Evaluating Performance

Transcription

1 5 Chapter Evaluatng Performance In Ths Chapter Dollar-Weghted Rate of Return Tme-Weghted Rate of Return Income Rate of Return Prncpal Rate of Return Daly Returns MPT Statstcs 5-

2 Measurng Rates of Return Measurng Rates of Return To evaluate the performance of a portfolo manager, you measure average portfolo returns. A rate of return (ROR) s a percentage that reflects the apprecaton or deprecaton n the value of a portfolo or asset. PERAscope offers the followng return methods: Dollar-Weghted Rate of Return Tme-Weghted Rate of Return Income Return Prncpal Return Dollar-Weghted Rate of Return The Dollar-Weghted Rate of Return shows an average return of all the dollars n the portfolo for the perod. It s also referred to as the Internal Rate of Return (IRR). If cash flows occur durng the perod, PERAscope uses the Internal or Dollar-Weghted Rate of Return for the perod. PERAscope calculates the Dollar-Weghted Rate of Return as follows: Internal or Dollar- Weghted Rate of Return Calculaton MVB ( + ) + n F ( + ) D D D MVE MVB The maret value at the begnnng of the perod, ncludng accrued ncome. The desred Internal or Dollar-Weghted Rate of Return. n The number of cash flows durng the perod. F The cash flow at tme durng the perod contrbutons are postve cash flows and wthdrawals are negatve cash flows. D The number of days wthn the perod. D The number of days after the end of the prevous perod the F cash flow occurred. MVE The maret value at the end of the perod, ncludng accrued ncome. 5-

3 Measurng Rates of Return For example, f the perod beng computed s the month of June and a flow occurred on June 5, D would be 5 and D would be 30. The Modfed BAI calculaton s an teratve process, the IRR s obtaned by selectng values for and solvng the equaton untl the result equals or s wthn a desred tolerance of MVE. The tolerance vares wth the degree of precson selected durng report generaton. Tme-Weghted Rate of Return For Tme-Weghted Rates of Return, PERAscope uses a Modfed BAI method that elmnates the mpact of cash flows on performance calculatons. Under ths method, PERAscope calculates returns for tme perods that have nown maret values at the begnnng and end of the perod. When calculatng the rate of return for a gven perod, PERAscope determnes the number (n) and tmng of days wthn the perod that have nown maret values. It then dvdes the perod nto n+ subperods, so that each subperod has nown maret values at the begnnng and end of the perod, and no nown maret values n between. Subperod rates of return are calculated dfferently dependng on whether cash flows occur durng the subperod beng measured. If there are cash flows, PERAscope uses the Internal or Dollar-Weghted Rate of Return calculaton. See the Dollar-Weghted Rate of Return secton on page 5- for detal. If there are no cash flows durng the perod, the Tme-Weghted Rate of Return s calculated as: Tme-Weghted Rate of Return Calculaton MVE MVB The Tme-Weghted Rate of Return for the subperod. MVE The maret value at the end of the subperod, ncludng accrued ncome. MVB The maret value at the begnnng of the subperod, ncludng accrued ncome. 5-3

4 Measurng Rates of Return Geometrc Lnng Any consecutve set of subperod returns can be lned together to create a Tme-Weghted Rate of Return for a longer perod. PERAscope lns the subperod rates of return geometrcally as follows: Geometrc Lnng (+ )(+ )...(+ n+ ) - The rate of return for the perod. The rate of return for subperod,, through n +. Annualzed Returns PERAscope reports the annualzed return or return per year for a mult-year perod as follows: Annualzed Return Calculaton AR ( + PR)/n - AR The Annualzed Return. PR The Perod Return. n The number of years n the perod. 5-4

5 Income Rate of Return Evaluatng Performance Measurng Rates of Return The Income Rate of Return calculates the return from ncome net of expense. Ths s done by dvdng the net ncome by the average maret value: Income Rate of Return Calculaton Income Income AvgMV AvgMV MVB + n F * D D D Income The sum of dvdend ncome and coupon ncome for the perod, plus the change n accruals from the begnnng to end of the perod, less expenses for the perod. MVB The maret value at the begnnng of the perod, ncludng accrued ncome. n The number of cash flows durng the perod. F The cash flow at tme durng the perod. D The number of days wthn the perod. D The number of days after the end of the prevous perod the F cash flow occurred. OTE: If returns are beng measured net of fees, then fees are also treated as an expense. 5-5

