Volatility

[2pac Shakur]

I am given a scenario set with monthly rates, various asset classes. I want to calculate the volatility for the 1 year, 5 year, 10 year, and 30 year returns. Do I annualize each scenario by given duration, then calc the variance/standard deviation? What's the correct methodology for calculating what the finance people will call "volatility". It seems kind of difficult to get a straight answer on this.

Thanking you in advance,
2pac Shakur

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[2pac Shakur]

I knew I should've used my alternative ID.
Professor Trump gets respect.

(Can you help me out? I know that you know, or at least your famous Uncle might know. Maybe you can call him for me. Pretty please?)

[bm1729]

It's always a treat to see you wander into a non-Cyberchat part of this forum, my friend. Drop by more often, won't ya.

This isn't my area of expertise here, Tupe, but my course 6 readings have given some cockiness that I might know what I'm talking about when it comes to some of these finance and investment-related matters.

Let me see if I can help you out here...

Volatility is defined as the standard deviation of the annual return. You said you have monthly rates, correct? Then you want to first annualize them by compounding them over twelve months. This will give you a set of annual rates. From there, it's just a simple matter of calculating the standard deviation of the annual returns. I gather you have to do this four times -- for the 1-year rate, the 5-year rate, the 10-year rate, and the 30-year rate.

Do I annualize each scenario by given duration, then calc the variance/standard deviation?By "duration," do you mean the length of the term of the investment (e.g., the 10-year rate has a duration of 10 years)? If so, no. Just annualize them to become yearly rates. (I know, it seems redundant to say that -- "annualize them to become yearly rates," but I'm just using the term annualize because you did).

Oh, and don't use the term duration to mean that. Duration has another meaning in the investments field.

Well, I hope that helps.

[2pac Shakur]

Volatility is defined as the standard deviation of the annual return. You said you have monthly rates, correct? Then you want to first annualize them by compounding them over twelve months. This will give you a set of annual rates. From there, it's just a simple matter of calculating the standard deviation of the annual returns. I gather you have to do this four times -- for the 1-year rate, the 5-year rate, the 10-year rate, and the 30-year rate.


What do you mean do it 4 times - for the 1 year, 5 year, 10 year, and 30 year? For example, on the 5 year, do I annualize the 5 year rate on each scenario, then take the standard deviation of that over each scenario?

(I know duration has another meaning - first derivative/slope of price wrt interest rates, I was just lazy in my posting.)

[bm1729]

What do you mean do it 4 times - for the 1 year, 5 year, 10 year, and 30 year? For example, on the 5 year, do I annualize the 5 year rate on each scenario, then take the standard deviation of that over each scenario?I understood your original post to mean that you had four different sets of interest rates -- each set applying to a different asset class: the 1-year, 5-year, 10-year, and 30-year asset class. (Are we talking about fixed income securities here?)

Did I misunderstand you? Maybe you can explain in more detail the data that you have to work with. I hate being of absolute no help to people.

[2pac Shakur]

I did a poor job of explaining. Of course, I'm not really a people person. Anyway...

I get N scenarios projecting out for Y years with A asset classes, each scenario has 12Y monthly rates for each of the A asset classes.

I want to calculate the volatility of the return for years (1-10).
What are the steps? It sounds like I should:

A. Calculate a total 10 year return for each scenario.
B. Annualize that return for each scenario. (take it to the 1/10 power)
C. Calculate the variance and standard deviation of the resulting N 10 year annualized returns.
D. Standard deviation is what is called the volatility.

If we wanted to check how good these scenarios are, we could compare the calculated volatility to the implied volatility from the market price of options that have a 10 year time horizon, like a 10 year S&P put. (for certain asset classes)

Correct?

[Colymbosathon ecplecticos]

Could you please show me a liquid market for ten year S&P puts?

[bm1729]

Sorry 2pac, more questions coming (I'm a bit slow tonight)...

I get N scenarios projecting out for Y years with A asset classes, each scenario has 12Y monthly rates for each of the A asset classes.Are these monthly rates that you speak of forward rates?

I want to calculate the volatility of the return for years (1-10).What about the 30-year return? You don't care about that anymore :-?

What are the steps? It sounds like I should:

A. Calculate a total 10 year return for each scenario.
B. Annualize that return for each scenario. (take it to the 1/10 power)
C. Calculate the variance and standard deviation of the resulting N 10 year annualized returns.
D. Standard deviation is what is called the volatility.Steps A and B don't seem right to me.

