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How to calculate a compound annual growth rate

Calculating an average rate of return over time provides a useful method of comparing the performance achieved by different firms or assets.

The formula to calculate a compound annual growth rate (CAGR) is:

[CAGR] = ( ( [Final Value] / [Inital Value] ) ^ ( 1 / ( [Ending Year] - [Starting Year] ) ) ) - 1

Key definitions

  • CAGR – Compound Annual Growth Rate
  • Ending Year – The final year of the period for which performance is being evaluated.
  • Final Value – The asset value at the conclusion of the evaluation period.
  • Initial Value – The asset value at the beginning of the evaluation period.
  • Starting Year – The initial year of the period for which performance is being evaluated.

What is a compound annual growth rate?

A compound annual growth rate ignores market volatility by providing a smoothed average return over a time period.

Why calculate a compound annual growth rate?

The compound annual growth rate provides a rough performance comparison method across firms. For example, how did the results of a group of fund managers or financial advisor compare over the last 5 years?

As with any statistic, using CAGR comes with health warnings. A compound annual growth rate provides a rear view mirror on what did happen but makes no attempt to project what may yet happen.

As an average the CAGR smooths away volatility, meaning the actual experience may have been a bumpier ride than the average makes it appear.

Always “trust, but verify” by being mindful of the context in which a CAGR is presented. A firm may boast of outperformance over the last 3 years, however their base may have been lower than their competitors after some dire results the year before the evaluation period.

Illustrative example

I want to calculate the 5-year compound annual growth rate of an investment property.

[CAGR] = ( ( [Final Value] / [Inital Value] ) ^ ( 1 / ( [Ending Year] - [Starting Year] ) ) ) - 1

[CAGR] = ( ( 684,000 / 492,500 ) ^ ( 1 / ( 2017 – 2012 ) ) ) – 1 = 6.79%

Next I want to compare that CAGR figure to the equivalent period return of the S&P500.

[CAGR] = ( ( 2,673.61 / 1,258.86 ) ^ ( 1 / ( 2017 - 2012 ) ) ) - 1 = 16.26%

Based upon these figures we can conclude that the compound annual growth rate of stock market outperformed the investment property over this period.

CAGR can tell a number of stories. Now consider the performance of the equity I contributed to the property, as opposed to the money borrowed from the bank.

[CAGR] = ( ( ( 684,000 – 443,250 ) / 49,250 ) ^ ( 1 / ( 2017 - 2012 ) ) ) - 1 = 37.35%

This time the property’s compound annual growth rate convincingly outperformed the stock market. When applied judiciously leverage can be a powerful growth accelerator. This highlights how important it is to understand the figures behind quoted CAGR values!

Next Steps

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How to make $1 million in one hour

Ever wondered how to make $1 million in one hour? Or to be more precise, with one hour’s effort? I’ll show you at the end of this post.

I’m not sure about you, but I find those 20/20 hindsight moan-alogues like “if only I’d bought Google shares at $50 back in 2004” singularly unhelpful.

My view? Either invent the time machine or just shut up.

If you invented the time machine there are much better uses for it than riding a rising stock!

History repeats, so listen to what it tells you

That said, history has a tendency to repeat, so there are some past patterns that are worth paying attention to.

Woulda/Coulda/Shoulda #1: If you’d invested USD$1 in a S&P500 index tracker in 1988 on “set and forget” basis, at the end of 2017 you would have an inflation-adjusted investment worth USD$4.76, a return of 476% over roughly 30 years.

S&P500 Price Return

S&P500 Price Return

Conclusion: Over the long term stock markets have tended to go up. However nothing is guaranteed, so Newton’s “what goes up must come down” experience with apples may hold true over the even longer term. Economists have the final say: “in the long run we are all dead“.

in the long run we are all dead

Woulda/Coulda/Shoulda #2: If you’d invested that same USD$1, and reinvested the dividends, then you would have an inflation adjusted USD$9.28 at the end of 2017, a return of 928% over roughly 30 years.

S&P500 Total Return

S&P500 Total Return

Conclusion: Opportunity cost is brutal and unforgiving. Money spent is gone forever. Money invested could attain an infinitely renewable flow of funds.

Opportunity cost is brutal and unforgiving

Woulda/Coulda/Shoulda #3: When it comes to longer-term returns, compounding dividend reinvestment has accounted more than half the total investment return over the last 30 years.

S&P500 Total Return Dividend Reinvestment

S&P500 Real Total Return split by Price Return and Dividend Reinvestment compounding.

Conclusion: To paraphrase something Albert Einstein apparently never actually said: “compounding is the most powerful force in the universe”.

Scientists

Great minds like simple investment approaches. Image credit: Brick Pimp and Jedipearse.

