This is not a simple question to answer, unfortunately. Different goods had drastically different price ranges back then due to differences in supply and technology. Since the prices of goods are the only real metric by which we can compare currencies, this means you could get a wide range of values.

Doug Smith claims that, specific to the price of bread, a denarius would be worth perhaps $20 USD. Comparing clothing, on the other hand, would result in a much higher value. Put simply, prices for everything were different back then, so there's no objective way to compare the currencies.

The others have covered the difficulties of converting Roman coinage. I would add one key question to your post - what time? The value of the denarius varied over time. Under Augustus the coin was comprised of 3.85g of silver but would begin to be debased starting with Nero (having to pay for the rebuilding of Rome after the Great Fire in 64 CE). By the 4th CE it had been so heavily devalued that inflation was rampant. Eventually Diocletian and Constantine just abandoned the silver coin and issued a new gold coin. So a denarius issued under Augustus has more value (based on amount of silver) than one under Aurelian which may only have a fraction of silver inside.

Generally classical economists abandon any attempts to work out a value equivalency but instead work out amounts using a "wheat equivalent" since we can often work out the how much a modius of wheat cost.