Risk free asset has a standard deviation of 0 ?

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I am looking at PM, page 421, question #4
1. Is this correct: risk free asset has a standard deviation of 0 ?
2. If one of the portfolio’s assets is risk free, then standard deviation is 0, then the correlation is 0 ? correlation = cov / (st A x st B)
3. If one of the portfolio’s assets is risk free, then the covariance is 0 ? covariance = correlation / (stA x st B)
 
Don’t have the curriculum in front of me, but that makes sense. Risk free means that you always get the expected return, no matter what. In other words, the probability distribution has all its mass in the point R=E(R) (meaning P(R=E(R))=1). Variance is zero, so is stdev.
For questions 2 and 3, I’m pretty sure you can argue that the correlation of the returns of two assets A and B, of which A is risk free, is zero. That is because even if R_B moves, R_A just won’t budge, so they are not correlated.
Does that help?
 
> 2. If one of the portfolio’s assets is risk free, then standard deviation is 0, then the correlation is 0 ? correlation = cov / (st A x st B)
If A’s std deviation = 0, then Cov (A,B) = 0. If the COV=0, then the correlation=0.
> 3. If one of the portfolio’s assets is risk free, then the covariance is 0 ? covariance = correlation / (stA x st B)
Not the covariance of the portfolio. Only the COV between those two pairs = 0.
 
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