Chapter 1 Asset Returns - Princeton University

1 Chapter 1 Asset ReturnsThe primary goal of investing in a financial market is to make profits withouttaking excessive risks. Most common investments involve purchasing financial assetssuch as stocks, bonds or bank deposits, and holding them for certain periods. Posi-tive revenue is generated if the price of a holding Asset at the end of holding periodis higher than that at the time of purchase (for the time being we ignore transactioncharges). Obviously the size of the revenue depends on three factors: (i) the initialcapital ( the number of assets purchased), (ii) the length of holding period, and(iii) the changes of the Asset price over the holding period. A successful investmentpursues the maximum revenue with a given initial capital, which may be measuredexplicitly in terms of the so-calledreturn. A return is a percentage defined as thechange of price expressed as a fraction of the initial price.

2 It turns out that assetreturns exhibit more attractive statistical properties than Asset prices it also makes more statistical sense to analyze return data rather thanprice ReturnsLetPtdenote the price of an Asset at timet. First we introduce various definitionsfor the Returns for the One-period simple Returns and gross returnsHolding an Asset from timet 1tot, the value of the Asset changes fromPt 1toPt. Assuming that no dividends paid are over the period. Then theone-periodsimple returnis defined asRt=(Pt Pt 1)/Pt 1.( )It is the profit rate of holding the Asset from timet 100Rt%, as 100 Rtis the percentage of the gain with respect to the initialcapitalPt 1. This is particularly useful when the time unit is small (such as a dayor an hour); in such casesRttypically takes very small values. The Returns for less2 Chapter 1 Asset Returnsrisky assets such as bonds can be even smaller in a short period and are often quotedinbasis points,whichis10, period gross returnis defined asPt/Pt 1=Rt+ 1.

3 It is the ratio of thenew market value at the end of the holding period over the initial market Multiperiod returnsThe holding period for an investment may be more than one time unit. For anyintegerk 1, the Returns for overkperiods may be defined in a similar example, thek-period simple returnfrom timet ktotisRt(k)=(Pt Pt k)/Pt k,and thek-period gross returnisPt/Pt k=Rt(k) + 1. It is easy to see that themultiperiod Returns may be expressed in terms of one-period Returns as follows:PtPt k=PtPt 1Pt 1Pt 2 Pt k+1Pt k,( )Rt(k)=PtPt k 1=(Rt+1)(Rt 1+1) (Rt k+1+1) 1.( )If all one-period returnsRt, ,Rt k+1are small, ( ) implies an approximationRt(k) Rt+Rt 1+ +Rt k+1.( )This is a useful approximation when the time unit is small (such as a day, an houror a minute). Log Returns and continuously compoundingIn addition to the simple returnRt, the commonly usedone period log returnisdefined asrt=logPt logPt 1= log(Pt/Pt 1) = log(1 +Rt).

4 ( )Note that a log return is the logarithm (with the natural base) of a gross returnand logPtis called the log price. One immediate convenience in using log Returns isthat the additivity in multiperiod log Returns , thekperiod log returnrt(k) log(Pt/Pt k) is the sum of thekone-period log Returns :rt(k)=rt+rt 1+ +rt k+1.( )An investment at timet kwith initial capitalAyields at timetthe capitalAexp{rt(k)}=Aexp(rt+rt 1+ +rt k+1)=Aek r, r=(rt+rt 1+ +rt k+1)/kis the average one-period log Returns . In thisbookreturnsrefer to log returnsunless specified that the identity ( ) is in contrast with the approximation ( ) which isonly valid when the time unit is small. Indeed when the values are small, the tworeturns are approximately the same:rt= log(1 +Rt) ,rt<Rt. Figure plots the log Returns against the simple Returns for theApple Inc share prices in the period of January 1985 February 2011.

5 The returnsare calculated based on the daily close prices for the three holding periods: a day, aweek and a month. The figure shows that the two definitions result almost the samedaily Returns , especially for those with the values between and Howeverwhen the holding period increases to a week or a month, the discrepancy betweenthe two definitions is more apparent with a simple return always greater than thecorresponding log of log Returns against simple Returns of the Apple Inc share prices inJanuary 1985 February 2011. The blue straight lines mark the positions where the tworeturns are log returnrtis also calledcontinuously compounded returndue to its closelink with the concept of compound rates or interest rates . For a bank depositaccount, the quoted interest rate often refers to as simple interest.

