[Returnanalytics-commits] r2147 - pkg/PortfolioAnalytics/sandbox/attribution/man
noreply at r-forge.r-project.org
noreply at r-forge.r-project.org
Thu Jul 12 11:16:32 CEST 2012
Author: ababii
Date: 2012-07-12 11:16:32 +0200 (Thu, 12 Jul 2012)
New Revision: 2147
Modified:
pkg/PortfolioAnalytics/sandbox/attribution/man/Attribution.Rd
pkg/PortfolioAnalytics/sandbox/attribution/man/Attribution.geometric.Rd
pkg/PortfolioAnalytics/sandbox/attribution/man/Attribution.levels.Rd
pkg/PortfolioAnalytics/sandbox/attribution/man/AttributionFixedIncome.Rd
pkg/PortfolioAnalytics/sandbox/attribution/man/Carino.Rd
pkg/PortfolioAnalytics/sandbox/attribution/man/Conv.option.Rd
pkg/PortfolioAnalytics/sandbox/attribution/man/DaviesLaker.Rd
pkg/PortfolioAnalytics/sandbox/attribution/man/Frongello.Rd
pkg/PortfolioAnalytics/sandbox/attribution/man/Grap.Rd
pkg/PortfolioAnalytics/sandbox/attribution/man/Menchero.Rd
pkg/PortfolioAnalytics/sandbox/attribution/man/Return.annualized.excess.Rd
Log:
- documentaion update (inline equations are supported)
Modified: pkg/PortfolioAnalytics/sandbox/attribution/man/Attribution.Rd
===================================================================
--- pkg/PortfolioAnalytics/sandbox/attribution/man/Attribution.Rd 2012-07-12 09:15:48 UTC (rev 2146)
+++ pkg/PortfolioAnalytics/sandbox/attribution/man/Attribution.Rd 2012-07-12 09:16:32 UTC (rev 2147)
@@ -29,7 +29,7 @@
the sector allocation. Interaction term is combined with
the security selection effect, \item bottom.up - the
priority is given to the security selection. Interaction
- term is combined with the sector allocation effect}. By
+ term is combined with the sector allocation effect} By
default "none" is selected}
\item{wpf}{vector, xts, data frame or matrix with
@@ -56,7 +56,9 @@
hedged into the base currency}
\item{bf}{TRUE for Brinson and Fachler and FALSE for
- Brinson, Hood and Beebower arithmetic attribution}
+ Brinson, Hood and Beebower arithmetic attribution. By
+ default Brinson, Hood and Beebower attribution is
+ selected}
\item{linking}{Used to select the linking method to
present the multi-period summary of arithmetic
@@ -68,10 +70,12 @@
linking method} By default Carino linking is selected}
\item{geometric}{TRUE/FALSE, whether to use geometric or
- arithmetic excess returns for the attribution analysis}
+ arithmetic excess returns for the attribution analysis.
+ By default arithmetic is selected}
\item{adjusted}{TRUE/FALSE, whether to show original or
- smoothed attribution effects for each period}
+ smoothed attribution effects for each period. By default
+ unadjusted attribution effects are returned}
}
\value{
returns a list with the following components: excess
@@ -92,34 +96,40 @@
returns at one level. If returns and weights are
available at the lowest level (e.g. for individual
instruments), the aggregation up to the chosen level from
- the hierarchy can be done using Return.level function.
- The attribution effects can be computed for several
- periods. The multi-period summary is obtained using one
- of linking methods: Carino, Menchero, GRAP, Frongello. It
- also allows to break down the geometric excess returns,
- which link naturally over time. Finally, it annualizes
- arithmetic and geometric excess returns similarly to the
- portfolio and/or benchmark returns annualization.
+ the hierarchy can be done using
+ \code{\link{Return.level}} function. The attribution
+ effects can be computed for several periods. The
+ multi-period summary is obtained using one of linking
+ methods: Carino, Menchero, GRAP, Frongello or Davies
+ Laker. It also allows to break down the geometric excess
+ returns, which link naturally over time. Finally, it
+ annualizes arithmetic and geometric excess returns
+ similarly to the portfolio and/or benchmark returns
+ annualization.
