $97.33 The note is pricing at discount (coupon rate lower than interest rate) and therefore its price is less than par value. Similarly, interest rate and number of period can be deduced: r = " " 1 (3)

2 n = " (" " ) " (4) The first four formulas apply when interest is paid once every single period. If it is compounded m times within each period, future value after n years: FV = PV* 1 (5) The above formula is not much different from (1) since it is just a transformation from single to multiple compounding period considering as i and m*n as n in the first four formulas. Using the money market investing example above, but now interest rate is paid semi-annually, future values after first, second, and third year: FV(1) = $1000* 1." FV(2) = $1000* 1." FV(3) = $1000* 1." = $ = $ = $ The more frequent interest is compounded, meaning that the interval between two interest paying points becomes smaller, the greater future value to the limit that this marginal increase does not make much difference, corresponding to continuously compounded interest (force of interest). Assuming m tends to infinity, future value: FV = lim " 1 Accordingly, remaining variables are defined: PV = FV*e -i*n i = n = " " " " " " Note: to calculate time period to double future value: = " " FV = 2PV è n = " " For i small and close to 0, ln(1i) i. Consider investing $1000 on Linkedin stock currently priced at $100 and annual return is 1%, so it takes " 69 years to double initial investment.."

3 II. Effective and nominal rate: Suppose one puts $1000 in saving account paying 4% interest per annum, convertible half-yearly. In other words, this equivalently means it pays 4% simple annual rate, or 2% period (semi-annual) rate. After 1 year, the investment accrues: FV(1) = $1000* 1." = $ However, using simple annual rate of 4% only gives FV = $1040 due to different compounding frequency. To get the same FV as in the first case: 1000*(1 r A ) = 1000* 1 è r A = 4.04% = 1000* 1." = $ Therefore, the effective annual rate is 4.04%, slightly greater than simple annual rate. Here, simple interest rate is called nominal interest rate; effective annual rate is the simple interest rate with annual compounding that pay the same future value as simple annual rate with m times compounding. The general formulas for effective annual rate r A and simple annual rate with m compounding are: r A = 1 1 (6) i = m[(1r A ) 1/m 1] (7) In case of continuously compounded interest rate, the relationship between effective annual rate and the aforementioned rate is: 1 r A = e i è r A = 1 e i è i= ln(1r A ) For example, an investment pays period interest rate 3% quarterly, corresponding to quoted annual percentage rate of 12%. The effective annual rate is then (1.03) 4 1 = 12.55%. if considering continuously compounded interest, the simple annual interest I = ln(1.1255) = 11.82%. III. Present values and annuity: a. Present value of series of payments made in advanced: A payment that is paid in advanced means it occurs at the beginning of each period (e.g start of month, year, etc.). To illustrate, suppose a company makes direct deposit to its employees saving account at the beginning of every month. This account pays i % monthly interest. V 0 V 1 V 2 V n n Present value of this series is:

4 L- Stern Group " = Ly Pham (1 ) (1 ) (1 ) (1 ) b. Present value of series of payments made in arrears: In contrary to the above case, a payment made in arrears occurs at the end of each period. Follow the same analysis. Present value of this series is: 0 V1 V2 Vn 1 2 n " = (1 ) (1 ) (1 ) (1 ) c. Annuity: Annuity is a series of same payments made at the end of every period. The present value of this series can be written in short form using geometric series: (1 ) (1 ) """#$%& = = 1 (1 ) (1 ) (1 ) 1 1 (1 ) = () d. Annuity due: The difference between annuity and annuity due is, in the later, payments are made at the beginning of each period. Hence, the series is discounted one period fewer than the above formula: 1 1 (1 ) """#$%& = = 1 (1 ) (1 ) (1 ) 1 e. Growing perpetuity: Perpetuity is a series of same payments that is made infinitely. Growing perpetuity is similar to perpetuity except that the payment grows at rate g infinitely: V V(1g) V(1g)2 Vn 0 ""#$%& "# = n 1 = = IV. Asset and portfolio returns: a. Asset returns: Return on asset over the holding period, which is time interval between buying and selling dates, is the rate of change of prices over the same period.

