## Lognormal Inflation Model in Monte Carlo Retirement Planning

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Topic Author
K72
Posts: 35
Joined: Wed Dec 05, 2018 8:04 pm

### Lognormal Inflation Model in Monte Carlo Retirement Planning

New user, first post. From the posts I've read this looks like a great forum. Here goes. I've gained a lot of insight using Monte Carlo simulations for retirement planning, especially the impact of volatility. I've used one online tool and have had two different financial planners with access to commercial tools create financial plans. In every case I've been surprised (and disappointed) that for the most part, inflation is treated as a fixed value for the duration of the simulation. One was with a well known financial institution (when I inquired about how to include volatility with inflation all I was told was that I could request any fixed value I wanted for inflation). Another was with a private financial planner who had a fancy tool with flashing arrows showing movement of \$ from one bucket to another but he wasn't able to explain anything at all about what the tool actually does. The online tool used either historical inflation (but no detail about what historical meant) or you could input a mean and standard deviation (for a normal distribution most likely but it wasn't clear).

Needless to say I decided to try it on my own. Plotting CPI values for the past 80 years (maybe less I don't remember exactly), the histogram sure looked lognormal to me so I went ahead and made my own Monte Carlo model in Excel using a normal distribution for investment return and lognormal for inflation (parameters were determined via least squares). I used separate random values for each computation. Using a normal distribution for inflation doesn't make sense to me because in reality we haven't seen many instances of deflation. What I did was cap the upper limit at 16% (I remember too well the early 80's) with no possibility of deflation. I did correlation analysis between CPI and S&P 500 return and found very weak correlation at best.

Even though my inflation model may be overly pessimistic, I'm actually quite happy with the result but I want to be cautious about thinking I'm doing something sensible if I'm really not. Does this approach seem reasonable? I'd appreciate comments either way. Thanks!

kolea
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### Re: Lognormal Inflation Model in Monte Carlo Retirement Planning

No, it does not seem reasonable. Inflation is not an independent process from equities or bonds. They both respond to what inflation is doing, although somewhat differently. For a MC simulation, the key is not finding the right PDF model for inflation, it is finding the right correlation between inflation and bonds and equities. That is not to say that treating inflation as a constant is the right thing to do either, after all, the historic nominal PDF for equity returns has a built-in dependence upon inflation. If I were writing a MC simulation, I would keep inflation a constant but use a real PDF for equity returns and a real PDF for bond yields. But even that may be flawed since equities, bond yields, and inflation all have memory, i.e., they have a non-zero auto-correlation. We see this as momentum in equities. Similar for bonds. It is really tricky to create a MC simulation that correctly models reality. That is why I don't trust them.
Kolea (pron. ko-lay-uh). Golden plover.

AlohaJoe
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### Re: Lognormal Inflation Model in Monte Carlo Retirement Planning

Ken72 wrote:
Wed Dec 05, 2018 11:23 pm
Using a normal distribution for inflation doesn't make sense to me because in reality we haven't seen many instances of deflation.
I don't understand this sentence. If I have a normal process with mean of 15% and standard deviation of 2% it will never go below 0 but will still be a normal distribution. Whether a distribution is normal or not is unrelated to whether values ever go negative. That said, inflation has been repeatedly shown to fail tests for normality, since it has significant skewness & kurtosis. (Here is one of many entries on the subject: https://www.nber.org/papers/w4837.pdf) That said, pretending it is normally distributed is also probably okay-ish, depending on what you're trying to learn from the model.
What I did was cap the upper limit at 16% (I remember too well the early 80's) with no possibility of deflation.
Since there have repeatedly been periods of deflation (more common than high inflation, for that matter), it seems weird to eliminate the possibility.