}
\details{
The arithmetic excess returns are decomposed into the sum
- of allocation, selection and interaction effects across n
- sectors:
+ of allocation, selection and interaction effects across
+ \eqn{n} sectors:
\deqn{R_{p}-R_{b}=\sum^{n}_{i=1}\left(A_{i}+S_{i}+I_{i}\right)}
The arithmetic attribution effects for the category i are
computed as suggested in the Brinson, Hood and Beebower
(1986): Allocation effect
- \deqn{A_{i}=(w_{pi}-w_{bi})\times R_{bi}} Selection
- effect \deqn{S_{i}=w_{pi}\times(R_{pi}-R_{bi})}
- Interaction effect
- \deqn{I_{i}=(w_{pi}-w_{bi})\times(R_{pi}-R_{bi})} r -
- total portfolio returns, b - total benchmark returns,
- w_pi - weights of the category i in the portfolio, w_bi -
- weigths of the category i in the benchmark, R_pi -
- returns of the portfolio category i, R_bi - returns of
- the benchmark category i. If Brinson and Fachler (1985)
- is selected the allocation effect differs:
- \deqn{A_{i}=(w_{pi}-w_{bi})\times (R_{bi} - R_{b})}
+ \deqn{A_{i}=(w_{pi}-w_{bi})\times R_{bi}}{Ai = (wpi -
+ wbi) * Rbi} Selection effect
+ \deqn{S_{i}=w_{pi}\times(R_{pi}-R_{bi})}{Si = wpi * (Rpi
+ - Rbi)} Interaction effect \deqn{I_{i}=(w_{pi}-w_{bi})
+ \times(R_{pi}-R_{bi})}{Ii = (wpi - wbi) * Rpi - Rbi}
+ \eqn{R_{p}}{Rp} - total portfolio returns,
+ \eqn{R_{b}}{Rb} - total benchmark returns,
+ \eqn{w_{pi}}{wpi} - weights of the category \eqn{i} in
+ the portfolio, \eqn{w_{bi}}{wbi} - weigths of the
+ category \eqn{i} in the benchmark, \eqn{R_{pi}}{Rpi} -
+ returns of the portfolio category \eqn{i},
+ \eqn{R_{bi}}{Rbi} - returns of the benchmark category
+ \eqn{i}. If Brinson and Fachler (1985) is selected the
+ allocation effect differs: \deqn{A_{i}=(w_{pi}-w_{bi})
+ \times (R_{bi} - R_{b})}{Ai = (wpi - wbi) * (Rbi - Rb)}
Depending on goals we can give priority to the allocation
or to the selection effects. If the priority is given to
the sector allocation the interaction term will be
@@ -133,45 +143,50 @@
can be summed up over time to provide the multi-period
summary:
\deqn{R_{p}-R_{b}=\sum^{T}_{t=1}\left(A_{t}'+S_{t}'+I_{t}'\right)}
- where T is the number of periods and prime stands for the
- adjustment. The geometric attribution effects do not
- suffer from the linking problem. Moreover we don't have
- the interaction term. For more details about the
+ where \eqn{T} is the number of periods and prime stands
+ for the adjustment. The geometric attribution effects do
+ not suffer from the linking problem. Moreover we don't
+ have the interaction term. For more details about the
geometric attribution see the documentation to
- \code{Attribution.geometric} Finally, arithmetic
+ \code{\link{Attribution.geometric}}. Finally, arithmetic
annualized excess returns are computed as the arithmetic
difference between annualised portfolio and benchmark
- returns: \deqn{AAER=r_{a}-b_{a}} the geometric annualized
- excess returns are computed as the geometric difference
- between annualized portfolio and benchmark returns:
- \deqn{GAER=\frac{1+r_{a}}{1+b_{a}}-1} In the case of
- multi-currency portfolio, the currency return, currency
- surprise and forward premium should be specified. The
- multi-currency arithmetic attribution is handled
- following Ankrim and Hensel (1992). Currency returns are
- decomposed into the sum of the currency surprise and the
- forward premium: \deqn{R_{ci} = R_{cei} + R_{fpi}} where
- \deqn{R_{cei} = \frac{S_{i}^{t+1} -
+ returns: \deqn{AAER=r_{a}-b_{a}}{AAER = ra - ba} the
+ geometric annualized excess returns are computed as the
+ geometric difference between annualized portfolio and
+ benchmark returns:
+ \deqn{GAER=\frac{1+r_{a}}{1+b_{a}}-1}{GAER = (1 + ra) /