5 L- Stern Group R(t) = Ly Pham ()() (8) () P(t) and P(t-1) are asset prices at time t and t-1, respectively. R(t) is referred to as simple net return. On the other hand, rewriting the above formula: R(t) = ()() () è 1 R(t) = = () () 1 () (9) () Formula (9) defines gross return of assets. Expanding gross return formula for multiple periods, it can be written as: 1R(k) = = = 1 = ( ) Based on this expression, k-period gross return is the geometric average of k consecutive single period returns. For instance, suppose one investor bought Apple stock in January for $600. At the end of January, it increased by $3 to $603, and ended up $608 at the end of February when the investor decided to sell it. Simple net returns: R1 = R2 = ()() () ()() R(2 )= = = () ()() () $"#$$"## $"## $"#$$"#$ = = = 0.5% = = 0.83% $"#$ $"#$$"## $"## = = 1.33% Two-month gross returns: 1R(2) = 1.005* = è R(2) = 1.33% This confirms the multiplicative property of 2-month gross return being geometric average 2 simple one-month gross returns To put it in detail, rewriting 2-month gross return: 1R(2) = () = () = 1 1 = 1 If is small and close to 0, 2-month simple net return is the sum of 2 simple 1-month net returns, that is, 2. However, the use of this statement should be careful since it might lead to very misleading result. b. Portfolio returns:

6 L- Stern Group Ly Pham Consider a portfolio of Google and Facebook stocks in which the investor invested xg and xf share of wealth in Google and Facebook, respectively. That is to say, xg and xf are fractions of total investment and xg xf = 1. The initial investment is assumed to be $W, so the dollar amounts invested in Google and Facebook are $W*xG and $W*xF, respectively. Let RG,1 be simple net return of Google from time t0 to t1; similarly, Rf,1 for Facebook. The value of portfolio at t1: V1 = W * [xg(1 RG,1) xf(1 RF,1)] Let Rp be portfolio simple net return; portfolio gross return is therefore: (1 Rp)*W = W*[xG(1 RG,1) xf(1 RF,1)] è 1 Rp = xg(1 RG,1) xf(1 RF,1) = xg xf xg*rg,1 xf*rf,1 = 1 xg*rg,1 xf*rf,1 è Rp = xg*rg,1 xf*rf,1 Thus, portfolio return is the weighted average of individual asset returns. In case of multi-asset portfolio, the general relationship of portfolio return and those of its constituents is: rp = (10) Following the portfolio example above, suppose 60% of initial investment ($100000) is allocated in Google and the remaining in Facebook. At the beginning of September, Google stock is priced at $610 and that of Facebook $25; at the end of the month, they soar to $615 and $27 for Google and Facebook, respectively. Returns on two stocks and portfolio return are as follows: RG = RF = $"#$$"#$ $"#$ $"#$"# $"# = = 0.08 Rp = = % The above portfolio return is called monthly simple return. To annualize the return, some calculations need proceeded: Annual simple return: RAnnual,p = 12*3.7% = 44.4% Annual monthly-compounded return: RAnnual, mth-cpd = (10.037)12 1 = 54.64% It is noted that if short sale is allowed, fractions of wealth are negatives values. In addition, the aforementioned formulas do not take into account dividends paid, in order to do so, dividend should be added to return calculations: R= () () = "#$%"& "#$ ("#"$% "#") (11) Asset return is therefore broken down into capital gain, which is due to price appreciation, and dividend yield. Suppose Google stock pays $1 dividend, its return is: RG = $"#$$"$"#$ $"#$ = % Recall the future value formula with continuously compounded interest FV = " and interest " rate is then i = " ". Now consider PV as value of asset (stock, ETF, portfolio, ) at the beginning of holding period and FV as that at the end of period; the number of period is now n = 1, and interest rate is similar to asset return. The continuously compounded return is thus: rt = " () () = ln(1rt) = ln(()) ln (()) (12) Therefore, continuously compounded return is calculated by taking natural log of gross return or difference in natural log of price. It can be shown by graphing and applying econometrics that stock

