Calculating Portfolio Standard deviation

Dear R helpers

I have following data

stocks <- c("ABC", "DEF", "GHI", "JKL")

prices_df <- data.frame(ABC = c(17,24,15,22,16,22,17,22,15,19), 
                                         DEF = c(22,28,20,20,28,26,29,18,24,21), 
                                          GHI = c(32,27,32,36,37,37,34,23,25,32),

                                          JKL = c(47,60,60,43,62,38,44,53,61,41))

sd_prices <- c(3.3483,3.9497,4.9721,9.3387)   # standard deviations say(sd1, sd2, sd3, sd4)
                       
weights <- c(0.10, 0.25, 0.20, 0.45)      # say (w1, w2, w3, w4)

I need to calculate the standard deviation of the portfolio. The formula is

stdev_portfolio = sqrt((w1*sd1)^2+(w2*sd2)^2+(w3*sd3)^2+(w4*sd4)^2 + 
                        2*w1*w2*sd1*sd2*correlation(ABC, DEF)+
  
                     2*w1*w3*sd1*sd3*correlation(ABC, GHI)+
                        2*w1*w4*sd1*sd4*correlation(ABC, JKL)+
                        2*w2*w3*sd2*sd3*correlation(DEF, GHI)+
                        2*w2*w4*sd2*sd4*correlation(DEF, JKL)+
                        2*w3*w4*sd3*sd4*correlation(GHI, JKL))
                  
     
                        
OR if we define 

P = sd_prices*weights

I need to calculate 

stdev_portfolio = sqrt((P1)^2+(P2)^2+(P3)^2+(P4)^2 + 
                        2*P1*P2*correlation(ABC, DEF)+
                        2*P1*P3*correlation(ABC, GHI)+
                        2*P1*P4*correlation(ABC, JKL)+
                        2*P2*P3*correlation(DEF,
GHI)+
                        2*P2*P4*correlation(DEF, JKL)+
                        2*P3*P4*correlation(GHI, JKL))
                        
In reality I will be dealing with not 4, but many stocks and hence I can't generalize this as

stdev_portfolio = sqrt((P[1])^2+(P[2])^2+(P[3])^2+(P[4)^2 + 
                        2*P[1]*P[2]*correlation(ABC, DEF)+
                        2*P1*P3*correlation(ABC,
GHI)+
                        2*P1*P4*correlation(ABC, JKL)+
                        2*P2*P3*correlation(DEF, GHI)+
                        2*P2*P4*correlation(DEF, JKL)+
                        2*P3*P4*correlation(GHI, JKL))                        
                        
Kindly advise as to how do I
calculate the portfolio standard deviation?

Thanking in advance

Amelia Vettori



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Dear Amelia,

If you have the actual data you should be able to use the variance covariance matrix to simplify this

Vdat<- cov(prices_df)

sum(diag(Vdat)) + 2*Vdat[upper.tri(Vdat)]

By using covariances instead of correlations you do not need to multiply by he standard deviations and by using variances there's no need to square. The only trick would be adding your weights back in.  See ?diag and ?upper.tri and ?vcov for relevant documentation.

Cheers,

Josh

On Jan 10, 2011, at 0:26, Amelia Vettori<amelia_vettori at yahoo.co.nz>  wrote:

Dear R helpers

I have following data

stocks<- c("ABC", "DEF", "GHI", "JKL")

prices_df<- data.frame(ABC = c(17,24,15,22,16,22,17,22,15,19),
                                          DEF = c(22,28,20,20,28,26,29,18,24,21),
                                           GHI = c(32,27,32,36,37,37,34,23,25,32),

                                           JKL = c(47,60,60,43,62,38,44,53,61,41))

sd_prices<- c(3.3483,3.9497,4.9721,9.3387)   # standard deviations say(sd1, sd2, sd3, sd4)

weights<- c(0.10, 0.25, 0.20, 0.45)      # say (w1, w2, w3, w4)

I need to calculate the standard deviation of the portfolio. The formula is

stdev_portfolio = sqrt((w1*sd1)^2+(w2*sd2)^2+(w3*sd3)^2+(w4*sd4)^2 +
                         2*w1*w2*sd1*sd2*correlation(ABC, DEF)+

                      2*w1*w3*sd1*sd3*correlation(ABC, GHI)+
                         2*w1*w4*sd1*sd4*correlation(ABC, JKL)+
                         2*w2*w3*sd2*sd3*correlation(DEF, GHI)+
                         2*w2*w4*sd2*sd4*correlation(DEF, JKL)+
                         2*w3*w4*sd3*sd4*correlation(GHI, JKL))



OR if we define

P = sd_prices*weights

I need to calculate

stdev_portfolio = sqrt((P1)^2+(P2)^2+(P3)^2+(P4)^2 +
                         2*P1*P2*correlation(ABC, DEF)+
                         2*P1*P3*correlation(ABC, GHI)+
                         2*P1*P4*correlation(ABC, JKL)+
                         2*P2*P3*correlation(DEF,
GHI)+
                         2*P2*P4*correlation(DEF, JKL)+
                         2*P3*P4*correlation(GHI, JKL))

In reality I will be dealing with not 4, but many stocks and hence I can't generalize this as

stdev_portfolio = sqrt((P[1])^2+(P[2])^2+(P[3])^2+(P[4)^2 +
                         2*P[1]*P[2]*correlation(ABC, DEF)+
                         2*P1*P3*correlation(ABC,
GHI)+
                         2*P1*P4*correlation(ABC, JKL)+
                         2*P2*P3*correlation(DEF, GHI)+
                         2*P2*P4*correlation(DEF, JKL)+
                         2*P3*P4*correlation(GHI, JKL))

Kindly advise as to how do I
calculate the portfolio standard deviation?

Thanking in advance

Amelia Vettori



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