+ (1 + ba) - 1} In the case of multi-currency portfolio,
+ the currency return, currency surprise and forward
+ premium should be specified. The multi-currency
+ arithmetic attribution is handled following Ankrim and
+ Hensel (1992). Currency returns are decomposed into the
+ sum of the currency surprise and the forward premium:
+ \deqn{R_{ci} = R_{cei} + R_{fpi}}{Rci = Rcei + Rfpi}
+ where \deqn{R_{cei} = \frac{S_{i}^{t+1} -
F_{i}^{t+1}}{S_{i}^{t}}} \deqn{R_{fpi} =
- \frac{F_{i}^{t+1}}{S_{i}^{t}} - 1} S^t_i - stop rate for
- asset i at time t F^t_i - forward rate for asset i at
- time t. Excess returns are decomposed into the sum of
- allocation, selection and interaction effects as in the
- standard Brinson model:
+ \frac{F_{i}^{t+1}}{S_{i}^{t}} - 1} \eqn{S_{i}^{t}}{Sit} -
+ spot rate for asset \eqn{i} at time \eqn{t}
+ \eqn{F_{i}^{t}}{Fit} - forward rate for asset \eqn{i} at
+ time \eqn{t}. Excess returns are decomposed into the sum
+ of allocation, selection and interaction effects as in
+ the standard Brinson model:
\deqn{R_{p}-R_{b}=\sum^{n}_{i=1}\left(A_{i}+S_{i}+I_{i}\right)}
However the allocation effect is computed taking into
account currency effects:
\deqn{A_{i}=(w_{pi}-w_{bi})\times (R_{bi} - R_{ci} -
- R_{l})} Benchmark returns adjusted fo the currency:
- \deqn{R_{l} = \sum^{n}_{i=1}w_{bi}\times(R_{bi}-R_{ci})}
- The contribution from currency is analogous to asset
+ R_{l})}{Ai = (wpi - wbi) * (Rbi - Rci - Rl)} Benchmark
+ returns adjusted fo the currency: \deqn{R_{l} =
+ \sum^{n}_{i=1}w_{bi}\times(R_{bi}-R_{ci})} The
+ contribution from the currency is analogous to asset
allocation: \deqn{C_{i} = (w_{pi} - w_{bi}) \times
(R_{cei} - e) + (w_{pfi} - w_{bfi}) \times (R_{fi} - e)}
where \deqn{e = \sum^{n}_{i=1}w_{bi}\times R_{cei}} The
final term, forward premium, is also analogous to the
asset allocation: \deqn{R_{fi} = (w_{pi} - w_{bi}) \times
- (R_{fpi} - d)} where \deqn{d = \sum^{n}_{i=1}w_{bi}\times
- R_{fpi}} and R_fpi - forward premium
+ (R_{fpi} - d)}{Rfi = (wpi - wbi) * (Rfpi - d)} where
+ \deqn{d = \sum^{n}_{i=1}w_{bi}\times R_{fpi}} and
+ \eqn{R_{fpi}} - forward premium
}
\examples{
data(attrib)
Modified: pkg/PortfolioAnalytics/sandbox/attribution/man/Attribution.geometric.Rd
===================================================================
--- pkg/PortfolioAnalytics/sandbox/attribution/man/Attribution.geometric.Rd 2012-07-12 09:15:48 UTC (rev 2146)
+++ pkg/PortfolioAnalytics/sandbox/attribution/man/Attribution.geometric.Rd 2012-07-12 09:16:32 UTC (rev 2147)
@@ -35,19 +35,27 @@
\code{\link{Attribution}} function. Geometric attribution
effects in the contrast with arithmetic do naturally link
over time multiplicatively:
- \deqn{\frac{(1+R_{p})}{1+R_{b}}-1=\prod^{n}_{t=1}(1+A_{t}^{G})\times\prod^{n}_{t=1}(1+S{}_{t}^{G})-1}
- Total allocation effect at time t:
+ \deqn{\frac{(1+R_{p})}{1+R_{b}}-1=\prod^{n}_{t=1}(1+A_{t}^{G})\times
+ \prod^{n}_{t=1}(1+S{}_{t}^{G})-1} Total allocation effect
+ at time \eqn{t}:
\deqn{A_{t}^{G}=\frac{1+b_{S}}{1+R_{bt}}-1} Total
- selection effect at time t:
+ selection effect at time \eqn{t}:
\deqn{S_{t}^{G}=\frac{1+R_{pt}}{1+b_{S}}-1} Semi-notional
fund: \deqn{b_{S}=\sum^{n}_{i=1}w_{pi}\times R_{bi}}
+ \eqn{w_{pt}}{wpt} - portfolio weights at time \eqn{t},
+ \eqn{w_{bt}}{wbt} - benchmark weights at time \eqn{t},
+ \eqn{r_{t}}{rt} - portfolio returns at time \eqn{t},
+ \eqn{b_{t}}{bt} - benchmark returns at time \eqn{t},
+ \eqn{r} - total portfolio returns \eqn{b} - total
+ benchmark returns \eqn{n} - number of periods
}
\details{
The multi-currency geometric attribution is handled
following the Appendix A (Bacon, 2004).