Pretty charts, but so what?

We’ve all seen these sort of charts before, and for most of us it hasn’t changed out behaviour one iota.

Now Iets bring it to life.

Some remarkable things occurred in the 1980s. Tim Berners-Lee invented the internet. Motorola released the first mobile phone. A generation of self-taught programmers cut their teeth on the Commodore 64 computer.

And in the UK Margaret Thatcher launched the tax-free Personal Equity Plan account (later rebranded the Individual Savings Account).

The basic idea behind an Individual Savings Account (ISA) is you invest after-tax savings up to an annual limit. Any subsequent income or capital gains earned are entirely tax-free. The money is available whenever you want it. There is no age restriction or pension/superannuation style lock-in.

American readers, this is the equivalent of a Roth IRA without the age restrictions.  Canadian readers, this is the equivalent of a Tax-Free Savings Account. Australian and New Zealand readers… in this case it sucks to be you, write to your local electoral representative asking them to implement a tax-free savings/investment account.

To illustrate how powerful an ISA can be, consider the chart below. It demonstrates the inflation adjusted results if an investor had fully utilised their annual ISA contribution limits over the past 30 years to invest in the S&P500 index, on the first day of each tax year, at the prevailing Bank of England spot rate, and reinvested all dividends.

Woulda/Coulda/Shoulda #4: Fully utilising the annual contribution limits available to every taxpayer potentially achieves remarkable results, such as a portfolio worth nearly USD$5 million (over GBP£3,600,00) at the end of December 2017.

S&P500 Total Return ISA Contribution

S&P500 Total Return ISA Contribution

Conclusion: Governments encourage the usage of legally tax-free savings accounts that, when used effectively, are more than capable of making an investor rich (enough) by most definitions of the word.

In the real world there would have been some brokerage, management and platform fees incurred. These would have created a drag on returns, however the point is that comparable results are achievable with minimal effort.

How to make $1 million in one hour

At the start of this story I promised to tell you how to make $1 million in one hour. The answer is at the start of each tax year fully fund your tax-advantaged savings account, and instruct your broker to invest the balance in a low-cost index tracker fund.

It will take you no more than 5 minutes per year, but from a financial perspective, it will probably be the best 5 minutes you spend all year.

After 12 years (1 hour / 5 minutes = 12) your investment will have compounded, potentially providing you entry to the “two comma club”.

Next Steps

  • As always, “trust, but verify”,  your mileage may vary.
  • If a tax-free savings account is available in your jurisdiction, make use of it!
  • If you liked this post then please share it with your friends.
Disclaimer: I may receive a (very) small commission from any purchase you make via links on this website.
How to unitise your portfolio

Deciding to unitise your portfolio allows you to consistently track and compare investment performance, regardless of capital flows.

The process to unitise a portfolio is:

  1. Choose an arbitrary initial “unit” price.
  2. Calculate how many “units” your portfolio initially contained using the formula:
[Unit Count] = [Initial Portfolio Value] / [Initial Unit Price]

Key definitions

  • Initial unit price – an arbitrary number you make up to act as a basis for comparison.
  • Portfolio value – the market value of the portfolio you wish to monitor for performance.
  • Unit Count – the number of units that your unitised portfolio contains.

What is unitisation?

Unitisation is a technique that converts the value of an asset or portfolio into “units”, the performance of which can then be tracked and compared with benchmarks or across asset classes.

Why unitise your portfolio?

If you have ever wanted to compare your returns against published benchmarks or the performance of investment managers then you are going to need unitisation.

Comparing investment returns across different asset classes or portfolios can be challenging. This is particularly true when the investor draws down capital or makes additional investments.

Unitisation allows investment performance to be tracked and compared irrespective of capital flows.

How to calculate unitised performance?

Tracking and comparing unitised performance requires the recalculation of the unit price based upon the current value of your portfolio.

The formula for calculating the current unit price is below.

[Current Unit Price] = [Current Portfolio Value] / [Unit Count]

To determine the portfolio performance you simply calculate the percentage change in unit price:

[Percentage Change] =  ( [Current Unit Price] - [Initial Unit Price] ) / [Initial Unit Price]

Now you can compare your portfolio’s performance to benchmark returns.

How to adjust for capital flows?

Capital flows do not impact the unit prices that you are using for monitoring and performance comparisons.

Capital flows do result in a change in the number of “units” that you hold. The calculation for that is:

[Unit Count Adjustment] = [Capital Flow] / [Current Unit Price]

To recalculate the total number of “units” held just apply the adjustment to your previous unit count.

[Revised Unit Count] = [Unit Count] + [Unit Count Adjustment]

How to handle fees, dividends, rent, taxes, etc?