Anyway, gotta run, 2pac. We'll resume this conversation tomorrow, which by then I'm sure someone would have come by to help you out. :D

Sorry I wasn't much help, dude.

[RobertNelson]

Volatility is quoted on an annul basis. If you have monthly numbers, then convert using the well known formula
Annual vol= monthly vol * SQRT(12)

In your OP, you referenced time frames -1, 5, 10 and 30 years. In usual capital market calculations, I think you are after the analog of Historic Vol. If you went to a Bloomberg and asked for Historic vol on a security, you would have to define the observation period. Some folks use 30 days, some folks use 5 years. It's up to you. The time frames you cited should probably refer to how many observations are in your calculation of the variance in the same way. If you do something else, like the nth root, then it is peculiar to your own purpose, and shouldn't be compared to vol quotes on options in the marketplace..

You DO NOT look at ten-year returns for the ten year vol. Decide now whether you intend ten year vol to mean "10 years worth of observations went into my calculation of the variance" or,you used the formula;

Ten year vol = 1 year vol * SQRT (10)..just like the monthly formula above.

The second definition is probably what most traders and cap market folks will interpret to be your meaning, fyi.

Since you are doing a calculation for each scenario, you will get a Vol for each scenario. You should end up with a Historic Vol for each asset. You can, I suppose, create one big old string of numbers, from all the scenarios and calculate that vol. Or you could just do an analysis of the 1000 odd vols recorded by scenario. You know, High, low, and average. That sort of thing.

Lastly, the standard practice is to compare the returns (change in price plus cash flow) in a period to the risk free rate. Commonly LIBOR is used as the risk free rate, instead of UST. But pick either one to suit your purpose. The vol is actually the difference between the short rate and the actually returns, not the absolute value of the returns. A lot of folks forget to do this, and they get very skewed results. So before getting your sequence of observation, be sure to deduct the "then monthly RFR" from the calculated returns. This is what a few other posts are getting at when they talk about the forward rates, I believe. Hopefully your model has a representation of the one month LIBOR or one month UST rate. If not, add it and all will be well.

Hope this helps.

[Ms. Re]

I am given a scenario set with monthly rates, various asset classes. I want to calculate the volatility for the 1 year, 5 year, 10 year, and 30 year returns. Do I annualize each scenario by given duration, then calc the variance/standard deviation? What's the correct methodology for calculating what the finance people will call "volatility". It seems kind of difficult to get a straight answer on this.

Thanking you in advance,
2pac Shakur

volatility is the standard deviation of logarithmic price changes...volatility is typically expressed as an annual number, and since std deviation is proportional to the sqrt of time, if you have monthly data, you multiply the std deviation of the monthly logarithmic price changes by the sqrt of 12

[Kenshiro]

The vol is actually the difference between the short rate and the actually returns, not the absolute value of the returns.

Uhh...what?

[RobertNelson]

I made a couple of presumptions about the OP. You're correct to call me on that point. If you are simply calculating the historic vol from actual data, just use the formulas without any subtraction, like on page 254 of Hull's textbook Futures and Options Market. My comment pertained to model results and I inferred a desire to calibrate the data.

Asset models will involve the relationship between asset prices from one time period to the next. Something like:

Price[t+1]= Price[t] * (rfr + risk Premia + N(0,1)* sigma)

where rfr is the risk free rate and N(0,1) is a random variable, and sigma is the volitility. (Make the obvious corrections to this formula for either discrete or continuous time, but I am too lazy to add the subscrpits and such)

If you modeled asset prices without the rfr, then there isn't anything stochastic about the volatility. The model result will be the same as the input. So the OP is pointless, since the output=input. Its just a matter of getting enough scenarios so that the sample approaches the input value.

Whether you use a risk premia or not, doesn't matter. If you add a constant in all periods, then it has no effect on the volatility.

So the sources of variation are in the rfr and the N(0,1) term. I always subtract out the rfr part so that I can get a nice clean read on my results.

If you do not use the rfr in your asset price derivations, then omit that part completely. I always include it, because it makes the calibration of the model a bit easier. (calibrate =does my model replicate the prices of a 5 year bond, the swap rates, or whatever security I am using as a benchmark)

[Old Actuary]

There's an article in the latest issue of Risks and Rewards that discusses how to calculate all the summary statistics for a stochastic forecast