The individual selection effects are computed using:
- \deqn{w_{pi}\times\left(\frac{1+R_{pLi}}{1+R_{bLi}}-1\right)\times\left(\frac{1+R_{bLi}}{1+b_{SL}}\right)}
+ \deqn{w_{pi}\times\left(\frac{1+R_{pLi}}{1+R_{bLi}}-1\right)\times
+ \left(\frac{1+R_{bLi}}{1+b_{SL}}\right)}
The individual allocation effects are computed using:
\deqn{(w_{pi}-w_{bi})\times\left(\frac{1+R_{bHi}}{1+b_{L}}-1\right)}
@@ -56,15 +64,17 @@
base currency were used: \deqn{b_{SH} =
\sum_{i}w_{pi}\times R_{bi}((w_{pi} - w_{bi})R_{bHi} +
w_{bi}R_{bLi})} Total semi-notional returns in the local
- currency: \deqn{b_{SL} = \sum_{i}w_{pi}R_{bLi}} Portfolio
- returns in the local currency: \deqn{R_{pLi}} Benchmark
- returns in the local currency: \deqn{R_{bLi}} Benchmark
- returns hedged into the base currency: \deqn{R_{bHi}}
- Total benchmark returns in the local currency:
- \deqn{b_{L}} Total portfolio returns in the local
- currency: \deqn{r_{L}} The total excess returns are
- decomposed into:
- \deqn{\frac{(1+R_{p})}{1+R_{b}}-1=\frac{1+r_{L}}{1+b_{SL}}\times\frac{1+b_{SH}}{1+b_{L}}\times\frac{1+b_{SL}}{1+b_{SH}}\times\frac{1+R_{p}}{1+r_{L}}\times\frac{1+b_{L}}{1+R_{b}}-1}
+ currency: \deqn{b_{SL} = \sum_{i}w_{pi}R_{bLi}}
+ \eqn{R_{pLi}}{RpLi} - portfolio returns in the local
+ currency \eqn{R_{bLi}}{RbLi} - benchmark returns in the
+ local currency \eqn{R_{bHi}}{RbHi} - benchmark returns
+ hedged into the base currency \eqn{b_{L}}{bL} - total
+ benchmark returns in the local currency \eqn{r_{L}}{rL} -
+ total portfolio returns in the local currency The total
+ excess returns are decomposed into:
+ \deqn{\frac{(1+R_{p})}{1+R_{b}}-1=\frac{1+r_{L}}{1+b_{SL}}\times\frac{1+
+ b_{SH}}{1+b_{L}}\times\frac{1+b_{SL}}{1+b_{SH}}\times\frac{1+R_{p}}{1+r_{L}}
+ \times\frac{1+b_{L}}{1+R_{b}}-1}
where the first term corresponds to the selection, second
to the allocation, third to the hedging cost transferred
Modified: pkg/PortfolioAnalytics/sandbox/attribution/man/Attribution.levels.Rd
===================================================================
--- pkg/PortfolioAnalytics/sandbox/attribution/man/Attribution.levels.Rd 2012-07-12 09:15:48 UTC (rev 2146)
+++ pkg/PortfolioAnalytics/sandbox/attribution/man/Attribution.levels.Rd 2012-07-12 09:16:32 UTC (rev 2147)
@@ -40,17 +40,16 @@
instruments). Benchmark should have the same number of
columns as portfolio. That is there should be a benchmark
for each instrument in the portfolio (possibly 0). The
- contribution to the allocation in the ith category for
- the dth level is:
- \deqn{\left(^{d}w_{pi}-^{d}w_{bi}\right)\times\left(\frac{1+^{d}R_{bi}}{1+^{d-1}R_{bi}}-1\right)\times\frac{1+^{d-1}R_{bi}}{1+bs^{d-1}}}
-}
-\details{
- The total attribution for each asset allocation step in
- the decision process is:
- \deqn{\frac{1+^{d}bs}{1+^{d-1}bs}-1}
-
+ contribution to the allocation in the \eqn{i^{th}}
+ category for the \eqn{d^{th}} level is:
+ \deqn{\left(^{d}w_{pi}-^{d}w_{bi}\right)\times
+ \left(\frac{1+^{d}R_{bi}}{1+^{d-1}R_{bi}}-1\right)
+ \times\frac{1+^{d-1}R_{bi}}{1+bs^{d-1}}} The total
+ attribution for each asset allocation step in the
+ decision process is: \deqn{\frac{1+^{d}bs}{1+^{d-1}bs}-1}
The final step, stock selection, is measured by:
- \deqn{^{d}w_{pi}\times\left(\frac{1+R_{pi}}{1+^{d}R_{bi}}-1\right)\times\frac{1+^{d}R_{bi}}{1+^{d}bs}}
+ \deqn{^{d}w_{pi}\times\left(\frac{1+R_{pi}}{1+^{d}R_{bi}}-1\right)
+ \times\frac{1+^{d}R_{bi}}{1+^{d}bs}}
}
\examples{
data(attrib)
Modified: pkg/PortfolioAnalytics/sandbox/attribution/man/AttributionFixedIncome.Rd
===================================================================
