RATES OF RETURN : UNDERSTAND, MEASURE AND COMPARE
What do rates of returns measure and why are they important?
A rate of return measures the $ profit divided by the $ invested. This is the metric most used to compare different investments. It is expressed as a percent because investment opportunities come in all sizes. Absolute dollars of profit do not allow for comparison, but a percentage is 'relative' to any size investment. The period of time measured is almost always presumed to be one year.
The profits from an investment can come from income received during the holding period, and also capital gains from the eventual sale. Together these are called the "total return". When comparing investments always use the total return. E.g. buyers of dividend stocks and preferred shares too often look only at the dividend yield, and ignore the potential for capital gains/losses. E.g. before investing in foreign countries consider both the security's return plus the potential for foreign exchange gains/losses.
Detailed instructions for measuring your own portfolio's rate of return are on the page Keep Track.
Are quoted rates of return comparable between investments? NO !
Each of the asset types in the box below has its returns normally calculated in a different way. For most uses the results are 'good enough' for comparison. But for any finetuning of a decision take the time to translate the 'normally' calculated return into a 'true' economic rate of return. The IIAC ( investment industry selfregulating body) has documented Canadian measurement conventions for fixed income products. Conventions may differ between countries.
Security Type  Method of Calculation 
Flat rate GIC, CD, CSB, term deposit  compound yearly, or earlier when interest payable, measured as simple interest 
Credit card debt, bank line of credit, bank acc't  accrue daily, compound monthly, measured as compound interest 
Bond yield  compound every 6 months, measured as simple interest 
Bond coupons valued at sale  accrue daily, measured as simple interest 
Stock index  ignores cash income, continually compounding on price alone 
Stocks, total return stock index, mutual funds  continually compounding on price plus all cash income flows, measured as compound interest 
Stock dividend  does not compound when payable, measured as simple interest 
Cdn mortgage  compound with payment schedule, measured as the 'true' economic rate equivalent of a bond's simple interest 
US mortgage  compound with payment schedule, measured as simple interest 
Real estate  measured as cumulative interest over holding period 
Gross domestic product (GDP)  compound yearly, measured as a 'real' rate 
Government TBills  have their own specific rules 
The 'true' economic rate of return is what most people's understanding of it would be. People refer to it as the Compound Annual Growth rate (CAGR), Effective Annual rate, Annual Equivalent rate, Internal Rate of Return (IRR), discount rate, geometric mean, or Annualized Compound rate.. Essentially these all refer to the same concept. Different terms are used in different contexts.
E.g. if $100 invested at the beginning of the year grows to $112 by the end of the year, then the rate of return was 12%. To be more specific;
 The period used is one year.
 The income paid, or payable, or accrued is included in the ending value,
 Any income paid early is reinvested to earn its own income for the remaining portion of the year, or considered to have done so.
 That incomeonincome is included in the endofyear value.
What do those terms mean?
There are many different words used to describe the measurement convention used in different situations. Unfortunately people use different words for the same thing, and use the same word to mean different thing. Always clarify in your mind what is being meant, without preconceptions.
 REAL vs. NOMINAL returns
 ANNUAL vs. CUMULATIVE returns (also called HOLDING PERIOD return)
 TOTAL return
 AVERAGE returns (arithmetic vs geometric)
 REALIZED profits vs. PAPER profits
 ACCRUED interest
 TO COMPOUND (verb)
 REGULAR vs. COMPOUND interest (adjective)
 SIMPLE vs. COMPOUND INTEREST (methodology)
 SIMPLE INTEREST (methodology)
 COMPOUND INTEREST (methodology)
 NOMINAL vs. EFFECTIVE rates
 ANNUAL PERCENTAGE RATE (APR)
 GOVERNMENT TBILL rates
 AFTERTAX returns
 REAL vs. NOMINAL returns:
 Real rates of return are what is left after the rate of inflation has been subtracted from the nominal rate. Much analysis of historical stock returns uses real returns because all investors demand at least the rate of inflation in order to justify deferring consumption, so it is premium above inflation that matters.
E.g. Longbond yields have historically paid 2% plus inflation. The '2%' is the real yield.
E.g. GDP is the yearly production of a country measured using the market value of items. Its yeartoyear change is heavily influenced by the inflation increases of the transactions. So the percent change in GDP is usually reported with the rate of inflation (GDP deflator) removed.
Instead of simple subtraction, you sometimes see the calculation of the real return as:
((1+return) / (1+inflation))  1.
E.g. when the nominal return = 5% and inflation = 3%,
((1+5%) / (1+3%))  1 = 1.9% real return (not 2%).
This is the technically correct math but the simple subtraction is good enough.
