Convert Annual rates to quarterly rates

General DSGE Modeling
1

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.

2

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}}

3

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}}

4
  1. Yes, if there is 0 inflation.
  2. My mistake. You are correct. I fixed the formula.