--- pkg/PortfolioAnalytics/sandbox/attribution/man/AttributionFixedIncome.Rd 2012-07-12 09:15:48 UTC (rev 2146)
+++ pkg/PortfolioAnalytics/sandbox/attribution/man/AttributionFixedIncome.Rd 2012-07-12 09:16:32 UTC (rev 2147)
@@ -61,33 +61,40 @@
risk-adjusted attribution. The arithmetic attribution is
handled using weighted duration approach (Van Breukelen,
2000). The allocation, selection and currency allocation
- effects for category i are: \deqn{A_{i} = (D_{pi}\times
- w_{pi}-D_{\beta}\times D_{bi}\times w_{pi})\times
- (-\Delta y_{bi} + \Delta y_{b})} \deqn{S_{i} =
+ effects for category \eqn{i} are: \deqn{A_{i} =
+ (D_{pi}\times w_{pi}-D_{\beta}\times D_{bi}\times w_{pi})
+ \times (-\Delta y_{bi} + \Delta y_{b})} \deqn{S_{i} =
D_{i}\times w_{pi}\times (-\Delta y_{ri} + \Delta
y_{bi})} \deqn{C_{i} = (w_{pi} - w_{bi})\times (c_{i} +
- R_{fi} - c')} where w_pi - portfolio weights, w_bi -
- benchmark weights, D_i - modified duration in bond
- category i. Duration beta:
- \deqn{D_{\beta}=\frac{D_{r}}{D_{b}}} D_r - portfolio
- duration, D_b - benchmark duration, D_bi - benchmark
- duration for category i, D_pi - portfolio duration for
- category i, Delta y_ri - change in portfolio yield for
- category i, Delta y_bi - change in benchmark yield for
- category i, Delta y_b - change in benchmark yield, R_ci-
- currency returns for category i, R_fi - risk-free rate in
- currency of asset i, \deqn{c'=
+ R_{fi} - c')}{Ci = (wpi - wbi) * (ci + Rfi - c')} where
+ \eqn{w_{pi}}{wpi} - portfolio weights, \eqn{w_{bi}}{wbi}
+ - benchmark weights, \eqn{D_{i}}{Di} - modified duration
+ in bond category \eqn{i}. Duration beta:
+ \deqn{D_{\beta}=\frac{D_{r}}{D_{b}}}{Dbeta = Dr / Db}
+ \eqn{D_{r}}{Dr} - portfolio duration, \eqn{D_{b}}{Db} -
+ benchmark duration, \eqn{D_{bi}}{Dbi} - benchmark
+ duration for category \eqn{i}, \eqn{D_{pi}}{Dpi} -
+ portfolio duration for category \eqn{i}, \eqn{\Delta
+ y_{ri}}{Delta yri} - change in portfolio yield for
+ category \eqn{i}, \eqn{\Delta y_{bi}}{Delta ybi} - change
+ in benchmark yield for category \eqn{i}, \eqn{\Delta
+ y_{b}}{Delta yb} - change in benchmark yield,
+ \eqn{R_{ci}}{Rci} - currency returns for category
+ \eqn{i}, \eqn{R_{fi}}{Rfi} - risk-free rate in currency
+ of asset \eqn{i}, \deqn{c'=
\sum_{i}w_{bi}\times(R_{ci}+R_{fi})} The geometric
attribution is adapted using Van Breukelen (2000)
approach for the arithmetic attribution. The individual
allocation and selection effects are computed as follows:
- \deqn{A_{i}=D_{i}w_{pi}-D_{\beta}D_{bi}w_{bi}}
+ \deqn{A_{i}=D_{i}w_{pi}-D_{\beta}D_{bi}w_{bi}}{Ai = Di *
+ wpi - Dbeta * Dbi * wbi}
\deqn{S_{i}=\frac{D_{pi}}{D_{bi}}\times (R_{bi} - R_{fi})
- + R_{fi}}
+ + R_{fi}}{Si = Dpi / Dbi * (Rbi - Rfi) + Rfi}
}
\examples{
data(attrib)
-AttributionFixedIncome(Rp, wp, Rb, wb, Rf, Dp, Db, S, wbf, geometric = FALSE)
+AttributionFixedIncome(Rp, wp, Rb, wb, Rf, Dp, Db, S, wbf,
+geometric = FALSE)
}
\author{
Andrii Babii
Modified: pkg/PortfolioAnalytics/sandbox/attribution/man/Carino.Rd
===================================================================
--- pkg/PortfolioAnalytics/sandbox/attribution/man/Carino.Rd 2012-07-12 09:15:48 UTC (rev 2146)
+++ pkg/PortfolioAnalytics/sandbox/attribution/man/Carino.Rd 2012-07-12 09:16:32 UTC (rev 2147)
@@ -27,19 +27,23 @@
effects so that they can be summed up over multiple
periods. Attribution effect are multiplied by the
adjustment factor: \deqn{A_{t}' = A_{t} \times
- \frac{k_{t}}{k}} where \deqn{k_{t} = \frac{log(1 +
- R_{pt}) - log(1 + R_{bt})}{R_{pt} - R_{bt}}} \deqn{k =
