Return on Equity Variance Analysis

[rerodgers]

In finance, I often analyze the rate or volume drivers of variances. I can
do it with a straight dollar variance, but I am trying to do it on a ratio,
like return on equity which is (Net Income/Average Equity). When I compare
two company's ROE %'s, I can explain how much of the variance is due to the
delta in net income and how much of the variance is due to the delta in
equity. I can also determine how much of the net income & equity deltas
are due to rate (i.e. profitability) changes and how much is to do
size/volume changes. I can then determine how much of the ROE change is
driven by the numerator, net income, and then break that down by rate &
volume changes in net income. I would like to do the same for the
denominator, equity. How do I calculate the rate and volume impact on a
denominator of a ratio. See my example below:

Variance Analysis
Co. A Co. B Variance Rate Volume Total
Net Income 50 150 100 25 75 100
Average Equity 500 1,000 500 100 400 500
ROE% 10.00% 15.00% 5.00%

Variance Analysis
ROE% Variance Analysis variance Rate Volume Total
Variance due to NI 20.00% 5.00% 15.00% 20.00%
Variance due to Equity -15.00% X? Y? -15.0%
ROE Variance 5.00% sum sum 5.00%

Each companies ROE is caclulated by dividing the net income by average
equity.
Assume that I have been able to calculate the amount of the variances due
to rate changes and volume changes as shown above.
The formula for calculating the variances are as follows:
20% Variance due to NI=Variance due to net income (100) divided by CO. A's
equity (500)
5% variance in NI due to rate=variance due to rate (25) divided by CO. A's
equity(500)
15% variance in NI due to volume=variance due to volume (75) divided by Co.
A's equity (500)
-15% variance due to equity=CO. B net income (150) divided CO. B equity
(1000) minus CO. B net income (150) divided by CO. A equity (500)
How do I calculate X & Y listed above? I am basically trying to
determine how to apportion the total variance due to the changes in equity
by rate and volume driven factors.

 

[JakeD]

Here is a way to apportion total variance so it is the sum of the indivdual variances. Let \(\displaystyle f(X,Y) = X/Y\) be ROE as a function of net income \(\displaystyle X\) and equity \(\displaystyle Y.\) The variance in income is \(\displaystyle X_2 - X_0\) which you have apportioned between variance due to rate \(\displaystyle X_1 - X_0\) and variance due to volume \(\displaystyle X_2 - X_1.\) Thus \(\displaystyle X_2 - X_0 = (X_2 - X_1) + (X_1 - X_0).\) Likewise for equity \(\displaystyle Y_2 - Y_0 = (Y_2 - Y_1) + (Y_1 - Y_0).\)

The total variance in ROE can be written

\(\displaystyle \begin{array}f(X_2,Y_2) - f(X_0,Y_0) &= (f(X_2,Y_2) - f(X_2,Y_1)) + (f(X_2,Y_1) - f(X_2,Y_0)) \\
&+ (f(X_2,Y_0) - f(X_1,Y_0)) + (f(X_1,Y_0) - f(X_0,Y_0)), \end{array}\)

and this is one way to apportion the total variance so it is the sum of the individual variances. For example, the variance due to the equity rate driver is \(\displaystyle f(X_2,Y_1) - f(X_2,Y_0),\) which can be converted to a percentage of the total variance.

Notice the variables are changed one at a time in a fixed, but arbitrary order. Changing the variables one at a time guarantees that the variances add up. Unfortunately, changing the order changes the variances slightly and there is no natural order to choose. But this method does apportion the total variance so it is the sum of the individual variances in a simple way.

 

[rerodgers]

Thanks for the reply. So, just to make sure I got it right. Please check my math below. I really appreciate your help.

Here is what I did, given your formulas and my example data from above:

To calculate the net income variance due to rate and volume I did the following. The total net income variance is 20% with the rate proportion equal to 5% = (50+25)/500 - 50/500 and the volume proportion equal to 15% = 150/500 - (50+25)/500.

To calculate the total equity variance due to rate and volume I did the following. The total equity variance is -15% with the rate proportion equal to -5% = 150/(500+100) - 150/500 and the volume proportion equal to -10% = 150/1000 - 150/(500+100).