Anyway, for a model it is fine-ish. It's not really any worse than assuming that equity returns are normal or lognormal. Or that there is no correlation between things. Or that there's no time-varying correlations. Or that there's no short-term momentum or long-term mean reversion. And all the other simplifying assumptions models make.

randomguy
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### Re: Lognormal Inflation Model in Monte Carlo Retirement Planning

AlohaJoe wrote:
Thu Dec 06, 2018 12:36 am
Ken72 wrote:
Wed Dec 05, 2018 11:23 pm
What I did was cap the upper limit at 16% (I remember too well the early 80's) with no possibility of deflation.
Since there have repeatedly been periods of deflation (more common than high inflation, for that matter), it seems weird to eliminate the possibility.

Anyway, for a model it is fine-ish. It's not really any worse than assuming that equity returns are normal or lognormal. Or that there is no correlation between things. Or that there's no time-varying correlations. Or that there's no short-term momentum or long-term mean reversion. And all the other simplifying assumptions models make.
he remembers the 80s. Very unlikely anyone remembers the long deflationary periods in US history

I agree fixed inflation is bad. The question is this remotely better or are we just spewing out more numbers. I haven’t actually looked at I would expect a pretty strong link between bond returns and nominal inflation. Stocks to some extent also.

AlohaJoe
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### Re: Lognormal Inflation Model in Monte Carlo Retirement Planning

randomguy wrote:
Thu Dec 06, 2018 12:57 am
AlohaJoe wrote:
Thu Dec 06, 2018 12:36 am
Ken72 wrote:
Wed Dec 05, 2018 11:23 pm
What I did was cap the upper limit at 16% (I remember too well the early 80's) with no possibility of deflation.
Since there have repeatedly been periods of deflation (more common than high inflation, for that matter), it seems weird to eliminate the possibility.

Anyway, for a model it is fine-ish. It's not really any worse than assuming that equity returns are normal or lognormal. Or that there is no correlation between things. Or that there's no time-varying correlations. Or that there's no short-term momentum or long-term mean reversion. And all the other simplifying assumptions models make.
he remembers the 80s. Very unlikely anyone remembers the long deflationary periods in US history
But why is "long periods" and "US history" the important parts? Much of Europe & Japan has had deflationary periods for the past decade, as has been much discussed over the past years. Why pretend that can never ever possibly happen in the US when modeling a multi-decade period? Especially when we had deflation as recently as March 2018? There's also been deflation in December 2017, October 2017, July 2017, November 2016, July 2016. Is it so hard to imagine that those short periods might stretch from a few months to 12 months? Meanwhile, where are the occurrences of inflation being 16%? The highest annual inflation ever got in the US was 13%. So why allow inflation to go higher than it ever has but not allow deflation to even reach the levels it actually did?

It just doesn't seem like an honest, impartial decision. It seems like one that is "I'm scared of inflation so I'm going to bias my model towards it". Which is fine. Models are made for that kind of exploration. But then you start getting on tricky ground when ask on the internet about how realistic your model is.

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### Re: Lognormal Inflation Model in Monte Carlo Retirement Planning

New member too. Great forum. Agree that zero correlation is very simplistic especially with respect to interest rates. I would assume in your model equity vol is much higher than inflation vol so pair correlation should matter less there.
Modeling real return as mentioned it’s a good alternative - (assuming normal distribution for real rates as they do go negative)
Regarding CPI distribution I would go for a shifted log normal to allow for sone negative inflation reading while maintaining a fatter tail on the right.

On the other hand constant inflation is even worse. Out of interest, what MonteCarlo tool are you using?

Valuethinker
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### Re: Lognormal Inflation Model in Monte Carlo Retirement Planning

Ken72 wrote:
Wed Dec 05, 2018 11:23 pm
New user, first post. From the posts I've read this looks like a great forum. Here goes. I've gained a lot of insight using Monte Carlo simulations for retirement planning, especially the impact of volatility. I've used one online tool and have had two different financial planners with access to commercial tools create financial plans. In every case I've been surprised (and disappointed) that for the most part, inflation is treated as a fixed value for the duration of the simulation. One was with a well known financial institution (when I inquired about how to include volatility with inflation all I was told was that I could request any fixed value I wanted for inflation). Another was with a private financial planner who had a fancy tool with flashing arrows showing movement of \$ from one bucket to another but he wasn't able to explain anything at all about what the tool actually does. The online tool used either historical inflation (but no detail about what historical meant) or you could input a mean and standard deviation (for a normal distribution most likely but it wasn't clear).