See also the "Nominal vs. Effective rates" section below, where 'nominal' is used in another context.
 ANNUAL vs. CUMULATIVE return (also called HOLDING PERIOD return) :
 Cumulative returns measure the total increase in the value of an investment over a number of years, not just one year. For example: if you bought your home for $100,000 and sold it 10 years later for $150,000, you had a 50% cumulative return.
Sometimes this measurement is the simplest, and perfectly valid, when comparing investments with the same time frame. But most times it is used to impress you because it produces a large number. You may not be told explicitly that it is cumulative  hoping you will think it is the annual rate earned each year of the investment. Even if they tell you, they count on you not being able to quickly convert it into a meaningful yearly return (only 4.1% in that example).
Most everyone thinks of rates of return in the context of a one year period. That percentage is 'meaningful' to people. They have certain benchmarks in their mind for comparison. They know the yearly rate for term deposits or for their bank's Line Of Credit. They know the yearly inflation rate. When comparing investments, yearly rates are the most logical, because investment terms may differ.
Some people use the term "total return" when referring to cumulative return.
 TOTAL return :
 You hear the term Total Return used most often to clarify that both the capital gains plus all dividend and interest income is being measured in total. E.g. Stock indexes measure only the price changes of their component companies. But some indexes publish their Total Return variant that includes dividends paid and the income earned by the reinvestment of those dividends. It is the Total Return Indexes that would be used to benchmark your own portfolio returns.
See also the "Annual vs. Cumulative" section above, where 'total return' is used in another context.
 AVERAGE returns (arithmetic vs geometric) :
 You know how to calculate an arithmetic average. But the question is: "Do investors WANT to find an average return of a multiyear period?" Consider the example
 Start with $100
 Earn 100% return the first year. ($100*100%=$100 profit)
 Lose 50% the second year. ($200*50%=$100 loss)
 End with the same original $100
The arithmetic average return of the two years would be (100 plus negative 50) / 2 = 25%. But over the whole period there was a 0% true return ("geometric mean"). The arithmetic average undervalues losses because they are calculated on the larger $ investment at the year's start  while gain are calculated on the smaller $ investment at the year's start. The arithmetic mean will always be larger than the geometric mean.
Fortunately, when you hear the term 'average' used by mutual funds or others in the finance industry, it almost always refers to the geometric mean that you DO want to use to compare investments. They use the term 'average' because that is the concept everyone understands.
The greater the volatility of individual year's returns, the greater the difference between the arithmetic and geometric means. The difference can be estimated by the equation:
Difference = 1/2 Variance = 1/2 StandardDeviationSquared
E.g. US equities historically had a 20% standard deviation with a 10% average (geometric) return. The expected difference would be 1/2 * 0.2 * 0.2 = 2%. The expected arithmetic mean would be 10% + 2% = 12%.
 REALIZED profits vs. PAPER profits :
 Realized profits have been converted to cash by a transaction. E.g. dividends have been received, or an asset has been sold. Paper profits have had no transaction to prove their value. E.g. increases in value have been estimated by transactions in the market of similar assets. This distinction does not affect the method chosen to measure the rate of return.
Many investors have a preference for high dividend stocks because they feel this cash is more 'real' than paper profits. But in the accumulation phase, those realized profits must be reinvested back into paper assets, leaving the investor no more sure of his wealth or gains.
 ACCRUED interest :
 Accrued interest is acknowledged as payable, eventually, but not yet booked (posted to your account). For example with credit cards, the interest expense for each day is calculated individually. Only at month end are they added together and posted to your account. The total accrued up to any midmonth date does not affect the calculation of interest for subsequent days. I.e. it has only accrued, not compounded.
Bank accounts and credit cards post all the daily accruals for the month, at the month end. Only then does it compound. With bonds, the accruals keep adding up for 6months, until they equal the interest payment due. When buying a bond or debenture, you pay the transaction price plus the portion of the next interest payment that has accrued since the bond's last payment.
 TO COMPOUND (verb) :
 Most investments have profits that compounds. E.g. a mortgage's interest compounds. That means that any unpaid interest that is due at month end, but not paid, is added to the balance of the debt ... so the subsequent interest is calculated on the nowbigger principal. If the mortgage payment is received, nothing is added to the principal.
Compounding reflects an activity that is factual (true or not). For example: Preferred shares have their attributes defined by the prospectus. The prospectus will state (e.g.) that the dividends are cumulative (accrue if unpaid), but none say that unpaid dividends compound (unpaid dividends never earn interest to compensate for their being paid late).
The frequency of compounding will always be at least as often as the scheduled cash flows. E.g. A monthlypay mortgage will compound monthly and a weeklypay mortgage will compound weekly. If it were not to compound, there would be no incentive to make the required payment  the eventual payment would be the same whether paid on time, or late. It is the compounding that creates the incentive to pay on time.