- \frac{log(1 + R_{p}) - log(1 + R_{b})}{R_{p} - R_{b}}} In
- case if portfolio and benchmark returns are equal:
- \deqn{k_{t} = \frac{1}{1 + R_{pt}}} where A_t' - adjusted
- attribution effects at period t, A_t - unadjusted
- attribution effects at period t, R_pt - portfolio returns
- at period t, R_bt - benchmark returns at period t, Rp -
- total portfolio returns, Rb - total benchmark returns, n
- - number of periods The total arithmetic excess returns
- can be explained in terms of the sum of adjusted
+ \frac{k_{t}}{k}}{At' = At * kt / k} where \deqn{k_{t} =
+ \frac{log(1 + R_{pt}) - log(1 + R_{bt})}{R_{pt} -
+ R_{bt}}} \deqn{k = \frac{log(1 + R_{p}) - log(1 +
+ R_{b})}{R_{p} - R_{b}}} In case if portfolio and
+ benchmark returns are equal: \deqn{k_{t} = \frac{1}{1 +
+ R_{pt}}}{kt = 1 / (1 + Rpt)} where \eqn{A_{t}'}{At'} -
+ adjusted attribution effects at period \eqn{t},
+ \eqn{A_{t}}{At} - unadjusted attribution effects at
+ period \eqn{t}, \eqn{R_{pt}}{Rpt} - portfolio returns at
+ period \eqn{t}, \eqn{R_{bt}}{Rbt} - benchmark returns at
+ period \eqn{t}, \eqn{R_{p}}{Rp} - total portfolio
+ returns, \eqn{R_{b}}{Rb} - total benchmark returns,
+ \eqn{n} - number of periods The total arithmetic excess
+ returns can be explained in terms of the sum of adjusted
attribution effects: \deqn{R_{p} - R_{b} =
- \sum^{n}_{t=1}\left(Allocation_{t}+Selection_{t}+Interaction_{t}\right)}
+ \sum^{n}_{t=1}\left(Allocation_{t}+Selection_{t}+
+ Interaction_{t}\right)}
}
\examples{
data(attrib)
Modified: pkg/PortfolioAnalytics/sandbox/attribution/man/Conv.option.Rd
===================================================================
--- pkg/PortfolioAnalytics/sandbox/attribution/man/Conv.option.Rd 2012-07-12 09:15:48 UTC (rev 2146)
+++ pkg/PortfolioAnalytics/sandbox/attribution/man/Conv.option.Rd 2012-07-12 09:16:32 UTC (rev 2147)
@@ -32,9 +32,11 @@
exposure}
}
\examples{
-option = matrix(c(1000, 1000, 1000, 300, 400, 10, 20, 30, 40, 50, 10, 11, 12, 13, 14, 12, 13,
-14, 15, 16, 0.1, 0.2, 0.3, 0.4, 0.5, 0.1, 0.1, 0.2, 0.2, 0.3), 5, 6)
-colnames(option) = c("Strike", "Number", "Current option", "End option", "delta", "returns")
+option = matrix(c(1000, 1000, 1000, 300, 400, 10, 20, 30, 40, 50, 10, 11,
+12, 13, 14, 12, 13, 14, 15, 16, 0.1, 0.2, 0.3, 0.4, 0.5, 0.1, 0.1, 0.2,
+0.2, 0.3), 5, 6)
+colnames(option) = c("Strike", "Number", "Current option", "End option",
+"delta", "returns")
rownames(option) = c("CVX", "XOM", "GE", "WMT", "FB")
Conv.option(option)
}
Modified: pkg/PortfolioAnalytics/sandbox/attribution/man/DaviesLaker.Rd
===================================================================
--- pkg/PortfolioAnalytics/sandbox/attribution/man/DaviesLaker.Rd 2012-07-12 09:15:48 UTC (rev 2146)
+++ pkg/PortfolioAnalytics/sandbox/attribution/man/DaviesLaker.Rd 2012-07-12 09:16:32 UTC (rev 2147)
@@ -26,16 +26,20 @@
time. This function uses Davies and Laker linking method
to compute total attribution effects. Arithmetic excess
returns are decomposed as follows: \deqn{R_{p} - R_{b} =
+ Allocation + Selection + Interaction}{Rp - Rb =
Allocation + Selection + Interaction} \deqn{Allocation =
\prod^{T}_{t=1}(1+bs_{t})-\prod^{T}_{t=1}(1+R_{bt})}
\deqn{Selection =
\prod^{T}_{t=1}(1+rs_{t})-\prod^{T}_{t=1}(1+R_{bt})}
\deqn{Interaction =
- \prod^{T}_{t=1}(1+R_{pt})-\prod^{T}_{t=1}(1+rs_{t})-\prod^{T}_{t=1}(1+bs_{t})+\prod^{T}_{t=1}(1+R_{bt})}
- R_pi - portfolio returns at period i, Rb_i - benchmark
- returns at period i, rs_i - selection notional fund
- returns at period i, bs_i - allocation notional fund
- returns at period i, T - number of periods
