in the notes, LOS9.e Explain how taxes affect investment risk the example is a 50/50 bond/stock portfolio, std of bond is 6% and stock is 16%, tax of bond is 30% and stock 20%. Assume correlation of 1.
Pretax std is easily calculated as 11%
However, after tax std is calculated as the simple sum up of after tax std of bond and equity
= 50%*6%*(1-30%)+50%*16%*(1-20%)=8.5
I dont understand , why it is not calculated using the portfolio average tax rate
= 50%*20%+50%*30%=25%
after tax std
=11%*(1-25%)=8.25
It’s calculated as the weighted average of the asset class’ after tax standard deviation.
0.5*(1-0.3)*(0.06)+0.5*(1-0.2)*(0.16) = 8.5%.
You’re trying to simplify the equation which you can do in the case since it’s equally weighted, but the simplification is:
0.5[(0.06*0.7)+(0.16*.8)] = 8.5%.
The tax rates that you’re trying to take the average of don’t impact the volatility proportionately because the total volatilities are different. I.e. by taking a 25% tax to 16% equity volatility (12%) you’re actually understating the after tax volatility, and by taking a 25% tax to the bond volatility of 6%, you’re actually overstating the after tax volatility. Of the total portfolio, equity constitutes a higher portion of overall volatility, so by understating it, your portfolio aftter tax standard deviation of 8.25% is also understated.
Remember that the formula is the sum of the weighted average after tax vol. of asset classes. Try and stick to the formula and weight them accordingly. On the exam we might see a 70/30 split or a 55/45 split and attempting to simplify the equations like you are is only going to make the calculations harder than they need to be!
Happy studying!
what would be a practical example of us needing to calculate the after tax standard deviation of returns?
For a taxable investor, you would want to analyze the post tax return and post tax risk objectives in the ips.