The more frequent the compounding the greater the true rate of return. This is because the income is put to work quicker, earning more of its own income.
 Principal at the start (P).
 Interest earned in 1st compounding period = (P * i%).
 Principal value then after the interest compounds = P + (P * i%) or P*(1 + i%).
 Interest earned in 2nd compounding period = P*(1 + i%) * i%.
 Principal value then after the 2nd period's interest compounds = P*(1 + i%)(1 + i%).
If the investment compounded monthly, then there would be twelve repeats of (1 + i%) in a year. The interest rate i% is not the yearly rate. It is the rate for only the compounding period  in this example the monthly rate. The 'true' yearly rate of return would be calculated from the the resulting (Principal at end) / (Principal at beginning)  1 ... as if the interest compounding each month does not get paid, whether it does get paid or not.
 REGULAR vs. COMPOUND interest :
 These terms are usually used to describe term deposits, GICs and CDs. They are meant to distinguish between
 products that pay out the interest earned when it becomes payable, for you to invest elsewhere (regular), and
 products that retain the interest and reinvest it within the product (compound).
 SIMPLE vs. COMPOUND INTEREST (methodology) :
 There are two ways to measure yearly interest, just like a distance can be measured in English miles or Metric meters. Or think of the two systems like two languages that both use the same words (interest rate), but to mean different things. When you hear the term 'compound' ask yourself first whether it is used as a verb or an adjective to indicate the measurement method.
When a product is described as "x% compounded monthly" (or weekly, etc) you know the rate is measured using simple interest methodology because none of that clarification is necessary when measuring with compound interest methodology.
The following two sections describe each in more detail.
 SIMPLE INTEREST (methodology) :
 To calculate the rate of return using simple interest methodology, add all the interest paid in a year together. No income earned on reinvested income is included. Divide that $ interest by the $ investment at the beginning of the year. Simple interest methodology ignores the timevalueofmoney (TVM means a dollar today is worth more than a dollar tomorrow). If a $100 investment pays $1 interest each month, simple interest methodology treats all the year's $12 as if paid at the year end. It would make no difference if all the $12 was paid after the second day. The return would still be measured as $12/$100 = 12%.
The APR or Annual Percentage Rate quoted for auto leases and credit cards are one example of simple interest discussed below.
 Bonds pay interest twice yearly. If they are quoted to pay 12%, then 6% is paid each 6 months.
 GICs and US mortgages may be paid every month. If their quoted rate is 12%, then 1% is paid/charged each month.
 Stocks' dividend yields are quoted as the total of all (normally 4) payments in the year, divided by the current stock price.
 COMPOUND INTEREST (methodology) :
 Compound interest takes into account the timevalueofmoney. It recognizes that being paid interest before the end of the year allows you to reinvest it to earn additional income (regardless what you actually do with the money). This is the methodology that calculates what has been referred to above as the 'true' economic rate. E.g. 12% simple interest that pays 6% twice a year would be quoted as 12.36% compound interest.
 Principal at the start (P = $100).
 Interest earned in 1st 6month compounding period = (P * i%) = $100 * 6% = $6.
 Principal value then after the interest compounds = P + (P * i%) or P*(1 + i%) = $106.
 Interest earned in 2nd 6month compounding period = P*(1 + i%) * i% = $106 * 6% = $6.36.
 Principal value at year's end = P*(1 + i%)(1 + i%) = $112.36 .
 Compound interest rate = 112.36 / 100 1 = 12.36%.
It makes no difference to the measurement process whether interest is paid out or reinvested within the original product. Interest paid out is presumed used to buy another investment that earns the same return.
Stock totalreturn indexes are measured using compound interest. Whenever the component stocks pay dividends, the dividends are considered to be reinvested to buy more index units. The value of the investment at the end of the year includes the value of the additional units bought plus any dividends subsequently earned by those units.
 NOMINAL vs. EFFECTIVE rates :
 The use of these terms is virtually synonymous with "simple vs. compound interest (methodology)" above. The terms are used in the context of quoting two different rates for the same product; one using simple interest methodology (nominal) and another using compounding interest methodology (effective). Warning, 'nominal' was also used above in the section "real vs. nominal returns".
 ANNUAL PERCENTAGE RATE (APR) :
 The APR quoted for auto leases and credit cards uses Simple Interest methodology. It deserves its own section here because the frequency of payments is almost never disclosed. While normally a Simple Interest quote will stipulate the payment frequency, sellers quoting APR assume you know the payment frequency. A quoted 12% APR assuming onceyearly compounding would give you a 'true' economic rate of 12%. A quoted 12% APR assuming monthly compounding would give you a 'true' 12.7% rate.
