Analyzing the Retirement Game

The previous post, “Playing the Retirement Game“, introduced you to a simple financial model for a salaried employee named Sal. If you have not played the Retirement Game yet, you are encouraged to visit github.io and figure out what is the best advice you can give Sal regarding his savings and investments.

To obtain a deeper analysis, I put a wrapper around the game to automatically run it multiple times and generate numbers and charts. Here is my experience with the game.

First observation is that randomness introduced in the game for debt and equity investments causes a large spread in outcomes for a given investment strategy.

Living Purely off Fixed Deposits

When all investments are in fixed deposits, how long does the money last? It depends on the level of savings. Without aggressive saving rates we see a shortfall of 20 to 60 years worth of starting salary. But the good news is that if Sal sets aside a fairly large fraction into savings, he will stay in positive zone.

rg1_exprate_score_purrefd

Score vs expense level, pure FD investment strategy

What I don’t like about fixed deposits is that they lose value with time. So, in one sense, Sal’s scrimping and saving and denying his family the money only goes to support the banker’s lifestyle.

Large Spread in Outcomes with Risky Investments

When the game is run many times with risky investments, most runs will have several loss events, but some lucky runs will have zero or very few loss events. These lucky runs, thanks to the power of compounding, will yield extremely large scores. Conversely, some unlucky runs will yield very low scores. Given below are histograms for pure debt and pure equity strategies.

rg2_puredebt_hist

Pure Debt Investment Strategy, 65% expense rate

Pure debt investment with 65% expense rate (i.e. 35% saving rate) gives a mean score of 1, which is approximately break even. Note that a 35% saving rate with pure debt strategy gives same results as a 50% savings rate with pure FD. To see this in another way, we find that pure debt strategy with 50% saving rate gives a mean score of 30 as against zero for pure FD at same saving rate. Score of 30 means retaining 30 years worth of starting annual salary at the end, a decent sum.

Now let us see what happens with a pure equity strategy. Thanks to the compounding effect, mean score is 3970 at 50% saving, 2624 at 35% saving rate, 380 at 10% saving rate! Thousands of years of annual salary is serious money: Sal can surely take early retirement!

However, there is a downside: although mean scores are high, so is the variability. Here is the histogram for 25% savings rate, which shows that there is a 45% chance that Sal won’t make much with this strategy, but a 35% chance that Sal will do very well, plus  20% chance of doing really really well.

rg3_pureequity_75expratePure Equity strategy with 75% expense rate

Debt to Equity Ratio

How does D/E allocation ratio impact results? We shall approach this question from a different angle: let’s measure only the chances of survival with a given mixed strategy. That is, given a certain allocation strategy, what fraction of trial runs result in a positive score? We will call this fraction the success rate of an allocation strategy.

The figure below charts success rate vs D/E ratio. The curves were generated for varying expense rates. The curves are not smooth because a large enough sample size was not used. This jaggedness is noise and should be ignored.

Debt to Equity ratio is given by a parameter k, D : E :: k : 20-k. Horizontal axis represents k as k varies from 0 to 20. Therefore, leftmost values are for pure equity moving to pure debt at the right.

rg4_successratio_vs_exprate

Success ratio (chance of positive score) vs debt/equity ratio

The figure above is very interesting. We can infer several things:

  • Success ratio is higher for higher saving rates (lower expense ratios). That’s obvious, really: more money Sal puts into savings, more likely will the money last until the end.
  • At lower saving rates, debt tends to reduce survival rate.
  • With higher saving rates, there is a sweet spot. best success rate is at about 60-70% debt. I suspect a mixed strategy tempers equity risk by not wiping out all the capital in less lucky runs (that have many loss years).

Remember, with the way the game is set up, if capital is wiped out post retirement, there is no scope for Sal to earn more money, resulting in an inescapable downward spiral into increasing debt.

Conclusion

I feel this extremely simple model has been successful in delivering several insights — or at least, suggesting some directions on where to explore further, perhaps with more realistic models.

In particular, this game does not allow exploration of a non-constant allocation strategy. Many books and blogs advice starting off with higher risk in earlier years and then shift the portfolio to less risky ones later. A hint of this can be seen in the previous chart — building net worth aggressively in initial years is equivalent to using a higher savings rate in the game.