+ \prod^{T}_{t=1}(1+R_{pt})-\prod^{T}_{t=1}(1+rs_{t})-
+ \prod^{T}_{t=1}(1+bs_{t})+\prod^{T}_{t=1}(1+R_{bt})}
+ \eqn{R_{pi}}{Rpi} - portfolio returns at period \eqn{i},
+ \eqn{R_{bi}}{Rbi} - benchmark returns at period \eqn{i},
+ \eqn{rs_{i}}{rsi} - selection notional fund returns at
+ period \eqn{i}, \eqn{bs_{i}}{bsi} - allocation notional
+ fund returns at period \eqn{i}, \eqn{T} - number of
+ periods
}
\examples{
data(attrib)
Modified: pkg/PortfolioAnalytics/sandbox/attribution/man/Frongello.Rd
===================================================================
--- pkg/PortfolioAnalytics/sandbox/attribution/man/Frongello.Rd 2012-07-12 09:15:48 UTC (rev 2146)
+++ pkg/PortfolioAnalytics/sandbox/attribution/man/Frongello.Rd 2012-07-12 09:16:32 UTC (rev 2147)
@@ -27,12 +27,15 @@
attribution effects so that they can be summed up over
multiple periods Adjusted attribution effect at period t
are: \deqn{A_{t}' =
- A_{t}\times\prod^{t-1}_{i=1}(1+r_{pi})+R_{bt}\times\sum^{t-1}_{i=1}A_{i}'}
- A_t' - adjusted attribution effects at period t, A_t -
- unadjusted attribution effects at period t, R_pi -
- portfolio returns at period i, R_bi - benchmark returns
- at period , Rp - total portfolio returns, Rb - total
- benchmark returns, n - number of periods
+ A_{t}\times\prod^{t-1}_{i=1}(1+r_{pi})+R_{bt}
+ \times\sum^{t-1}_{i=1}A_{i}'} \eqn{A_{t}'}{At'} -
+ adjusted attribution effects at period \eqn{t},
+ \eqn{A_{t}}{At} - unadjusted attribution effects at
+ period \eqn{t}, \eqn{R_{pi}}{Rpi} - portfolio returns at
+ period \eqn{i}, \eqn{R_{bi}}{Rbi} - benchmark returns at
+ period, \eqn{R_{p}}{Rp} - total portfolio returns,
+ \eqn{R_{b}}{Rb} - total benchmark returns, \eqn{n} -
+ number of periods
}
\examples{
data(attrib)
Modified: pkg/PortfolioAnalytics/sandbox/attribution/man/Grap.Rd
===================================================================
--- pkg/PortfolioAnalytics/sandbox/attribution/man/Grap.Rd 2012-07-12 09:15:48 UTC (rev 2146)
+++ pkg/PortfolioAnalytics/sandbox/attribution/man/Grap.Rd 2012-07-12 09:16:32 UTC (rev 2147)
@@ -26,18 +26,20 @@
function uses GRAP smoothing algorithm to adjust
attribution effects so that they can be summed up over
multiple periods Attribution effect are multiplied by the
- adjustment factor \deqn{A_{t}' = A_{t} \times G_{t}}
- where
+ adjustment factor \deqn{A_{t}' = A_{t} \times G_{t}}{At'
+ = At * Gt} where
\deqn{G_{t}=\prod^{t-1}_{i=1}(1+R_{pi})\times\prod^{n}_{t+1}(1+R_{bi})}
- A_t' - adjusted attribution effects at period t, A_t -
- unadjusted attribution effects at period t, R_pi -
- portfolio returns at period i, R_bi - benchmark returns
- at period i, Rp - total portfolio returns, Rb - total
- benchmark returns, n - number of periods The total
- arithmetic excess returns can be explained in terms of
- the sum of adjusted attribution effects: \deqn{R_{p} -
- R_{b} =
- \sum^{n}_{t=1}\left(Allocation_{t}+Selection_{t}+Interaction_{t}\right)}
+ \eqn{A_{t}'}{At'} - adjusted attribution effects at
+ period \eqn{t}, \eqn{A_{t}}{At} - unadjusted attribution
+ effects at period \eqn{t}, \eqn{R_{pi}}{Rpi} - portfolio
+ returns at period \eqn{i}, \eqn{R_{bi}}{Rbi} - benchmark
+ returns at period \eqn{i}, \eqn{R_{p}}{Rp} - total
+ portfolio returns, \eqn{R_{b}}{Rb} - total benchmark
+ returns, \eqn{n} - number of periods The total arithmetic
+ excess returns can be explained in terms of the sum of
+ adjusted attribution effects: \deqn{R_{p} - R_{b} =
+ \sum^{n}_{t=1}\left(Allocation_{t}+Selection_{t}+
+ Interaction_{t}\right)}
}
\examples{
data(attrib)
Modified: pkg/PortfolioAnalytics/sandbox/attribution/man/Menchero.Rd
