Finding Out How Much is Enough

In our last post we described projecting a set of 100 possible time paths for Total Assets after retirement. Each time path is associated with a different set of possible future rates and prices drawn from historical prices and rates. We are going to evaluate alternative retirement plans by comparing the possible time paths they imply. The time paths are going to look sort of like the final chart on the last post. Lots of lines.

So we need a systematic way to compare the sets of lines. We do this by drawing particular attention to four of the lines on the chart by making them heavier and different colors – the 5th worst (colored red), the 25th worst (colored magenta), the 25th best (colored dark purple), and the 5th best (colored dark blue). We are interested in these cases as some sort of summary of the whole set of outcomes. Specifically, half of the outcomes are between the 25th worst and the 25th best (so there is 50/50 chance the outcome will be in that range), almost all, 95%, of the outcomes are better than the 5th worst (so unless we are really unlucky we can expect and outcome at least that good), and the 5th best is better than almost all of the outcomes (so we would have to be really lucky to do better than that).

Now suppose I retire just after I turn 62 (Judy will be 58). We can reasonably expect to have about $300,000 in total assets after paying off the houses, $26,000 annual income and expect to spend about $45,000 per year. We start by supposing the income will be fully adjusted for inflation each year, that we will keep 75% of our assets in stocks, and each year we put 20 months of the expected "draw" into a money market account. By "draw" I mean spending minus income which we will need to draw out of the money market account. We then spend out of the money market account for the following year. Each year we rebalance accounts – putting 75% of our assets into stocks and 20 months of draw into the money market account. The following chart shows the implied results.

These results don't look very good (there is too high a likelihood that we will run out of assets before we expect to die). In fact, for all of the cases except the best 7 or better, our total assets don't last 42 years after retirement (when I would be 104 and Judy 100). And for most of the cases the assets run out in just a few years.

   So instead, suppose I wait until full retirement age, 66, (judy will be 62). Then we can reasonably expect to have about $600,000 in total assets after paying off the houses, $33,000 annual income and still would expect to spend about $45,000 per year. If we make the same assumptions about how the money will be managed the results are as shown in the following chart.
 This is better, but still not good. For the worst 20 cases the assets are exhausted before 38 years after retirement( when I would turn 104 and Judy 100). I probably won't live that long -- but there is a good chance Judy will come close to 100 -- and for fully one fifth of the cases the assets don't last 38 years. 

    So we need more -- how much more? It turns out that we would need about one million in total assets before the results for all but the worst 4 cases have the assets lasting at least 38 years after retirement. That seemed pretty discouraging because I don't see any sure way of getting one million.  

   Then I had a thought. If we look at the charts carefully we see that each line has lots of ups and downs, which is realistic – it is the way real world results look; but it also means that for each case, when an interval with bad outcomes comes along as it almost always does, the total assets begin to get rapidly used up. Suppose we adjust our spending based on how well our investments fare. When we have bad outcomes we reduce our spending some. It turns out that doing that makes a lot of difference in how long the assets last. 

   So we plan to reduce our spending (adjusted for inflation) each year when we rebalance accounts, by multiplying by 95% if our total assets (adjusted for inflation) are below the value at retirement; and on the other hand we will increase our spending by multiplying by 103% each year if our total asset value is higher than 125% of the value at retirement. But we will only reduce spending to minimum of 75% of the basic planned amount and raise spending to a maximum of 150% of the basic planned amount.

   Then if we retire after I turn 66 (Judy 62) with $600,000 the results are as shown on the following chart. 

  This is good!! The results for all of the cases are successful. Even for the worst case there is about $140,000 worth of assets remaining 38 years after retirement and for the 25th worst case (the magenta line) there is over $500,000 left 38 years after retirement and we have a 75/25 chance of doing better than that.

It looks like this is enough to retire. In fact, maybe we don't need quite that much. Maybe we could retire a few years earlier -- if things go well. 

Prices and Rates: Past and Future

Our last post explained that we can calculate how long after retirement our assets will last  –  If we make assumptions about future inflation and rates of return on assets and if we have a well defined plan for income we expect and spending. We also saw that seemingly modest changes in assumptions about future inflation and rates of return result in big changes in how long our assets will last. 