Needless to say I decided to try it on my own. Plotting CPI values for the past 80 years (maybe less I don't remember exactly), the histogram sure looked lognormal to me so I went ahead and made my own Monte Carlo model in Excel using a normal distribution for investment return and lognormal for inflation (parameters were determined via least squares). I used separate random values for each computation. Using a normal distribution for inflation doesn't make sense to me because in reality we haven't seen many instances of deflation. What I did was cap the upper limit at 16% (I remember too well the early 80's) with no possibility of deflation. I did correlation analysis between CPI and S&P 500 return and found very weak correlation at best.

Even though my inflation model may be overly pessimistic, I'm actually quite happy with the result but I want to be cautious about thinking I'm doing something sensible if I'm really not. Does this approach seem reasonable? I'd appreciate comments either way. Thanks!
Do you escape this problem by using real returns?

At least on an annual basis, CPI and S&P 500 would have very weak correlation. Dimson & Marsh in one of their Credit Suisse presentations (available online) has a slide on that. My guess would be that rising inflation actually has a negative correlation with S&P 500 (a guess).

(I think the intellectual confusion comes because of this: nominal bonds very clearly have a negative correlation with inflation, especially rising inflation - it's bad for bondholders; stocks pay high real returns but that does not mean inflation is good for stock returns (rising inflation in particular) because it tends to bring higher interest rates, which are generally bad for stocks (directly due to the alternative use of capital, indirectly due to the impact on the economy in general) also there is an "inflation drag" in corporate profits and taxes (inflation causes an inflation of accounting and taxable profits, but not in an economic sense).

It's very easy to confuse the fact that stocks pay high real returns with stocks as an inflation hedge. The important thing about stock returns is that they are *volatile* - that real return is paid for with a lot of risk. You can lose 25% of your portfolio in one day (fortunately in 1987 I did not have a lot of investments). And you can lose 50% in a very short period of time (in 1973-75 in the UK, you lost something like 80% in real terms). It's a well argued fact about stock markets that returns have been excessive (Mehra and Prescott 1986) -- the best guess I have seen is that investors are averse to short term losses in real wealth, thus safe assets are overvalued and risky assets are undervalued.

TIPS bonds should be strongly correlated with CPI, real estate a higher correlation than stocks (rents are linked to inflation, so are land prices in the long term). I am not sure the conditions which produced the general inflation of the 1970s following large price rises in commodities would be repeated - I am less sure a commodity is an inflation hedge.

Raybo
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### Re: Lognormal Inflation Model in Monte Carlo Retirement Planning

I've written a couple Monte Carlo simulations of my retirement assets and market returns. Like you, I used a normal distribution for both bonds and stock returns. I chose to use a normal distribution for inflation, as well. I capped it a -2% but let the upside run, as high inflation is a real retirement worry. I made no attempt to correlate any of these. I just pulled a value from each distribution when I needed one.

Some thoughts:

I knew normal distributions wasn't close to accurate. But, when I did these simulations, I didn't have access to real market data going back decades. Now, I would likely gather the data on a monthly basis and simply put it into a customized step function for each distribution (stock, bond, inflation). I have been musing with also adding a modifying factor by analyzing the how one month's returns influences the next month's returns. I was never comfortable using normal distributions as my underlying data sources.

While investment returns and inflation are prime drivers of retirement survival, there are other important variables to consider. For example, I added a simplified federal tax system to my program. As I took money out of tax-deferred accounts to pay for retirement expenses, I calculated how much tax I'd owe on it. Taxes are not insignificant when funding retirements out of tax-deferred funds.