===================================================================
--- pkg/PortfolioAnalytics/sandbox/attribution/man/Menchero.Rd 2012-07-12 09:15:48 UTC (rev 2146)
+++ pkg/PortfolioAnalytics/sandbox/attribution/man/Menchero.Rd 2012-07-12 09:16:32 UTC (rev 2147)
@@ -26,25 +26,29 @@
function uses Menchero smoothing algorithm to adjust
attribution effects so that they can be summed up over
multiple periods Attribution effect are multiplied by the
- adjustment factor \deqn{A_{t}' = A_{t} \times (M +a_{t})}
- where
- \deqn{M=\frac{\frac{1}{n}(R_{p}-R_{b})}{(1+R_{p})^{\frac{1}{n}}-(1+R_{b})^{\frac{1}{n}}}}
- \deqn{a_{t} =
- \left(\frac{R_{p}-R_{b}-M\times\sum^{n}_{t=1}(R_{pt}-R_{bt})}{\sum^{n}_{t=1}(R_{pt}-R_{bt})^{2}}\right)\times(R_{pt}-R_{bt})}
+ adjustment factor \deqn{A_{t}' = A_{t} \times (M
+ +a_{t})}{At' = At * (M + at)} where
+ \deqn{M=\frac{\frac{1}{n}(R_{p}-R_{b})}{(1+R_{p})^{\frac{1}{n}}-
+ (1+R_{b})^{\frac{1}{n}}}} \deqn{a_{t} =
+ \left(\frac{R_{p}-R_{b}-M\times\sum^{n}_{t=1}(R_{pt}-
+ R_{bt})}{\sum^{n}_{t=1}(R_{pt}-R_{bt})^{2}}\right)\times(R_{pt}-R_{bt})}
In case if portfolio and benchmark returns are equal the
limit of the above value is used: \deqn{M = (1 +
- r_{p})^\frac{n-1}{n}} A_t' - adjusted attribution effects
- at period t, A_t - unadjusted attribution effects at
- period t, R_pt - portfolio returns at period t, R_bt -
- benchmark returns at period t, Rp - total portfolio
- returns, Rb - total benchmark returns, n - number of
- periods
+ r_{p})^\frac{n-1}{n}}{M = (1 + rp)^((n - 1) / n)}
+ \eqn{A_{t}'}{At'} - adjusted attribution effects at
+ period \eqn{t}, \eqn{A_{t}}{At} - unadjusted attribution
+ effects at period \eqn{t}, \eqn{R_{pt}}{Rpt} - portfolio
+ returns at period \eqn{t}, \eqn{R_{bt}}{Rbt} - benchmark
+ returns at period \eqn{t}, \eqn{R_{p}}{Rp} - total
+ portfolio returns, \eqn{R_{b}}{Rb} - total benchmark
+ returns, \eqn{n} - number of periods
}
\details{
The total arithmetic excess returns can be explained in
terms of the sum of adjusted attribution effects:
\deqn{R_{p} - R_{b} =
- \sum^{n}_{t=1}\left(Allocation_{t}+Selection_{t}+Interaction_{t}\right)}
+ \sum^{n}_{t=1}\left(Allocation_{t}+Selection_{t}+
+ Interaction_{t}\right)}
}
\examples{
data(attrib)
Modified: pkg/PortfolioAnalytics/sandbox/attribution/man/Return.annualized.excess.Rd
===================================================================
--- pkg/PortfolioAnalytics/sandbox/attribution/man/Return.annualized.excess.Rd 2012-07-12 09:15:48 UTC (rev 2146)
+++ pkg/PortfolioAnalytics/sandbox/attribution/man/Return.annualized.excess.Rd 2012-07-12 09:16:32 UTC (rev 2147)
@@ -29,8 +29,8 @@
scale by raising the compound return to the number of
periods in a year, and taking the root to the number of
total observations:
- \deqn{prod(1+R_{a})^{\frac{scale}{n}}-1=\sqrt[n]{prod(1+R_{a})^{scale}}-1}{prod(1
- + Ra)^(scale/n) - 1}
+ \deqn{prod(1+R_{a})^{\frac{scale}{n}}-1=\sqrt[n]{prod(1+R_{a})^{scale}}-
+ 1}{prod(1 + Ra)^(scale/n) - 1}
where scale is the number of periods in a year, and n is
the total number of periods for which you have
@@ -40,10 +40,11 @@
benchmark we can compute annualized excess return as
difference in the annualized portfolio and benchmark
returns in the arithmetic case: \deqn{er = R_{pa} -
- R_{ba}}
+ R_{ba}}{er = Rpa - Rba}
and as a geometric difference in the geometric case:
- \deqn{er = (1 + R_{pa}) / (1 + R_{ba}) - 1}
+ \deqn{er = (1 + R_{pa}) / (1 + R_{ba}) - 1}{er = (1 +
+ Rpa) / (1 + Rba) - 1}
}
\examples{
data(attrib)
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