# Why start in a fixed, constant real rate?

Why do we use a fixed, constant real rate?

When others (outside of a tax advantaged account) make the real rate a variable.

In contrast to others, we start with real, purchasing power rate and then convert to nominal. Others start with nominal and then try to convert to real. Those investors who buy nominal interest rate instruments know the nominal rate of return they will receive. But, they cannot know the real rate of return they will receive because it is a variable to them. That is because they cannot predict the rate of change in the consumer price index (CPI-U) which is the usual measure of erosion of purchasing power. Because of this, the realized real rate of return,

rn = ((1 + in ) − 1) / (1 + cn )), defined as the quotient less one of one plus the nominal rate over the quantity one plus the CPI rate, is volatile from period-to-period. The CPI-U volatility makes it so. For them to know the realized real rate they must first obtain the realized rate of inflation. There is no

avoiding this with a fixed nominal rate of interest.

In contrast, investors who buy fixed real rate of interest instruments do not know, and have no reason to care, what the nominal rate of interest is on them. They know that they will receive a contractually guaranteed, constant rate of real interest. Their expected real return will equal their realized real rate of return.

Nominal Interest Rate = Real Interest Rate + Expected Inflation Rate + Risk Premium

The real interest rate is the true, purchasing power, interest rate.

By going from the real rate to the nominal we have the corresponding calculated nominal rate of interest in = cn + rn + cn× rn where rn is the contractual fixed, real rate. This calculation picks up the cross product term cn× rn from the calculation rn = (1 + rn)(1 + cn).

Others work in the opposite direction, calculating the real rate as x# = in + cn. This is not precise. It is in error because it ignores the cross product term. The effect of this can have a significant cumulative effect over the life of a contract.

In summary, we use a fixed real rate, with the corresponding nominal being variable. Others take the nominal as fixed and calculate a corresponding real rate that is variable.