So we need to think carefully about what future inflation rates and rates of return are likely to be. There isn't much to go on but what has happened in the past. 

The following chart shows (for each month since 1934) the annual interest rate on long term bonds (the green line), the annual interest rate on short term treasury bills (the yellow line - which I am assuming is a good approximation for money market rates), and the average dividend rate on the stocks in the S&P index (the orange line that is at the very bottom of the chart). Remember that you can click on the chart to see it better.

We see that the average dividend rates on stocks were much smaller than interest rates and that long-term bond rates were usually (but not always) higher than short-term rates. We see that there was a period in the 1980's when interest rates got higher. We also see that depending on what period we average over we can get almost any average value we want between about 3% and 8% for the interest rate. And remember that our previous post showed that changes that big are big changes indeed. 

What about prices? The next chart shows what happened to the Consumer Price Index (CPI, shown as the dark blue line) the S&P price index (SPI, shown as the red line) and the Bond price index (BPI, shown as the green line) during the same interval.

We see that the change in stock prices was remarkably bigger than inflation (the change in CPI). That is why stocks have been such a good investment -- not because of their dividends but because of their appreciation in value (price). While inflation seems small compared to the change in stock prices it did increase steadily over the period with some short intervals of relatively rapid increase. In the 1980's the inflation rate (the slope of the blue line) was about the same as the rate of increase in stock prices (the slope of the red line). Notice also that bond prices didn't change all that much and that they generally declined until the 1980's and generally increased after the 1980's. If you compare the two charts you see that bond prices decline when interest rates rise and bond prices rise when interest rates decline. It has to be that way otherwise investors wouldn't be willing to hold both new and old bonds in their portolios -- bond prices adjust so that both new and old bonds offer the same market rates of return.

Now we know what rates and prices have been in the past. But what we see is that they have been pretty much all over the place. Not much help in deciding what rates we ought to use to calculate how long our retirement assets will last. 

We have to give up the idea that we can find a single set of rates that are the "right" rates to use in our calculations. Instead we need to think in terms of ranges of possible rates that will have associated with them ranges of possible results. Each set of possible rates implies a possible result about how long our assets will last. Since there are many possible sets of rates there are many possible results about how long our assets will last. That is what we meant by planning with uncertainty.

This makes calculating how long our assets will last more complicated because we need to do the same calculations many times and keep track of the different answers we get each time so that we can somehow consider all the different answers. While the calculations are more complicated because they are done over and over again they really aren't harder -- we just do the same thing over and over. That is what computers are good at. Doing the same thing over and over.

And since we are going to need (and are going to use) a computer to do the calculations over and over we may as well add another complication. Instead of assuming that future rates of inflation and rates of return will be constant through the future we can assume they will change from month to month, like they obviously have in the past (look at the charts above). The computer is just as happy with rates that change from month to month as ones that are constant -- we simply need to have numbers for it to use.

So what we did was construct 100 different cases by randomly drawing chunks of historical rates from the past and stringing them together. None of these cases taken as a whole actually did occur in the past but they are each made up of intervals that actually did occur sometime since 1934. We regard them collectively as describing the range of what is likely possible for inflation rates and rates of return in the future. 

Now given a plan about future income and spending, total assets and how they will be managed after retirement; we can calculate how our total assets will fare in each case. And we can display the results as lines on a chart like the following, where each of the 100 light purple lines represents a time path for total assets after retirement implied by the same specific plan (described in the text above the chart) and one of our sets of projected future rates. 

(You really do need to click on this chart to see it better).

For some of the cases the assets last at least 40 years after retirement and for others they don't last very long at all. 

In evaluating a plan, we are not so much concerned about each individual outcome – we are concerned about the entire range of possibilities the set of outcomes describes because we think that that range describes the range of possibilities for the future. We don't know, and can't know which of these will happen -- in fact, probably the future won't be any of them exactly, but we expect that the future results will probably lie within the range of outcomes described by these cases.

So now we have a technique that we can use to examine the consequences of alternative plans and select which has consequences we like. So that should be the subject of our next post. Comparing alternative plans to find out "How Much is Enough?"