When I wrote my programs, I was still working. I added contributions to tax-deferred accounts into the model as well. If you are currently retired and not earning any income or putting money into IRAs, then this isn't important. But, what about Roth conversions?

I also added social security, both income and taxes on that income, to my calculations.

The data my model produced was a table of years I ran out of money and the (average) returns and inflation values that led to them (capped at 100 years of age). This way I not only got a graph of my possible outcomes, but I could also see what conditions would lead to the failure of my retirement funds.

I haven't revisited this simulation in many years as it was designed to help me plan my retirement. Since I retired in 2000, I haven't found it necessary. But, of late, the programming bug has bit me again and now that market return data is readily available and should the level of interest rise above my level of indolence, I might take another crack at it.
No matter how long the hill, if you keep pedaling you'll eventually get up to the top.

Topic Author
K72
Posts: 35
Joined: Wed Dec 05, 2018 8:04 pm

### Re: Lognormal Inflation Model in Monte Carlo Retirement Planning

I'd like to thank everyone that responded. It was helpful for pointing out that (1) I hadn't provided sufficient background/explanation as to what I was doing and (2) I'd made some assumptions w/o realizing I had. In event, I'd like to respond with clarification/comment:

1. I'm not retired but plan to be within a few months. My model includes estimates for expenses and includes income from pensions, Social Security, and RMD's from IRA's. I have no Roth.

2. I'm using inflation for the sole purpose of adjusting projected annual expenses. I really didn't want to use a fixed value because I don't think it sufficiently represents future reality.

3. For the inflation model, I used the U.S. CPI from about 1940 to present (no particular rationale for choosing 1940), which is how I got the lognormal looking distribution. I now realize I could go much further back (I did find some estimates of inflation back to 1800) and see the distribution is much more normal-ish with plenty of deflationary periods. This begs a question as to what period is reasonable to use for modeling inflation. I don't know the answer but I wanted to use something other than a fixed value.

4. I realize that a normal distribution of 15 with a sigma of 3 won't see much negative but I wouldn't think a model with this as inflation would be very useful. I would probably choose more like a mean of 4 with sigma of 2 or 3 but that would have plenty of negative. I just can't see using much negative inflation in a retirement model even though I know it could very well happen. I don't mind a bias toward higher inflation for the purpose of expense projection.

5. I made my own MC sim using Excel (so I could include the complexity of pensions and social security with cost of living assumptions, RMD's from IRA accounts and taxes). The online tool I used was https://www.portfoliovisualizer.com/mon ... simulation which has a variety of other tools that seem to be useful.

6. If I don't use MC then I'm not sure what to do for retirement planning to estimate if the money will last for 30 years (also an arbitrary choice on my part). I first tried fixed investment return and fixed inflation but quickly determined that all would be well as long as the investment return was higher than inflation by some margin. Also, I don't think I have the ability to predict investment returns or inflation for any future period of time. In the end I found MC to be a satisfying way to gain insight into retirement planning. Of course I realize whatever model is used is based on a lot of assumptions, but again, I'd rather do something rather than nothing and don't know what other approach to take.

Topic Author
K72
Posts: 35
Joined: Wed Dec 05, 2018 8:04 pm

### Re: Lognormal Inflation Model in Monte Carlo Retirement Planning

I also have a question. Several responses mentioned "real return". Can someone explain what this is? If it means investment return minus inflation, then I understand.

randomguy
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Joined: Wed Sep 17, 2014 9:00 am

### Re: Lognormal Inflation Model in Monte Carlo Retirement Planning

AlohaJoe wrote:
Thu Dec 06, 2018 2:40 am
randomguy wrote:
Thu Dec 06, 2018 12:57 am
AlohaJoe wrote:
Thu Dec 06, 2018 12:36 am
Ken72 wrote:
Wed Dec 05, 2018 11:23 pm
What I did was cap the upper limit at 16% (I remember too well the early 80's) with no possibility of deflation.
Since there have repeatedly been periods of deflation (more common than high inflation, for that matter), it seems weird to eliminate the possibility.

