# Rebalancing and the Marine Chronometer

In the autumn of 1707, one of the worst British maritime disasters occurred near the Isles of Scilly (formerly known as the Cornish Shires), an archipelago located in Great Britain’s Cornish peninsula. In very bad weather, several ships of war returning from a military mission in the Mediterranean were wrecked. Between 1,400 and 2,000 sailors and soldiers perished. It was determined that the main cause of the disaster was the inability of navigators to calculate their position at sea.

Following this event, the British government launched a competition for boats to find their location during navigation. The missing data, at the time, was the longitude (relative east-west position), which combined with the latitude (north-south position, already known) would have allowed a ship to easily determine its location.

After 31 years of research, a clockmaker named John Harrison found a way to determine longitude with a clock. We already knew that the Earth rotates 15 degrees per hour (360 degrees per 24 hours). You could then solve the equation if we knew the exact time between the moment of departure from the port and the present. Ultimately, it took a reliable clock that still worked well despite the movements of a ship. John Harrison was the inventor of this clock, which today is called a maritime chronometer.

A few years ago in finance something similar in usefulness to longitude was invented; rebalancing (or "volatility pumping" in a colloquial English). The investor can find this term in the description of the mutual funds and exchange traded funds (ETFs) when the frequency of rebalancing the underlying assets is described. Here is the basic principle:

## Rebalancing Table

Portfolio composed of A and B, each initially representing 50% of the total
Period
(1)
Yield (A)
(2)
\$100
(3)
Yield (B)
(4)
\$100
(5)
Average (A+B)
(6)
Rebalancing
(7)
0 –––––– 100 –––––– 100 100 ––––––
1 5 105 10 110 107.50 107.50
2 -25 78.75 35 148.50 113.63 112.88
3 35 106 -25 111 109 119
4 -25 80 35 150 115 125
5 35 108 -25 113 110 131
6 -25 81 35 152 116 138
7 35 109 -25 114 112 145
8 -25 82 35 154 118 152
9 35 110 -25 116 113 160
10 10 121 5 121 121 172

In the table, we show a portfolio of two assets, A and B. These could be two ETFs, for example, one that holds shares and the other, bonds, or an index of emerging market securities and one of first world country securities. A and B each initially represent 50% of the portfolio and are worth \$100 each.
In the first column, we have 10 periods, each representing one year. In the second column, you can see the annual yields of asset A. The third column lists the cumulative value of asset A. After 10 years the value is \$121.
Column 4 shows the yearly performance of asset B, as in column 2 for asset A. Column 5 is the cumulative value of asset B. The total after ten years is \$121, just like asset A.
Column 6 shows the average value of A plus B for each of the ten periods. For example, in period 1, the value of asset A is \$105 and asset B is \$110. The amount is \$215, which, divided by 2 is \$107.50.
Column 7 is that of rebalancing. Note that at the end of column 7, the result obtained through portfolio rebalancing is \$172, while at the bottom of the column 6 the value is \$121. Rebalancing has led to an increase in value of 42% of the portfolio ((172-121) /121). This shows that rebalancing can significantly increase the performance of a portfolio.

## The Logic of Rebalancing

The logic of rebalancing is as follows: initially asset A and asset B are equal in dollar value because the investor thinks this is the optimal distribution for the risk they want to take. Over time, one of the two assets grows faster than the other creating an imbalance with respect to the initial distribution of 50%. As the saying goes, "No tree can grow to heaven." The asset that grew fast may slow quickly, stop growing and eventually recede. In contrast, the asset that grew more slowly could accelerate its growth. In order to benefit from the growth of the latter, one would sell part of it and purchase the other asset. The underlying theory is, "Buy low and sell high." This is done at the end of each period for the beginning of the next period.

Naturally, rebalancing assumes that we would like to keep what we have in our portfolio.

## The Mechanics of Rebalancing

After one year, period 1, the total value of the portfolio increased (2 x \$107.50 = \$215 or \$15 more than the start). That's good, but the relationship between A and B is not exactly 50%: B grew faster than A. We sell the part of B that exceeds \$107.50 (or \$2.50) and the proceeds are invested in A. This way, at the start of period 2, A and B are equal again at 50/50 or \$107.50 each.

During period 2, we invested \$107.50 in asset A but lost 25% (\$107.50 x 0.75 = \$80.625); while we invested \$107.50 asset B and gained 35% (\$107.50 x 1.35 = \$145.125). The new portfolio value is \$225.75, which divided by 2 gives more or less \$112.88 (column 7).

So far, we do not see a significant difference between column 6 (the average of the two assets without rebalancing) and column 7 (the result of rebalancing).

At the start of period 3, the excess part of asset B is sold and the proceeds are used to purchase asset A so that we are back to 50/50. We now have \$112.88 of A and \$112.88 of B.

During period 3 (for simplicity, values are rounded), we invest \$113 at 35% in A (\$113 x 1.35 = \$153) and \$113 at -25% in B (\$113 x 0.75 = \$85). The sum of the two results divided by 2 gives \$119, which we show in column 7 of rebalancing.

At the end of the period 3, we begin to see a difference in favour of rebalancing: \$119 in column 7 versus \$109 in column 6. The latter is the sum of the value of A and B without the effect rebalancing.

At the start of the period 4 we invest \$119 in A and \$119 in B after reducing the amount of A at the end of the period 3 and increasing that of B.

And so on, one period after another. We see that at the end of the ten periods, the cumulative benefit of rebalancing has dramatically increased the overall yield of the portfolio. This is why institutions apply rebalancing in the management of their portfolios. The benefit to performance is obvious, in addition to reducing the overall risk of the portfolio.

Each act of rebalancing involves costs: selling shares of the surplus asset and buying those in deficit involves commission charges. The more often a portfolio is rebalanced, the greater the cost. One way to reduce costs would be to invest new funds in the portfolio. Another disadvantage could be the tax on gains from the excess shares sold.

## Frequency of Rebalancing

Most investors do not rebalance their portfolios. However, for investors who would like to do so, three methods can be used:

1. Once a year, for example on a birthday;
2. When the amount of an asset exceeds, for example, 5% of the other(s);
3. A combination of the two approaches.

Method 2 is better than method 1 since it reacts quickly when an imbalance is obvious. By contrast, method 1 is good because an investor may forget to rebalance, but it is unlikely that they forget their own birthday.

The goal of a set rebalancing frequency is to reconcile the cost associated with this activity with the advantages it provides.

In our example, the portfolio consists of only two assets. If one has several assets, the principle of rebalancing remains the same.

It is also good to know that there is software available which calculates the rebalancing of a portfolio for you.

## The author

### Charles K. Langford

PhD, Fellow CSI

Charles K. Langford is President of Charles K. Langford, Inc, Portfolio Managers. He teaches portfolio management at School of Management (École des Sciences de la Gestion), University of Québec (Montréal). He is the author of 14 books on portfolio management, derivatives strategies and technical analysis.

Until 2007 he has been vice-president overlay risk management for Visconti Venosta Teaspoon Approach Management, Ltd. Until 1990 he was portfolio manager for Refco Futures (Canada) Ltd.

He has received a Bachelor degree from Université de Montréal, a Master degree and the PhD from McGill University (Montreal); he is Fellow of CSI (Canadian Securities Institute).