Risks & Rewards
... a continuation of Part II

In the previous Part we talked about various Risk/Reward definitions, ending with this one:
 We look at all the 1year Returns and see
which ones were less than 3%, then calculate the sum of the absolute value of the
deviations from 3%:
That is, for all Returns R% < 3% we calculate
Σ  R  3 
(This gives us our measure of Risk. Then we repeat this for
2year Returns,
3year Returns etc.)
 Then, we look at all the 1year Returnsyear Returns and see
which ones were greater than 8%, then calculate the sum of the absolute value of the
deviations from 8%:
That is, for all Returns R% > 8% we calculate
Σ  R  8 
(This gives us our measure of Reward and we repeat this for
2year Returns,
3year Returns etc.)
 The Ratio of these two numbers is our
Risk / Reward Ratio,
for 1year,
2year,
3year etc.)
>Where the 3% and the 8% are parameters, selected by us, right?
Right.
However, as we saw in the previous Part, this Risk/Reward Ratio has problems.
For example, if we consider two stocks which
have never had a return less than 3% then the Risk would be zero for both hence this Ratio would be
zero for both so ...
>So you could't use this Ratio to decide between them, eh?
Exactly! The Reward may be greater for one, but we wouldn't be able to tell from the Ratio.
It'd be better if we wrote the Ratio in the form 0/7 or 0/9 for a portfolio with zero Risk,
but returns of 7% or 9%.
Anyway, what we're looking for is a function which we can use to decide whether it's
better to go with this stock or that, this mutual fund or that, this portfolio allocation or
that, this ...
>Okay, okay! I get it! So?
So we look elsewhere, like so:
Note that the Risk
and the Reward, as defined above, depend upon the sequence
of 1year, 2year, 3year etc. Gain Factors which we'll call
G_{1}, G_{2}, G_{3} etc.
>Gain factors?
Yes. If $1.00 grows to $1.34 after 1 year, then G_{1} = 1.34 and if it grows
to $2.17 after 3 years then G_{3} = 2.17 and so on. That's what we mean by
Gain Factors.
Note that, in the above definition of
Risk/ Reward,
we can consider the 3% as an inflation rate so the
Risk, above, was the risk of getting a return
less than inflation (which may be something other than 3%, of course). So, whatever our new
definition will be, inflation should play a role.
Indeed, it'd be better NOT to assume a constant
inflation rate (like 3%) but consider the sequence of
of 1year, 2year, 3year etc. Inflation Factors which we'll call
I_{1}, I_{2}, I_{3} etc.
>I take it that if inflation increases by 2.5% over 3 years then ... uh ...
Then the 3year Inflation Factor is I_{3} = 1.025 so ...
>So the Inflation Factors are like the Consumer Price Index, eh?
Yes, assuming we start at CPI = 1, at the beginning of year 1.
So far we have our function (the one that'll replace the Risk/Reward Ratio)
depending upon the sequence of Gain Factors and Inflation Factors and, of course, the number of years.
In fact, although we'd be happy if the Gain Factors were huge, it seems reasonable
that the influence of the Gain Factor for year 4 (for example) will be reduced if the
corresponding Inflation Factor (for year 4) is large. This suggests that perhaps it's the Ratio
G_{4}/I_{4} which should enter into our
function
(thereby reducing the Gain Factor ... due to Inflation for that year).
In fact, it's not unreasonable to consider a sum, like:
G_{1}/I_{1} + G_{2}/I_{2}
+ G_{3}/I_{3} + ... + G_{N}/I_{N}
>That looks familiar!
Yes, a similar sum occurred when we talked about the Maximum Rate of Withdrawal,
here. The formula
for f, the Maximum Rate of Withdrawal (MRW)
from a Portfolio (so that it's reduced to $0 after N years) is:
f
= 1 / (
I_{1}/G_{1}
+ I_{2}/G_{2}
+ I_{3}/G_{3}
+ ...
+ I_{N}/G_{N}
)

>And that's your function?
No, not at all. That's our Reward. Our Risk will be tied up with the probability of
achieving a MRW less than, for example, the inflation rate.
>The probability of achieving less? What does that mean?
Well, we might consider the distribution of MRWs a la Monte Carlo, selecting random
returns for the period 1950  2000 and a sequence of actual inflation rates and running
a thousand 30year simulations and seeing how many gave a Maximum Rate of Withdrawal
greater than 2% or 4% or 6% etc. with a particular asset allocation:
>So what's your Risk/Reward?


Well, we could consider a thousand 30year Monte Carlo simulations and set:
Risk / Reward = (Percentage of years that the MRW was less than the Inflation Rate)
/(Average MRW)
>And what's that?
Well it'll depend upon our asset allocation and the years from which we select our returns and ...
>And?
If we select returns from the period 19502000 (and actual 30year sequences of
inflation rates from 1928) and we assume
a 4 x 25 portfolio and ...
>4 x 25?
Yes, 25% each of Small and Large Cap Growth and Value stocks.
Anyway, we'd get a Ratio of 5/11.
>Meaning that 5% of the MRWs were less than the Inflation rate and ...
And the average MRW was 11%.
>Is that good?
It depends. We only use this Ratio to compare ...
>So ... compare!
Okay. If we use the same time period but change our allocation to:
25% Large Cap Growth + 20% Large Cap Value
+ 45% Small Cap Value + 10% Govt Long Bonds
then we'd get a Ratio of 4/12.
>The Risk is that 4% of MRWs were less than inflation and ...
And the Reward was that the average MRW was 12%.
>Smaller Risk. Greater reward. I'll take it
... but how do I do these calculations? I mean ...
Go here and download the spreadsheet
which has this ratio, somewhere.
Look for it ...


Here's a picture where the return and inflation data is selected (a la Monte Carlo) from the
period 1928  2000:
>Like I said ... smaller risk, greater reward.
But remember that one can always define Risk in such a way that the Risk/Reward ratio in
maximized if ...
>If you keep your money in a shoe box.
Uh ... yes. Here's more pictures illustrating the effect of changing the asset allocations. We
look at all years and inflation rates from 1928 to 2000 and plot the Maximum Rate of Withdrawal
over all 30year periods.
You'll see that the minimum MRW (which defines some sort of Safe Withdrawal Rate) changes
as well as the Average and the Maximum and ...
>Just show the picture.
Okay, but note that 5T means 5year Treasuries so you'll notice that ...
>Just show the picture!