6 Measurng Rates of Return Prncpal Rate of Return The Prncpal Rate of Return calculates the return based on changes n value of the assets. Ths s done by subtractng the ncome return from the total return: Prncpal Rate of Return Calculaton Prncpal Total - Income Total The total (tme-weghted) return for the perod. Income The ncome return for the perod. 5-6

7 Daly Returns Evaluatng Performance Daly Returns Cash flows and maret volatlty can have a substantal mpact on rate of return calculatons. To ensure an accurate return, you can value your portfolos daly or whenever cash flows occur. Ths s nown as daly valuaton. Whether you value your portfolos monthly, daly, or whenever cash flows occur, PERAscope calculates the Dollar-Weghted, Tme-Weghted, Income, and Prncpal Rate of Returns exactly the same way (that s, as descrbed n the prevous secton on Measurng Rates of Return). 5-7

8 MPT Statstcs MPT Statstcs Geometrc Mean MPT Statstcs are statstcs common to a dscplne nown as Modern Portfolo Theory. PERAscope uses the followng MPT statstcs to evaluate rs and other factors relatng to portfolos or assets: Geometrc Mean Arthmetc Mean Beta Alpha R-Squared Standard Devaton Sharpe Rato Treynor Rato PERAscope uses the geometrc mean or average n all calculatons nvolvng the mean or average rate of return. It s the nth root of the product of n numbers and s used to measure the compound rate of return over tme. Snce the geometrc mean represents the constant rate of return you would need to earn n each year to match actual performance over some past nvestment perod, t s an excellent measure of past performance. Geometrc Mean Calculatons PERAscope calculates the geometrc mean as follows: î The rate of return. The number of data tems. 5-8

9 MPT Statstcs The product of the monthly rates of return s further calculated as follows: * * 3 * L * Alternately: î Exp Ln ( + ) Arthmetc Mean The rate of return. Exp The exponental functon. Ln The natural logarthm. PERAscope uses the arthmetc mean to calculate standard devaton. See the Standard Devaton secton on page 5-4 for addtonal detal. The arthmetc mean or average s your standard, average calculaton. If your focus s on future performance, you would use the arthmetc mean because t provdes an unbased estmate of the portfolo s expected future return, assumng the expected rate of return does not change over tme. Arthmetc Mean Calculaton PERAscope calculates the arthmetc mean as follows: The rate of return. The number of data tems. 5-9

10 MPT Statstcs Beta Beta provdes a measure of the tendency of an asset or portfolo s returns to respond to swngs n the maret. It s the slope of the securty characterstc lne obtaned from lnear regresson. Lnear regresson s the process of fndng the lnear equaton that best approxmates the relatonshp between two varables. PERAscope calculates the beta as follows: Beta Calculaton ( P * M ) M P M M P The th nstance of the portfolo s return. M The th nstance of the maret s return. The number of data tems. P The th nstance of the portfolo s excess return ( P R ). M The th nstance of the maret s excess return ( M R ). Ths statstc reflects only the maret-related porton of an asset s or a portfolo s rs. It s a narrower measure than standard devaton whch reflects the total rs (maret related and unque.) In general, the volatlty of the relevant maret ndex s consdered to be.00, so a beta of.50 would ndcate a volatlty level 50% greater than that of the maret. Snce ths statstc s relatve to the maret, betas for assets or portfolos wth lttle or no correlaton to the maret are less sgnfcant. You may wsh to consder R-Squared as a measure to determne the sgnfcance of ths statstc. 5-0