Anyway, for a model it is fine-ish. It's not really any worse than assuming that equity returns are normal or lognormal. Or that there is no correlation between things. Or that there's no time-varying correlations. Or that there's no short-term momentum or long-term mean reversion. And all the other simplifying assumptions models make.
he remembers the 80s. Very unlikely anyone remembers the long deflationary periods in US history
But why is "long periods" and "US history" the important parts? Much of Europe & Japan has had deflationary periods for the past decade, as has been much discussed over the past years. Why pretend that can never ever possibly happen in the US when modeling a multi-decade period? Especially when we had deflation as recently as March 2018? There's also been deflation in December 2017, October 2017, July 2017, November 2016, July 2016. Is it so hard to imagine that those short periods might stretch from a few months to 12 months? Meanwhile, where are the occurrences of inflation being 16%? The highest annual inflation ever got in the US was 13%. So why allow inflation to go higher than it ever has but not allow deflation to even reach the levels it actually did?

It just doesn't seem like an honest, impartial decision. It seems like one that is "I'm scared of inflation so I'm going to bias my model towards it". Which is fine. Models are made for that kind of exploration. But then you start getting on tricky ground when ask on the internet about how realistic your model is.
Because you don't notice short periods of time and obviously the US is exceptional:) And again the deflation is at the point that nobody notices it as 4 months of 1% deflation (roughly what we had at the end of 2008) doesn't get much notice. It just isn't a big enough value. You could use international deflation but to fair then you should also end up using international inflation. And probably international bond and stock returns. And yes to be honest, you should use what the data is not what you wish it to be.

Personally I think all of this complexity is going to buy you basically nothing. You want to know what your SWR for a portfolio is for the next 30 or so years? About 5.5%+-2%. I don't think we can actually get much more accurate than that. Since you want to be conservative pick something down in the 3.5-4% range and plan on adjusting as life happens. One of the toughest things to accept is that you can't control everything.

AlohaJoe
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### Re: Lognormal Inflation Model in Monte Carlo Retirement Planning

Ken72 wrote:
Thu Dec 06, 2018 10:39 pm
6. If I don't use MC then I'm not sure what to do for retirement planning to estimate if the money will last for 30 years (also an arbitrary choice on my part).
I agree with randomguy that there is limited value in trying to be too precise. I don't think there's anything wrong with using Monte Carlo. It is one tool among many.

There are tons of retirement calculators out there. By all means, build your own for fun. But most people are suited by just using http://cfiresim.com/ or https://www.i-orp.com or all the other ones you can Google. Most people use several and hope they all agree

kolea
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### Re: Lognormal Inflation Model in Monte Carlo Retirement Planning

Ken72 wrote:
Thu Dec 06, 2018 11:37 pm
I also have a question. Several responses mentioned "real return". Can someone explain what this is? If it means investment return minus inflation, then I understand.
Yes, real return is the return that has been adjusted for inflationary effects. It is not strictly "subtracting" inflation since it is polynomial.
Kolea (pron. ko-lay-uh). Golden plover.

Valuethinker
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### Re: Lognormal Inflation Model in Monte Carlo Retirement Planning

Ken72 wrote:
Thu Dec 06, 2018 11:37 pm
I also have a question. Several responses mentioned "real return". Can someone explain what this is? If it means investment return minus inflation, then I understand.
Simple formula

nominal return = real return + inflation

Correct formula (the Fisher formula)

(1+nominal) = (1+ real) times (1+ inflation)

ie 7% inflation becomes 1+0.07 = 1.07 in the second term of that equation

or

real return= ((1+nominal)/(1+inflation))-1 answer will be a decimal (multiply by 100 to get a percentage)

The simple formula is fine for talking about it or thinking about it, but errors will creep in when compounded over a number of years. So in a spreadsheet, the Fisher formula is better.