7 L- Stern Group Ly Pham price time series is not stationary while that of stock return is more stationary and thus easier to analyze. That is why return is more favorable, not to mention return reflects the growth of asset better than price and it is unit-free. Accordingly, price at time t is: Pt = Pt-1er (13) Hence, continuously compounded return on portfolio is: = ln 1 = ln (1 ) (14) Continuously compounded portfolio return is not weighted average of its constituents simple returns, which is the case of simple portfolio return. In addition, it can be shown that multi-period continuously compounded return is equal to the sum of one-period continuously compounded return: rt(k) = ln. = rt rt-1. rt-k1 = V. Adjusting for inflation: In order to take into account the effect of inflation, it only needs to divide gross inflation rate for n periods by price or value being considered and the remaining formulas as mentioned earlier follows the same argument. Let f be annual inflation rate. Present value at current price level is then: " " = (15) It is common that people use CPI (consumer price index) to measure inflation; hence, to calculate real price/value, nominal amount is divided by current CPI, which is period return and gross return are: "#$ = 1 "#$ =,"#$,"#$,"#$ =,"#$ "# "# "#. Thus, real simple one- 1 = (16) "# "# (17) Similarly, real continuously compounded one-period return: "#$ = ln 1 "#$ = ln "# "# = " ln "# "# = "#$%& (18) πc = ln "# ln "# = ln (1 ) and πc is continuously compounded one-period inflation rate while π is simple inflation rate. VI. Project and R applications:

8 In this section, we will go through step by step to perform exploratory data analysis and return calculations in R. Amazon and Wal-Mart are two stocks in concern. The procedure is outlined as follows: Get historical stock prices: Historical data can be downloaded directly from Yahoo Finance by navigating to its web page and doing it manually; however, it is more efficient and faster to get the data with one line of code in R (as shown below). In this project, we consider monthly adjusted closing prices of Amazon and Wal-Mart stocks. At first, some packages need to be loaded into R, including: zoo, tseries, quantmod, and PerformanceAnalytics; then historical stock series of Amazon and Wal- Mart are downloaded and assigned to objects in R: library(zoo); library(tseries); library(quantmod), library(performanceanalytics); amazon.price = get.hist.quote(instrument="amzn", start=" ", end=" ", quote="adjclose", provider="yahoo", origin=" ", compression="m", retclass="zoo") walmart.price = get.hist.quote(instrument="wmt", start=" ", end=" ", quote="adjclose", provider="yahoo", origin=" ", compression="m", retclass="zoo") The get.hist.quote() function is used to get historical prices from data providers (usually Yahoo) and it has several arguments as indicated above. In particular, quote specifies whether the downloaded data is Open, High, Close, or Adj. Close ; if left blank, all of them will be downloaded. Compression governs the frequency of data, such as daily (d), weekly (w), or monthly (m); retclass specifies class of retrieved data: zoo (data indexed by Date), its (indexed by POSIXct), and ts (numeric index). In order to obtain stock charts, we can use chartseries() function: getsymbols("amzn") chartseries(amzn,theme=charttheme('white')) getsymbols("wmt") chartseries(wmt,theme=charttheme('white'))