11 MPT Statstcs Alpha Alpha Calculatons Alpha s a measure of rs adjusted performance that quantfes the dfference between an asset s or portfolo s actual performance and the performance antcpated n lght of the asset s or portfolo s rs (beta) and the maret s (relevant maret ndex) behavor. Alpha tells us how much better, or worse, a portfolo or asset dd relatve to what t was expected to do based on ts rs posture. A postve alpha ndcates a portfolo or asset s return has been more than commensurate wth ts rs posture. Alpha s the Y ntercept of the securty characterstc lne obtaned from lnear regresson. Lnear regresson s the process of fndng the lnear equaton that best defnes the relatonshp between two varables. OTE: If the bult-n lnear regresson functon n your spreadsheet applcaton uses the arthmetc mean, you wll not get the same result. You must use the formula that follows snce PERAscope uses the geometrc mean. PERAscope calculates alpha as follows:. Pˆ * Mˆ ^ P The geometrc mean of the portfolo s excess return. β The slope of the securty characterstc lne obtaned from the lnear regresson of the portfolo s excess return over the maret s excess return. ^ M The geometrc mean of the excess return of the benchmar. PERAscope s Modern Portfolo Theory report contans an annualzed alpha that s calculated as follows: OTE: ( ) Ths calculaton assumes you are calculatng alpha for a monthly return. 5-

12 5- Evaluatng Performance MPT Statstcs R-Squared R-Squared s the percentage of a portfolo s movement that can be explaned by movement n ts benchmar or n the maret. Ths statstc ndcates the percentage of an asset s or portfolo s rs whch cannot be elmnated through further dversfcaton. In precse percentage terms, ths fgure ndcates just how closely an asset s or portfolo s performance varaton paralleled the maret over the same tme perod. Lower fgures ndcate less correlaton wth the maret, and hence lower sgnfcance of the beta statstc. A relevant maret ndex s used as a proxy for the maret when measurng R-Squared. R-Squared Calculaton PERAscope calculates R-Squared as follows: P The th nstance of the portfolo s return. M The th nstance of the maret s return. R The th nstance of the rs free benchmar. P The th nstance of portfolo s excess return ( P R ). M The th nstance of maret s excess return ( M R ). The number of data tems. ( ) P P M M M P M * P R

13 MPT Statstcs Sharpe Rato Sharpe Rato Calculaton The Sharpe rato or measure s the rato of the portfolo s excess return to standard devaton. See the Standard Devaton secton on page 5-4 for addtonal detal. It s calculated by dvdng the geometrc mean of the portfolo s excess return over a perod by the standard devaton of returns over that perod. Hgher values are desrable and ndcate a greater return per unt of rs. PERAscope calculates the Sharpe rato as follows: Sharpe Pˆ T ^ P The geometrc mean of the portfolo s excess return. σ T The total return standard devaton. 5-3

14 MPT Statstcs Standard Devaton Standard devaton provdes a statstcal measure of the daly, monthly, quarterly, or annual ups and downs of an asset s or portfolo s return. Moneymaret assets, whch have stable asset values, have standard devatons near zero. Volatle, aggressve-growth portfolos can have standard devatons that are substantally hgher. Total Return Standard Devaton Total return standard devaton measures the volatlty of the total return of the portfolo. PERAscope calculates total return standard devaton as follows: T ( P -P) P The total return for the th perod. P The arthmetc mean of the portfolo s returns over all ncluded perods. The number of perods of returns ncluded n the computaton. Excess Return Standard Devaton Excess return standard devaton measures the volatlty of the excess return of the portfolo. PERAscope calculates excess return standard devaton as follows: X ( P - Px) P The excess return for the th perod. P x The arthmetc mean of the portfolo s excess return over all ncluded perods. 5-4

15 MPT Statstcs Treynor Rato Treynor Rato Calculaton The Treynor s a gauge of rs-adjusted performance that s calculated by dvdng the excess return of a portfolo above the rs-free return by ts beta. Hgher values are desrable and ndcate a greater return per unt of rs. PERAscope calculates the Treynor rato as follows: Pˆ Treynor β ^ P The geometrc mean of the portfolo s excess return. β The slope of the securty characterstc lne. Ths s usually nterpreted as the level of undversfable rs of the portfolo. 5-5

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