# Convert Annual rates to quarterly rates

Hi.
In many DSGE models we use quarterly time series but some information of the model for calibration are in annual rates.For example depreciation rate or average annual steady-state interest rate.
I want to know how we can convert annual interest rate to quarterly rates ?
For example suppose that in the calculation of discount factor \beta average annual steady state growth rate is 5 percent. How we can convert this rare to quarterly interest rate ?
Can we employ the following formula?

1.05=(1+r)^4
1.05^{0.25}=1+r

Then we can calculate discount factor.

\beta=\frac{1}{1+r}=\frac{1}{1.05^{0.25}}

Another problem is calculation of quarterly depreciation rate with annual depreciation rate.
Suppose that annual depreciation rate is 10 percent. How we can calculate quarterly depreciation rate ?
Can we use the following formula?

1.1=(1+\delta)^4
1.1^{\frac{1}{4}} -1 = \delta

and \delta is quarterly depreciation rate.

The geometric mean is the correct approach. But for depreciation, it should be
(1-\delta)^4=(1-\delta_{ann})\Leftrightarrow \delta=1-(1-\delta_{ann})^{\frac{1}{4}}

Thanks professor for your comment. But my approach for calculation of discount factor or \beta is correct ?

Your second formula for depreciation rate should be

\delta=1- (1-\delta_{annual})^{\frac{1}{4}}

1. Yes, if there is 0 inflation.
2. My mistake. You are correct. I fixed the formula.