9 As shown above, the charts consist two parts with the upper panel being historical prices while the lower indicating volume traded in millions of shares. This is similar to those often seen on financial websites. Currently, Amazon is traded at around $250 while Wal-Mart in $70 range. Plot historical stock prices: Suppose we are observing monthly data for Amazon and Wal-Mart for the period The downloaded data is saved in files contained in working directory. To check current working directory, use getwd() function and use setwd() to set new working directory. To read file into R workspace, read.csv() is used; this depends on the file extension being imported to R (read.csv, read.table, etc.). loadpath='/users/lmp406/my Office/Lp papers/' amzn.df=read.csv(file=paste(loadpath,'amzn.csv',sep=''),header=t,stringsasfactors=f) walmart.df=read.csv(file=paste(loadpath,'walmart.csv',sep=''), header=t,stringsasfactors=f) plot(amzn.df$adj.close,type='l',lwd=2,ylab='price',main='amazon & Walmart Stock Price') lines(walmart.df$adj.close,lty='dashed',lwd=2,col='blue') legend(x='topleft',legend=c('amzn','wmt'),col=c('black','blue'),lwd=2, lty=c('solid','dashed')) The data is read as data frame object, which is rectangular with observations in rows and variables in columns, in R workspace. Following is a sample of data frame object: > head(amzn.df) Date Adj.Close 1 2/1/ /1/ /2/ /1/ /1/ /2/ > head(walmart.df) Date Adj.Close 1 2/1/ /1/ /2/ /1/ /1/ /2/

10 The above code generates the monthly prices graph of two stocks. Note that to plot additional series in the existing one, function lines() is used instead on plot()" since plot() will initiate new chart panel. Amazon stock displays a seemingly exponential growth in price and so does that of Wal-Mart yet it is not clear in case of the later since Wal-Mart is trading at much lower price level compared to Amazon s. Simple 5-year monthly return of Amazon and Wal-Mart can be computed: n=nrow(amzn.df) #number of observations ret5yrs.amzn=amzn.df[n,2]/amzn.df[1,2]-1 #last obs/first obs -1 ret5yrs.walmart=walmart.df[n,2]/walmart.df[1,2]-1 > ret5yrs.amzn [1] > ret5yrs.walmart [1] Within 5 years from 2007 to 2012, Amazon stock has surged from $40 to $250 price range, indicating a simple return for the entire holding period of 520% while that of Wal-Mart is 63%. In the above code, the data frame still contains two variables: Date and Adj Close. In order for the data frame to have only one factor Adj Close, we need to change Date as factor to character by specifying rownames of each series as its Date as illustrated: rownames(amzn.df)=amzn.df$date rownames(walmart.df)=walmart.df$date amzn.df=amzn.df[,'adj.close',drop=f] walmart.df=walmart.df[,'adj.close',drop=f] According, the slicing rule as shown in return calculations above has changed and is simpler: n=nrow(amzn.df) #number of observations ret5yrs.amzn=amzn.df[n]/amzn.df[1]-1 #last obs/first obs -1 ret5yrs.walmart=walmart.df[n]/walmart.df[1]-1

11 > ret5yrs.amzn [1] > ret5yrs.walmart [1] Compute simple returns: As derived and formulized in theoretical section aforementioned, simple net return is just the rate of change in prices between two ends of period and can be calculated in several ways in R. However, firstly, we need to create zoo objects from data frame objects so that it can take into account the date associated with each stock price and the follow-up graphs will be indexed by dates instead of numbers. #as.yearmon reformats dates in downloaded data to desired monthly data #as.yearmon(" ") => [1] "Mar 2007" dates.amzn=as.yearmon(amzn.df$date,'%m/%d/%y') dates.walmart=as.yearmon(walmart.df$date,'%m/%d/%y') amzn.z=zoo(x=amzn.=df$adj.close, order.by=dates.amzn) walmart.z=zoo(x=walmart.df$adj.close,order.by=dates.walmart) To compute simple monthly return, we can use slicing and indexing techniques in R or other built-in functions: #simple return calculations using slicing and indexing amzn.ret=amzn.df$adj.close[2:n]/amzn.df$adj.close[1:n-1]-1 walmart.ret=walmart.df$adj.close[2:n]/walmart.df$adj.close[1:n-1]-1 #simple return calculations using diff and lag built-in functions amzn.ret.z=diff(amzn.z)/lag(amzn.z,k=-1) walmart.ret.z=diff(walmart.z)/lag(walmart.z,k=-1) #simple return calculations using Return.calculate() or CalculateReturns() #functions of PerformanceAnalytics package amzn01.ret.z=return.calculate(amzn.z,method='simple') walmart01.ret.z=return.calculate(walmart.z,method='simple') Compute continuously compounded returns: Similar to simple net return, continuously compounded returns are calculated using one of the following methods: #cc return calculations using log gross return amzn.ccret01=log(1amzn.ret) walmart.ccret01=log(1walmart.ret) #cc return calculations using log prices amzn.ccret02=log(amzn.df$adj.close[2:n]/amzn.df$adj.close[1:n-1]) walmart.ccret02=log(walmart.df$adj.close[2:n]/walmart.df$adj.close[1:n-1])