For unitisation purposes treat portfolio related cash flows as capital flows.

Wherever income, regardless of source, remains within the portfolio treat it as a capital inflow, which increases the number of units held.

If an expense is met from within the portfolio then treat it as a capital outflow, which reduces the number of units held.

Illustrative example

I want to compare the performance of an investment property against the S&P500 Price Return over 2017.

Asset class comparison

Asset class comparison. Image credit: Sean Kenney.

First calculate the number of units held at the start of the comparison period, using an arbitrary unit price of 100.

[Units] = [Initial Portfolio Value] / [Unit Price]

[Units] = [£672,000] / [100] = 6,720

The net income generated by the property partially supports my cost of living, so there are no capital flows to adjust for.

Next calculate the unit price at the end of the comparison period.

[Current Unit Price] = [Current Portfolio Value] / [Unit Count]

[Current Unit Price] = [£684,000] / [6,720] = [101]

Finally determine the annual return.

[Percentage Change] =  ( [Current Unit Price] - [Initial Unit Price] ) / [Initial Unit Price]

[Percentage Change] = ( 101 – 100 ) / 100 = 1%

The S&P500 price return in 2017 was 19.42%.

The investment property price return during the same period was 1.00%.

Based upon that analysis the property underperformed during 2017.

Once inflation has been accounted for this property’s value actually declined in real terms 2017.

Next Steps

  • Unitise and compare your investment performance across asset classes.
  • Analyse how you performed compared to industry benchmarks.
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Disclaimer: I may receive a (very) small commission from any purchase you make via links on this website.
How To Adjust For Inflation

Adjusting historical amounts for inflation requires the use of inflation index values published by statistical collection agencies such as the Office of National Statistics.

The formula to adjust a historical nominal amount for inflation is:

[Real Amount] = [Nominal Amount] * ( [Later Period Index Value] / [Earlier Period Index Value] )

Key definitions

There are several ways of describing the value of an item:

  • Nominal Amount – the amount that was actually paid for an item when it was purchased.
  • Real Amount – the inflation-adjusted amount paid for an item, what it would cost using today’s money.

What is inflation?

Inflation represents the reduction over time of the purchasing power of a unit of currency.

In other words, a dollar from the past used to purchase a larger quantity of goods than a dollar from today.

Why adjust for inflation?

People are prone to remembering prices in the past being lower than they are today. Adjusting prices for inflation allows us to compare amounts across different time periods using a common basis. Only then can we assess whether things have gotten more (or less) expensive.

Illustrative example

The battery box that powers my kid’s Lego train recently died. Was the replacement part more or less expensive today than it had been 40 years ago?

In 1978 a Lego model train battery box cost £1.35.

In 2018 the equivalent part cost  £11.99.

Lego prices 1978 to 2018

Lego price comparison 1978 to 2018. Historical data sourced from Brickset.com.

I looked up the Office of National Statistic’s Retail Price Index values for 1978 and 2016 (the most recently published figure available at the time of writing). I then plugged those values into the inflation adjustment formula.

Real Amount = [£1.35] * ( 258.56 / 43.11 ) = £8.09

Today’s price of £11.99 is greater than the inflation-adjusted historical amount of £8.09.

Therefore we can conclude that the price of this item has increased at a faster rate than inflation.

Real economists adjust for inflation

Any good comparative analysis of financial values across time periods will always adjust values for inflation. Failing to do so distorts the facts and tells an incorrect story.

Consider the two charts below. The Department of Education recently issued a report highlighting that university graduates typically earn more than non-graduates. On the left is a chart displaying the median earnings in nominal amounts. On the right is a chart displaying those same median earnings, but this time as inflation-adjusted real amounts.

Adjusting for inflation

Adjusting for inflation

Both charts highlight the earnings difference. The trend lines on both charts finish at identical points.

Now compare the direction of the lines between the charts.

The nominal values chart happily shows earnings increase over time. It tells the story that investing in yourself by completing university will result in higher earnings that increase over time.

The real values chart tells a very different story. What jumps out is the fact that inflation-adjusted earnings have been falling over the last 10 years, at a faster rate for graduates than non-graduates!

While it is still true that graduates earn higher incomes than non-graduates, the context for the invest in yourself decision has changed. Would you be as likely to take on lots of student debt if you knew your standard of living (funded by real wages) would fall rather than rise?

Whenever you see period-on-period comparisons or time series trends presented, always “trust, but verify” that the figures being discussed are using real and not nominal values. If they do not then critically assess the conclusions being drawn.

Next Steps

  • Validate your trends, forecasts, and projections to ensure you are accounting for inflation.
  • If you liked this post then please share it with your friends.
Disclaimer: I may receive a (very) small commission from any purchase you make via links on this website.