converting nominal stock returns into real stock returns

[Lisa Wilson]

Hi

This is a basic question but thought someone in macroeconomics or finance (or any related discipline) may help provide an appropriate answer.

I understand that converting a nominal level series into a real level series (i.e., price adjusted) can be done by deflating the nominal series

gen real= nominal/price_deflator

However, if the series is nominal stock returns and I want to convert it into real stock returns, would it make sense to simply subtract inflation:

gen inflation=(price_deflator-L.price_deflator)/L.price_deflator

gen real_stock_return=nominal_stock_return - inflation

I hope someone can help

[Joro Kolev]

Hi Lisa,

Note that the first expression that you use results in something that we do not know in what units it is. So it is better to deflate in levels more like this, say you want everything to be in 2021 year real units. Then

(1) RealPrice{t} = (NominalPrice{t}/Deflator{t})*Deflator{t=2021}.

Regarding your question, yes, it makes sense to deflate returns by subtracting the inflation rate from the nominal return, and we do it in finance all the time. For example the expression (MarketReturn - RiskFree) appears all the time in finance.

A) I think this is an approximation, but an accepted approximation. I think the exact expression is RealReturn = (Gross Nominal Return/Gross Inflation Rate) - 1. Lets call the price P, and the deflator D, and disregard the multiplication by a constant that I showed in eq(1) because it does not matter in returns. Then the real return by definition is

(2) Real Return = (Pt/Dt - Pt-1/Dt-1)/(Pt-1/Dt-1) = (Pt/Pt-1)/(Dt/Dt-1) - 1, which is Gross Nominal Return divided by Gross Inflation Rate minus 1.

When I start from the expression that you are asking for, which written out is

(3) [(Pt - Pt-1)/Pt-1] - [(Dt - Dt-1)/Dt-1] = Pt/Pt-1 - Dt/Dt-1

and as you can see the right hands sides of eq.(3) and eq.(2) are not exactly equal. Lets try some numbers, say nominal return is 0.12 and inflation is 0.05. Then the exact expression eq.(2) gives us

(1.12/1.05) - 1 = 0.0666.

The approximate expression in eq.(3) gives us

0.12 - 0.05 = 0.07, about which approximation the popular among financial economist Greek philosopher Mediocrates might have said "Eeeeh, good enough..."

In short, what you are doing is an acceptable approximation which is used all the time in finance and macroeconomics.

B) When you are doing this be careful to match the time convention properly. Say returns might be calculated both as (Pt - Pt-1)/Pt-1, or forward as (Pt+1 - Pt)/ Pt. So be careful that the returns timing convention matches the inflation timing convention.

[Lisa Wilson]

Thanks a lot Joro. I find this very useful indeed.

Regarding

(1) RealPrice{t} = (NominalPrice{t}/Deflator{t})*Deflator{t=2021}.

,
This means that the real price will be in 2021 terms. Therefore, if I want to get the real values, say in 1990 terms I use RealPrice{t} = (NominalPrice{t}/Deflator{t})*Deflator{t=1990}. Indeed, I did not know that we should also multiply by the deflator for a selected year. Is it incorrect if I do not multiply by the deflator for the selected year?

Regarding

For example the expression (MarketReturn - RiskFree) appears all the time in finance

,
This will be excess returns (or equity premium), right? For real returns, I will subtract inflation and not the risk-free rate. Do you agree?

Thank you again

[Joro Kolev]

"Incorrect" is a big word... When you only divide by the deflator you still get some real values, but it is not clear what is the meaning of these real values, that is, it is not clear in what unit of measurement they are expressed. They still can be used for some things and we already saw an example where it does not matter whether you multiply or not -- if you calculate returns the constant multiple drops out.

But if you want for example to interpret a raw number Price/Deflator, and lets say that Price is in USD, and you get Price/Deflator = 666, we do not know what is this 666.

On the other hand if we calculate (Price{t}/Deflator{t})*Deflator{t=1990} = 666, then we know that our quantity is in 666, year 1990 US dollars.

And yes, in your case you subtract inflation

Real Return ≅ (Nominal Return) - (Inflation Rate).

where ≅ means "approximately equals".

Thanks a lot Joro. I find this very useful indeed.

Regarding ,
This means that the real price will be in 2021 terms. Therefore, if I want to get the real values, say in 1990 terms I use RealPrice{t} = (NominalPrice{t}/Deflator{t})*Deflator{t=1990}. Indeed, I did not know that we should also multiply by the deflator for a selected year. Is it incorrect if I do not multiply by the deflator for the selected year?

Regarding ,
This will be excess returns (or equity premium), right? For real returns, I will subtract inflation and not the risk-free rate. Do you agree?

Thank you again