12 #cc return calculations using diff log prices amzn.ccret.z=diff(log(amzn.z)) walmart.ccret.z=diff(log(walmart.z)) #cc return calculations using Return.calculate amzn01.ccret.z=return.calculate(amzn.z,method='compound') walmart01.ccret.z=return.calculate(walmart.z,method='compound') Plotting notes: #plot simple return (Figure 1) par(mfcol=c(1,2)) plot(amzn.ret,type='l',main='amazon smpl mthly returns',ylab='return',lwd=2) plot(walmart.ret,type='l',main='walmart smpl mthly returns',ylab='return', col='blue',lwd=2) #plot continuously compounded return (Figure 2) par(mfcol=c(1,2)) plot(amzn.ccret01,type='l',main='amazon CC monthly returns', ylab='cc Return',lwd=2) plot(walmart.ccret01,type='l',main='walmart CC monthly returns', ylab='cc Return', col='blue',lwd=2) #plot stock price with date index (Figure 3) plot(amzn.z,type='l',lwd=2,xlab='months',ylab='prices') lines(walmart.z,lty='dashed',lwd=2,col='blue') legend(x='topleft',legend=c('amzn','wmt'),col=c('black','blue'),lwd=2, lty=c('solid','dashed')) #merge 2 series and make multiple plots with date index (Figure 4) amznwalmart.z=merge(amzn.z, walmart.z) plot(amznwalmart.z, plot.type='multiple',lwd=c(2,2),col=c('black','blue'), lty=c('solid','dashed'),ylab=c('amzn','wmt'),main='')

13 Figure 1 Figure 2 Figure 3 Figure 4 As mentioned earlier, changing data frame object to zoo object allows us to represent and plot data indexed by date. In addition, it is also easier too merge and subset data. For example: #merge data > head(merge(amzn.ret.z,amzn.ccret.z)) amzn.ret.z amzn.ccret.z Mar Apr May Jun Jul

14 #subset data > window(amzn.z, start=as.yearmon('jan 2012'),end=as.yearmon('Dec 2012')) Jan 2012 Feb 2012 Mar 2012 Apr 2012 May 2012 Jun 2012 Jul 2012 Aug 2012 Sep 2012 Oct Nov 2012 Dec There are many R packages with useful functions that help obtain financial data easier and faster, as well as facilitate the financial analysis. For instance, getfinancials() allows us to download financial statements; periodreturn(), including: dailyreturn(), monthlyreturn(), weeklyreturn(), quarterlyreturn(), annualreturn(), allreturn(), helps calculate arithmetic returns. > getsymbols('goog') [1] "GOOG" > monthlyreturn(goog,subset='2012') monthly.returns > periodreturn(goog,period='yearly',subset='2009::') yearly.returns > getdividends('msft',from=' ',to=sys.date()) [,1]

15 References Adams, Andrew T., et al. Investment Management. West Sussex, Print. Zivot, Eric. Return Calculations. University of Washington. n.p. n.d L-Stern Group. All Rights Reserved. The information contained herein is not represented or warranted to be accurate, correct, or complete. This report is for information purposes only, and should not be considered a solicitation to buy or sell any security. Redistribution is prohibited without written permission.