It works, but I just want to make sure I am right.

 

[JakeD]

Thanks for the reply. So, just to make sure I got it right. Please check my math below. I really appreciate your help.

Here is what I did, given your formulas and my example data from above:

To calculate the net income variance due to rate and volume I did the following. The total net income variance is 20% with the rate proportion equal to 5% = (50+25)/500 - 50/500 and the volume proportion equal to 15% = 150/500 - (50+25)/500.

To calculate the total equity variance due to rate and volume I did the following. The total equity variance is -15% with the rate proportion equal to -5% = 150/(500+100) - 150/500 and the volume proportion equal to -10% = 150/1000 - 150/(500+100).

It works, but I just want to make sure I am right.

You've got it. Though the results look good, the raw formulas do look strange at first glance. So don't show them to anyone. :wink:

I'm glad I could help.

 

[Denis]

I remember (years ago, when I worked for a financial institution) making up a
"Variance Report" for each major products.
The analysis was quite similar to what you're doing: the differences in the actual
and budgeted interest revenue were broken down between volume and rate.

To show a simple example, this could be (in 000's) the analysis of a certain loan category:

         ACTUAL                         BUDGET                     VARIANCE
Average Interest Rate          Average Interest Rate          Net : Volume  Rate 
 2000     180     9%            2400     192     8%           -12 :   -36   +24

Budgeted (or projected) was 2.4 million as an average for the year, for a revenue
of 192 thousand, so an average rate of 8%.

The year ended up with actuals of 2.0 million / 180 thousand / 9%.

This was a shortage in revenue of 12 thousand, made up of -36 thousand due
to a shortage of 400 thousand in size, but +24 thousand due to higher rate.

ONLY one "real calculation" was made:
180 / 2000 * (2000 - 2400) = -36 : the variance due to size

The variance due to rate was simply defaulted: -12 - (-36) = 24.

OK; you stated this:
> To calculate the net income variance due to rate and volume I did the following.
> The total net income variance is 20% with the rate proportion equal to
> 5% = (50+25)/500 - 50/500 and the volume proportion equal to
> 15% = 150/500 - (50+25)/500.

I don't see the need for the 2nd (volume proportion) calculation, as it can also
be simply defaulted; or did I miss something?

Hope this makes sense :wink:

 

[rerodgers]

You are right. If you know the total variance and you know one piece there is no need to calculate the second piece, because it is the difference between the total variance and the rate variance that you calculated.

Rate and volume variances are a classic cost/manufacturing accounting convention to shed more insight into variances. It is a very useful tool to understand what drove variances other than just knowing that you have a total cost/profit/revenue/etc. variance. I use it a lot in finance and I teach it to our junior analysts as a way to peel back the onion. One slight variation on the analysis is to incorporate a third element, mix. So you can then break down the total variance by changes in rate, changes in volume, and changes in mix (i.e. products, labor, etc.). Here is a summary of how to calculate the 3 variance drivers. Assume just like in the example above, you have an actual interest rate being compared to a budgeted interest rate given actual and budgeted interest and assets.

1) The volume variance is calculated by taking the difference in asset sizes/volume and multiplying it by the total budgeted interest rate % for the entire portfolio of assets.

2) The rate variance is calculated by taking the difference in interest rates and multiplying it by actual asset size/volume for each asset class.

3)The mix variance is calculated by taking the difference in % of total assets/volume that each asset class represents between the actuals and budgets multiplying that by the total actual asset/volume and then multiplying that by the difference in that budgeted interest rate % for the asset class and the total budgeted interest rate % for all asset classes.

As a result, you can know determine how much of the total interest variance is due to changes in asset class interest rates, the size/volume of all the asset classes and finally, the mix of the different type of assets.

I do this type of analysis all the time, the issue I had is that I ran into trouble trying to calculate the rate and volume variances on a ratio like ROE. It adds a slight complexity to the analysis because you have changes in the numerator and denominator that can effect the ratio.

Thanks for the replies. I ran into this problem about a year ago and left it and it occured to me to ask the question on the web in order to have someone better at math than me solve it.

Thanks.