no - I kept all numbers constant.
If you visit our site, you will find that I do not defend the system. It's economic returns are terrible, and getting worse. The problem that I have with privatization plans is the overstatement of what they can do. This claim : a low wage worker will do better in the private market even at the lowest rate of return even cashing out at a market bottom is off the charts.
In you case, you are using 10% which is more than what is withheld for SS today (8.6%). Your worker is not poor - on an inflation adjusted basis he looks more like a high-wage worker who took-up a low-paying hobby. This person's results will be the worst possible for SS. The economic returns that you use are almost the highest in the last 100 years. They are in fact higher than my sources say occurred in the 20th century - Crestmont takes out transactions fees. Mind you we aren't even touching the idea that wage-weighted investment will materially underperform the market averages. We aren't looking at the possibility that the low wage worker is a bad investor - well other than cashing out at a market bottom. We are ignoring the EITC which repaid Joe for much of his contribution.
To your claim: we will say 10% withheld for a workers whose wage rose from $25,000(2012 dollars) to 34,670 in 2012. Assuming a 4.92 rate of return before a crash of 40% (the trough was actually 60% fall). Assuming 6% unemployment. At the end of 2012, he would have about $153,000. Assuming the crash occurred, at that point, he would have about $90,000 or a monthly payment of $225 a month at 5% interest. All of this ignores the fact that there is a cap on wages and the fact that you worker would have to buy life insurance that comes free with SS. This doesn't mean that SS was a good deal. You need to look at what the value of a $1,000 annuity inflation protected with survivor benefits was worth at the age of 65. It may well not be $90,000. Given that the life expectancy of a retiree is about 17 years my guess is that it is more than 90K.
I could very well question any number of your claims. Your figures increase SS portion of payroll taxes by 2.4% - entirely to fund a personal account. Without payroll taxes, existing benefits would exhaust the Trust Fund in about three years. So Joe will be worse off to the extent that the Federal Government has to increase debt to fund benefits to existing people. Joe will have to pay that debt over his lifetime. In a worst case example, the Joe's portion of the increased debt is more than his entire Personal account.
This doesn't say that SS is better than a privatized system. If the results today aren't superior for every case, they will be at some point. The economic returns of Social Security have fallen every year since its creation. The man who ran the system in 1944 told Congress, if you do not raise taxes now, the hidden costs of Social Security will force future generations to increase payroll taxes to a rate at which they would be able to get comparable insurance at a lower price from a private business. His testimony is an interesting read.