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 1-year 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 2-year Returns, 3-year Returns etc.)
• Then, we look at all the 1-year Returns-year 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 2-year Returns, 3-year Returns etc.)
• The Ratio of these two numbers is our Risk / Reward Ratio, for 1-year, 2-year, 3-year 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 1-year, 2-year, 3-year etc. Gain Factors which we'll call G1, G2, G3 etc.

>Gain factors?
Yes. If \$1.00 grows to \$1.34 after 1 year, then G1 = 1.34 and if it grows to \$2.17 after 3 years then G3 = 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 1-year, 2-year, 3-year etc. Inflation Factors which we'll call I1, I2, I3 etc.

>I take it that if inflation increases by 2.5% over 3 years then ... uh ...
Then the 3-year Inflation Factor is I3 = 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 G4/I4 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: G1/I1 + G2/I2 + G3/I3 + ... + GN/IN

>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 / ( I1/G1 + I2/G2 + I3/G3 + ... + IN/GN )

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 30-year 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 30-year 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 1950-2000 (and actual 30-year 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 ...