# Analysis of Mixed Duration Passive Equity Portfolios

## Investment Thesis

Can individual investors expect high yield on short term equity portfolios?

# Problem Specification

Suppose you want to build a sound investment portfolio. Suppose, further, that you’ve settled on a long-term strategy. A problem arises:

How should short term funds be handled?

Short-term investment requirements are often disjoint from their long-term counterparts. Short term investment often acts to:

- Save for a particular event.
- As a safety buffer.
- Have higher liquidity & stability requirements.

Here I consider the viability of adopting a long term passive diversified portfolio approach to short term asset allocation. Backed by a statistical enquiry to assess performance in the expectation.

The argument is built on the premise that arbitrary samples from diversified market segments outperform the appalling yield on cash. This is interesting as the benefits of market diversification need not only accrue on very long time horizons.

# Hypothesis & Premise

Let’s compare two portfolio options:

**Invest in cash.****Invest in a randomly sample portfolio of stocks**.

Disclaimer: costs & intricacies are ignored — which in our cases is probably a pessimistic bias if anything.

## Invest in Cash

- Banks & equity firms offer a flat 2.5% compounded annually on idle cash.

## Randomized Index Portfolios

We want to represent the market in an arbitrary sense, independent of time, specific equity bundles, global events etc — to capture expected behaviour of the market (in the limit). We also wish to represent actual portfolios that inidividuals can build using cheap investment platforms.

We can then create simple aggregate portfolios & bootstrap their performance to access the performance of portfolios in the space of all possible portfolios, over the time span of the data.

These great level of randomisation significantly removes bias.

The objective is to measure the performance (& variation) of possible equity portfolios over short durations, & compare this portfolio distribution to that of the return on cash.

# Experimental Design

To achieve this, we implement a simple algorithm:

- Fetch data of the market over the last 20 years. ETFs are a great cheap tool to invest in the market for individuals, & thus a great fit for our experiment. We thus use ETFs & not individual equities. I’ve opted to use the Satrix ETFs on offer in the public markets, as they offer both niché & diversified market segments (capuring all portfolio types).
- Now that we have the data, for a giving number of years (ranging from 1-to-10) we assess the return of all possible portfolios. This is done by:

- Sampling 3 assets (ETFs at random) to represent a portfolio.
- Sampling a random starting date.
- Computing the return & standard deviation of returns over the duration.
- Repeating 2'000 times to bootstrap — representing all possible portfolios (if you’re mathematically inclined: think of this as
*the portfolio space*). - Bootstrapping also captures a distribution over the portfolios, not simply a sample statistic, thus we capute variation in these estimates.
- The process is repeated over many time durations (1-to-10 years).

Finally, we constast the distribution over portfolios with an investment in cash at 2.5% compounded anually. In aid of answering the question:

Is market yield worth its risk in the shortrun?

# Implementation

Now we can implement this algorithm in R.

# Analysis

Now that we’ve generated the samples, we can contrast the findings with cash investments. We begin by looking at the distribution of returns for each time period.

The code below generates the distribution of return for each duration period. The y-axis gives the number of years invested (1-to-10) & the x-axis gives the ROI on the portfolios sampled.

Each histogram/distribution is thus a bootstrap estimate of all possible portfolios for the given time duration. Note:

- Where the 0 is on the x-axis (no change in capital gains).
- The longer the duration, the wider the variation in returns.
- The variation in returns is greatly positively skewed, greatly increasing the expecte return over longer time horizons.

It appears clear that high ROI can be achieved by long term holdings in arbitrary equity portfolios.

## Now we can compare the return distributions to the cash investment

Above, we see:

- Y-axis: The distrbution of returns (ROI) on the y-axis.
- X-axis: Year holding period.
- Blue dashed line: The mean (expected) portfolio return.
- Pink dashed line: The cash investment

Interpretation:

- The equity portfolio returns consistantly & considerably beats cash investments in the expectation (average).
- The distribution is skewed above the cash investment, though does persist far below the cash investment.
- The difference in yields increase overtime, as portfolios compound faster.

Although longer duration portfolios have higher varation, in the long run (9 and 10 year holdings) portfolio variation decreases, whilst yield continues to increase — this is indicative of some convergence to stable high yield ROI in equity markets of long time horizons.

Risk tolerance offers some decisive power about the investment vehicle of choice, however it is unclear if portfolios are worth their trouble for very short durations (under 2 years) as the sampling varation appears to outweight the benefits of the additional expected yield.

**Assets in Question**

Our market indices were as random as possible, offering the most pessimistic estimates. Here are some figures ot better understand the assets in used in the experiment. Below we see the return on each asset over the full 20 year span.

Finally, here we provide the assets by correlation, as one wishes to build decorrelated portfolios to achieve market diversity. The assets have further been clustered by correlation (utilising heirarchical clustering).

# Final

Tada! Understanding the natural variation in equity portfolios goes a great way in building intuition around investment vehicles & diversified portfolio returns. We explored this from a statistical viewpoint, bootstrapping samples to estimate population behaviour in the limit.

The full code is available here.