This is also the right place for a strong disclaimer:

Sal’s world is very simple and in no way matches the complexities of real life. It would be extremely foolhardy to use the numbers presented here to take real life decisions. So don’t do that, and if you still do, don’t blame me!

So for example, Sal can put 50% of his salary in FDs and expect the money to last until age 85. But would you get the same results in real life? Who knows?

Nevertheless, some important lessons can be derived from the Retirement Game (notice these ideas are not at all novel — most financial planning blogs and experts will give similar advice).

  1. Risk free investments do not provide sufficient protection in the long term.
  2. One has to, especially from early on, grow money faster albeit taking on more risk.
  3. Provided you have enough money in play, you can reduce the risk with a mixed portfolio. Although it was not allowed in Sal’s world, one can also use a time-varying strategy: start with high risk high reward strategy but gradually reduce high risk investments in later years.

If nothing more, even if this post encourages you to seriously look at investment options beyond bank fixed deposits, I think this Retirement Game will have served its purpose.

Further Reading

Estimate how much you will need at retirement based on current expenses: https://www.bigdecisions.com/calculators/retirement-spend-calculator

Mathematical models of the stock market – https://www.scientificamerican.com/article/can-math-beat-financial-markets/

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Playing the Retirement Game

How do I get out of the salary trap?

As should be clear from the previous posts, especially the ones titled Salary Curve, Life Expectancy, and Middle Class Saving Mantras, almost all of us middle class people are squarely in a salary trap. The trap has four walls:

  • Limited (and declining) real income from salary
  • Long remaining  life span after salary income stops
  • Lack of know-how on how to grow one’s net worth
  • Limiting mindset regarding financial matters

The question, of course, is this: is there a way out of the salary trap? Well, there must be at least four ways out of the salary trap — one path for each of the four walls listed above!

Most people will focus on breaking through the first two walls:

  • Increase salary income: job hopping / industry hopping / retraining / second job
  • Continue working beyond normal retirement age

Although these two walls may be pushed back a bit, so to speak, they are very difficult to truly break through.

In this post, we shall focus on the third wall, and examine the effect of increasing one’s know-how regarding savings growth. Please note that this post does not tell you how to increase your know-how. Instead, the focus is on analyzing the impact on one’s life if better saving decisions are taken from the beginning.

To that end, let us look at a simple simulation, what we shall call the Retirement Game. The Retirement Game is an extremely simple financial model of a salaried person named Sal. In the game, you are Sal’s financial advisor and must help Sal get out of the salary trap by advising him on his investment decisions.

An F-minus cartoon

You will basically provide Sal an allocation strategy: what fraction of Sal’s net worth is to be held in each asset class (more details below). At the end of each year, Sal’s portfolio is rebalanced so as to continue adherence to the allocation strategy. This means, for example, if one asset class grows very well, its value as a fraction of total worth will exceed the value decided in the allocation strategy. Then Sal books profits to bring this asset class fraction back to the desired level and buys more of the underrepresented asset class.

Based on your investment suggestions, Sal’s savings will grow or shrink. After retirement, Sal lives off his savings for the rest of his lifespan.

A game needs a way to keep score. The score is the amount of real savings (inflation-adjusted) that Sal has left over at the end of the game. If Sal runs out of money, he will live off borrowed money, then his net worth may become negative at the end. Notice that once Sal gets into debt after retirement, he has no chance to recover, and his net worth will keep spiraling downward.

The objective of the Retirement Game is to understand the impact of taking different investment decisions over Sal’s lifetime.

Now let us get into the details of the game. Remember, we want to keep the game as simple as possible to meet its objective, but no simpler. So most parameters are kept constant, although they will not be constant in real life. Also unlike real life, Sal does not face shocks to his expenses such as sending a child abroad for education, children’s marriages, or expensive illness in family. Nevertheless, with a simpler model, we hope to get improved understanding of the shape of the solution.

Thus, there are just a few knobs and levers to operate in the game.

Constant Parameters

Sal gets a job at age 20, his career plateaus at 40, he retires at 60, and has a natural life span of 85.

Starting post-tax salary is 3,00,000 per year. Salary grows at 15% pa for first 20 years, then grows at 2.5% pa less than inflation rate for the remaining part of his career. Actually, starting salary gets factored out from the final score, so don’t agonize over the starting salary being too low or too high.