6 For example,an interest rate of 5% payable every six months will be quoted as a simple interestof 10% per annum in the market. However if an account with the initial capital $1is held for 12 months and interest rate remains unchanged, it follows from ( ) thatthe gross return for the two periods is1 (1 + )2= , the annual simple return is 1= , which is called thecompound4 Chapter 1 Asset Returnsreturnand is greater than the quoted annual rate of 10%. This is due to the earningfrom interest-on-interest in the second six-month suppose that the quoted simple interest rate per annum isrand is un-changed, and the earnings are paid more frequently, say,mtimes per annum (atthe rater/meach time of course). For example, the account holder is paid everyquarter whenm=4,everymonthwhenm= 12, and every day whenm= to increase, and the earnings are paid continuously the gross return at the end of one year islimm (1 +r/m)m= generally, if the initial capital isC, invested in a bond that compounds con-tinuously the interest at annual rater, then the value of the investment at timetisCexp(rt).

7 Hence the log return per annum isr, which is the logarithm of the gross indicates that the simple annual interest raterquoted in the market is in factthe annual log return if the interest is compounded continuously. Note that if theinterest is only paid once at the end of the year, the simple return will ber,andthelog return will be log(1 +r) which is always smaller summary, a simple annual interest rate quoted in the market has two interpre-tations: it is the simple annual return if the interest is only paid once at the end ofthe year, and it is the annual log return if the interest is compounded Adjustment for dividendsMany assets, for example some blue-chip stocks, pay dividends to their share-holders from time to time. A dividend is typically allocated as a fixed amount ofcash per share. Therefore adjustments must then be made in computing Returns toaccount for the contribution towards the earnings from dividends.

8 LetDtdenotethe dividend payment between timet 1andt. Then the Returns are now definedas follows:Rt=(Pt+Dt)/Pt 1 1,rt= log(Pt+Dt) logPt 1,Rt(k)= Pt+Dt+ +Dt k+1 Pt k 1,rt(k)=rt+ +rt k+1=k 1 j=0log Pt j+Dt jPt j 1 .The above definitions are based on the assumption that all dividends are cashed outand are not re-invested in the Bond yields and pricesBonds are quoted in annualized yields. A so-called zero-coupon bond is a bondbought at a price lower than its face value (also called par value or principal), withthe face value repaid at the time of maturity. It does not make periodic interestpayments ( coupons), hence the term zero-coupon . Now we consider a zero-coupon bond with the face value $1. If the current yield isrtand the remainingduration isDunits of time, with continuous compounding, its current priceBtshould satisfy the conditionBtexp(Drt)=$1, the price isBt=exp( Drt) dollars.

9 Thus, the annualized log- return of thebond islog(Bt+1/Bt)=D(rt rt+1).( )Here, we ignore the fact thatBt+1has one unit of time shorter maturity that we have two baskets of high-yield bonds and investment-gradebonds ( the bonds with relatively low risk of default) with an average duration years each. Their yields spread ( the difference) over the Treasury bond withsimilar maturity are quoted and plotted in Figure The daily Returns of bonds canthen be deduced from ( ), which is the change of yields multiplied by the daily changes of treasury bonds are typically much smaller. Hence, the changesof yield spreads can directly be used as proxies of the changes of yields. As expected,the high-yield bonds have higher yields than the investment grade bonds, but havehigher volatility too (about 3 times). The yield spreads widened significantly in aperiod after the financial crisis following Lehman Brothers filing bankrupt protectionon September 15, 2008, reflecting higher default risks in corporate 1 Asset ReturnsFigure series of the yield spreads (the top panel) of high-yield bonds (blue curve)and investment-grade bonds (red curve), and their associated daily Returns (the 2nd and3rd panels) in November 29, 2004 December 10, Excess returnsIn many applications, it is convenient to use anexcess return , which is definedin the formrt r t,wherer tis a reference rate.

10 The commonly used referencerates are, for example, bank interest rates,LIBOR rates (London Interbank OfferedRate: the average interest rate that leading banks in London charge when lendingto other banks), log Returns of a riskless Asset ( , yields of short-term governmentbonds such as the 3-month US treasury bills) or market portfolio ( the S&P500 index or CRSP value-weighted index, which is the value-weighted index of allstocks traded in three major stock exchanges, created by the Center for Research inSecurity Prices of University of Chicago).For bonds,yield spreadis an excess yield defined as the difference between theyield of a bond and the yield of a reference bond such as a US treasury bill with asimilar of financial return Behavior of financial return dataIn order to build useful statistical models for financial Returns , we collect someempirical evidence first.