 GOVERNMENT TBILL rates :
 Tbills are very short term Treasuries that pay no interest. They are bought at a discount to their face value. The discount is considered to be the interest. Their % return is measured using a variant of Simple Interest. E.g. a $100 Tbill is purchased for $99. Its holding period return is 1 / 99 = 1%. If it has a 90 day maturity its quoted % return is 1% prorated by the day count. 1% * 365 / 90 = 4.1%.
 AFTERTAX returns :
 Whenever profits are taxed at different rates for different types of profit (interest, dividends, capital gains), any comparison of investments must measure net returns after all taxes are paid. This will be a personal metric when tax rates are progressive because personal marginal rates differ.
Start by calculating your effective tax rate for the asset's profits.
The calculation involves three steps. First determine the tax rates for each income type at your own personal tax bracket. Second revise the effective rate for any capital gains that will be lowered by holding for long periods that allow compounding taxfree. Third find the weighted average rate for the asset given the % portion of the total return coming from each type of profit. The Tax Burden spreadsheet has tabs for all those steps.
Once you have your effective tax rate, multiply the asset's % return by (1tax%). So an asset (A) earning a 10% profit taxed at 20% will have a higher after tax return than asset (B) earning 12% but taxed at 40%.
(A) 10% * (1  20%) = 8.0%
(B) 12% * (1  40%) = 7.2%
Buy a financial calculator
A financial calculator is the only way to deal with these inconsistencies. A $40 model will last you a lifetime and be used every day. Although not stocked most of the year, they are available during backtoschool sales in September. At other times of the year they can be found in university book stores and on line.
There are now phone apps that do the job. Look for those with the five basic inputs below. Many have built in situationspecific modules, but you never know what assumptions they have used in the background. The best calculator on the web is from Finance Calculator. Keep this link at the top of your Favorites list. If you want a hardcopy printout you can use these tables. These are what we used 'in the good old days'.
For a more comprehensive list of equations that derive from the timevalueofmoney read this page from Wikipedia. Damodaran has a pretty good webpage explaining Time Value Of Money.
Time Value of Money
The three timelines below exemplify most investments. Match the cash flow arrows of the time lines to the cash flows of your problem. Each uses 4 of the 5 variables below. You can solve for any of the variables, using a calculator, by inputting the other 3.
 Variables:
 1) n : The number of time periods. Each period can be a day, week, month or year, etc. The interest earned within the period will be considered to compound at the period end.
 2) i% : The interest rate is applied to the principal value at the beginning of each time period. If the time periods represent months, then the interest is the '% per month', etc. The rate is the same for all the periods. If this is not the case in your particular analysis, you cannot use these functions.
 3) PV : The initial investment (loan) is represented by the arrow for Present Value. The directions of the arrow represents cash going into, or coming out of the investment. The directions can be reversed with no change in the functions.
 4) FV : The value of the investment at the end of the time series is represented by the arrow for Future Value. Same as for PV, the direction of the cash flow can be switched.
 5) Pmt : The regular Payments of an annuity are all the same value and made at the time of compounding. You must input into the calculator whether the Payments happen at the beginning of the periods or at their end. The time lines below show both options: B) has Payments at the end of the period, C) has Payments at the beginning of the period.
A)
This time line shows the equality of cash at different times. There are no cash flows during the intervening time. Your calculator may call this function "interest", or something else. The variables for input are PV, FV, n and i%. Common uses for this timeline are
 Convert a quoted rate that uses simple interest to the true rate.
 Find the interest rate of a paydayloan.
 Find the price you should pay for a strip bond.
 Find the capital gains you realized from owning real estate over many years.
 Examples
B)
This time line shows a lumpsum investment (or loan) at the start, followed immediately be a series of equal payments that continue for a set period of time. This diagram shows the Payments happening at the end of the period. Alternately, the first payment may happen at t=0. Your calculator may call this function "loan", or something else. The variables for input are PV, Pmts, n and i%. Common uses for this timeline are
 Find the payments required for a mortgage.
 Find the rate of return implicit in an annuity purchased for retirement.
 Find the present value of an oil well.
 Examples
C)
This time line shows a series of equal payments that continue for a set period of time, until a lumpsum cash flow at the end. This diagram shows the Payments happening at the beginning of the period. Alternately, the first payment may happen at t=1. Your calculator may call this function "saving", or something else. The variables for input are FV, Pmts, n and i%. Common uses for this timeline are
 Determine how much must be saved every year in order to have $$ when retire.
 Find the rate of return implicit in the cost of life insurance.
 Examples