Inflation is 7.5% throughout. Cost of borrowing money is 10% pa.

There are only three Investment options with fixed risks and returns (% returns are post-tax). That is, there are no long term boom-bust economic cycles in this game.

  1. Bank Fixed Deposits, 5.5% pa, zero risk
  2. Debt instruments, expected yield 8% pa, risk of 10% loss
  3. Equity, expected yield 15% pa, risk of 30% loss

Yields and risks for Debt and Equity need to be explained. The model is as follows:

An investment may either give a bad return of x2 < 1.0 or a good return of x1 > 1.0 with probability of p and 1-p, respectively, per year. Then

(1 + expected yield) = (1-p) * x1 + p * x2

Or,

x1 = [(1 + expected yield) – p * x2 ]/(1-p)

Now,  expected yield and x2 are known. As value of p is varied (higher p represents more volatility), x1 will be change correspondingly to keep expected yield at the set level. For example,

1.15  = 0.90 * x1 + 0.10 * 0.70,  then x1 = 1.20 (i.e 20% profit)

1.15 = 0.80 * x1 + 0.20 * 0.70,  then x1 = 1.2625 (i.e. 26.25% profit)

By keeping p as the independent variable, we can better study the effect of market volatility on playing the retirement game.

We also need to model expenses incurred by Sal every year. The model we will use is not too complicated, but should reflect some of the ups and downs in expenditure that happen in real life.

We start with fixing a base level of expenditure at the start of the game, when Sal starts out a bachelor, but over the next 20 years his expenditure grows at the same rate as his salary does, that is, at 15%. However, after age 40 Sal switches to a constant lifestyle, so until retirement his expenses just keep pace with the inflation rate. At retirement, Sal cuts down his lifestyle by 50%. However, with time, expenses again rise with inflation. Since level of savings is an important decision, base level of expenditure is a tunable parameter. Yes, I agree, this is an extremely simplistic model!

Knobs and Levers

You can set the following knobs at the beginning of the game:

  1. Base level of expenditure, as a percentage of starting salary.
  2. Investment allocation decisions. What percent is put in Fixed Deposits, Debt and Equity. In this simple game, allocation strategy is constant throughout.
  3. Number of runs. Since debt and equity investments have randomized returns, play the game many times to get a better understanding.

There is only one lever: hit the [start] button to start the runs!

Note: Although one could chose a design where the expenditure and investment decisions can be changed every year, I have chosen to keep them constant for two reasons. First, it keeps the game simple so it’s easier to measure the effectiveness of a particular strategy that is applied consistently. Second, because random elements are involved, a proper analysis will require the same game to be run many times, which is easy to do when all the knobs are set only once at the start of the game.

Running the Game

We run the game in one year time steps.

At the end of each time step, do the following:

  • Compute changes for one time step
    • Case positive net worth: Calculate returns on existing investments
    • Case negative net worth: calculate interest on borrowings
    • Add gains or losses into current net worth
  • Compute current savings = salary – expenses
  • Compute new net worth = current net worth + current savings.

Note that current savings and net worth can each be negative. Returns on investments may also be negative.

End the game at age 85.

At end of the game, compute score as the following dimensionless number.

score = real net worth / starting salary  , or, in other words,

score = net worth adjusted for inflation / starting salary

Note: Since this game always ends at age 85, we did not really need to discount net worth. But if we ever make the lifespan variable, inflation-adjusted net worth will be required. Also, inflation-adjusted values are easier to understand.

One can interpret a score of N as ending at age 85 with a net worth equal to N years of annual starting salary.

Play the Game

The Retirement Game is hosted at github.io.Please visit this site where you will find the link for playing the retirement game.

If you want to study the code and perhaps modify it, it is also available for download:  download and unzip RetirementGame.zip to get a folder named RetirementGame. Its major functionality is in the associated JavaScript file.

In a few days, I shall publish the next piece, which analyzes the Retirement Game using more powerful tools.

Acknowledgments

Many thanks to:

Manas Sathe provided invaluable help with detailed review and a critique of the model.

Sandeep Ranade pointed me to Github Pages where the Retirement Game could be hosted (WordPress effectively does not allow JavaScript).