# CapitalTime

Articles on investing and capital management, with a quantitative focus.

#prp - Permanent Risk Parity (how I invest)

# Derivation of a Risk Parity Portfolio

### 2019-08-12

After learning about the risk parity concept, I tried deriving my own portfolio using three uncorrelated assets that I like: stocks, bonds, and gold.

I used the following risk data from the Portfolio Visualizer:

Asset class Stdev MaxDD
Bonds (10Y) 8.07% -15.76%
Stocks 15.40% -50.89%
Gold 20.12% -61.78%

Standard deviation and maximum drawdown (the worst decline) are two different ways to measure volatility and risk. Using historical data since 1972, bonds (10 year maturity) are the least risky, stocks are more risky, and gold is the most risky.

Note that using other datasets, perhaps covering longer periods or more/less granular data might give different results. Stocks had much more volatility during the 1930s for example, but gold was also controlled by the government and did not have a market price. For consistency, I'm using the same dataset for all assets, starting in 1972 when the US Dollar became a fiat currency.

There is a proper and thorough way to calculate risk parity. I am not using this method. Mine is a simplified approach that works when the assets being mixed have very low correlations to each other, or relatively equal correlations. This volatility-weighted approach is described on page 7 of the linked paper. I actually found this paper long after trying my own method; it turns out that my equations are equivalent to the method in the paper due to the low correlations between my three assets. Because I'm trying to describe how I derived my asset allocation, I will continue explaining my original thought process.

Using my own statement of the risk weighting problem, I tried to solve for asset class weights x, y, z so that (with Stdev values)

8.07 x = 15.40 y = 20.12 z
where x + y + z = 100%

See Dr. Kazemi's paper for more elegant equations which are easier to solve. Solving the equations, I found the following weights. RP1 is from Stdev and RP2 is from MaxDD:

Asset class Weight RP1 Weight RP2
Bonds (x) 51.95% 63.91%
Stocks (y) 27.22% 19.79%
Gold (z) 20.83% 16.30%

The two results are in the same ballpark. Rounding and quantizing these into multiples of 10%, we get the following possible configurations:

Asset class Weight
Bonds 50% or 60%
Stocks 20% or 30%
Gold 20%

We can feed the weights back into the Portfolio Visualizer and empirically test the characteristics of the resulting portfolio. Once gold is fixed at 20% there are only two possible configurations:

Asset weights CAGR MaxDD Sortino
60, 20, 20 8.69% -14.1% 0.89
50, 30, 20 9.11% -13.5% 0.92

The second portfolio is the winner, hands-down. It has higher performance, milder maximum drawdown, and better Sortino ratio (a measure of risk-adjusted return which considers downward volatility to be harmful).

We have derived the PRP allocation: 50% bonds, 30% stocks, 20% gold.

To gain further confidence that this mathematically derived result is actually a good design, one can compare to other solid portfolios such as balanced funds and the Permanent Portfolio. From the back-tests that I've done, PRP has the highest Sortino ratio among these (the best risk-adjusted return). Only the Permanent Portfolio does as well on maximum drawdown, but PRP still has higher performance and a higher Sortino ratio.

I will admit that my original derivation approach was a bit of a guess based on my intuition of weighting volatility. However, it turned out that my equations were identical to those in Dr. Kazemi's paper, after performing some mathematical transformations!

Jem Berkes