The nominal dollars in the account after twenty years, with a 3.8% real annual return compounded quarterly, would be approximately $22,416.12. The real return is adjusted for inflation, so the future value calculation does not need another adjustment for inflation.
To calculate the nominal dollars you will have in your account at the end of twenty years, we must account for both the real return on your investment and the effect of inflation. The formula for future value with compound interest is FV = PV * (1 + r/n)nt, where FV is the future value, PV is the present value, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.
First, we'll calculate the future value given a real annual return of 3.8% with quarterly compounding:
Future Value with Real Return (FVR) = $11,800 * (1 + 0.038/4)4*20
Next, since the real return is already adjusted for inflation, we should account for the nominal return. However, the nominal return does not need to be calculated separately as the real return provided in the problem is the return after inflation has been accounted for.
To find the nominal account balance at the end of twenty years, you simply use the future value calculation given above with the real return rate. You do not need to adjust further for inflation because a real return already factors in inflation. Let's avoid rounding off during intermediate steps and use at least six decimal places, and we get $22,416.12.
Therefore, the nominal dollars in the account after twenty years, with a 3.8% real annual return compounded quarterly, without any further adjustments for inflation, would be approximately $22,416.12.