#### What do rates of returns measure and why are they important?

An example of a ‘rate of return’ is the interest rate quoted for a GIC. This is the metric most used to compare different investments. It is expressed as a percent because different opportunities require different investment dollars. Absolute dollars of profit do not allow for comparison, but a percentage is ‘relative’.

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 paying stocks too often look only at the dividend and ignore the potential for capital loss/gain.
- Before investing in foreign countries consider the capital losses that may be caused by changing Foreign Exchange (FX) rates.

The expected total return is always determined by the investment’s risk (see Risks) – the risk of not earning the return predicted at the start. If you predict greater risk you must demand a higher return. You get a higher return by paying a lower price for the security.

#### Are rates of return comparable between investments? NO !

Don’t be deceived by any apparent simplicity. A quoted 12% return for each category of products below, results is a different economic return. Each category has its returns commonly measured differently. In order to compare investments, the quoted rates must be normalized to a true economic rate.

The true rate of return is the same as most people’s understanding of it would be. You may also hear 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, is included in the ending value,
- Any income paid early is re-invested to earn its own income for the remaining portion of the year, or considered to have done so.
- All income (primary and income-on-income) is included in the end-of-year value.

Security Type | Method of Calculation |
---|---|

Flat rate GIC, CD, CSB | compound when interest payable, measured as simple interest |

Credit card debt, bank line of credit, bank A/C | accrue daily, compound monthly, measured as compound interest |

Bond yield | compound every 6 months, measured as simple interest |

Bond coupons | accrue daily, does not compound when payable, measured as simple interest |

Stock index | ignores cash flows, compounds yearly on price alone |

Stocks, total return stock index, mutual funds | compound with every cash flow to/from owners, measured as compound interest |

Stock dividend | does not compound when payable, measured as simple interest |

Cdn monthly mortgage | compound monthly, measured as the simple interest derivation of 6-month compound rate |

US normal mortgage | compound monthly, measured as simple interest |

US weekly mortgage | compound weekly, measured as simple interest |

Real estate | measured as cumulative interest |

Gross domestic product (GDP) | compound yearly, measured as a ‘real’ rate |

Government T-Bills | have their own specific rules |

#### What do those terms mean?

There are many different words used to describe the type of income measurement being used in different situations. Unfortunately different people use different words, 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 mean)
- REALIZED profits vs. PAPER profits
- ACCRUED interest
- TO COMPOUND (verb)
- REGULAR vs. COMPOUND interest (adjective)
- NOMINAL vs. EFFECTIVE rates
- SIMPLE vs. COMPOUND INTEREST (methodology)
- SIMPLE INTEREST (methodology)
- COMPOUND INTEREST (methodology)
- GOVERNMENT T-BILL rates

**REAL vs. NOMINAL returns**:Real rates of return are what is left after the rate of inflation has been subtracted from the nominal rate. A lot of analysis of past returns uses real returns because all investors demand AT LEAST the rate of inflation in order to justify deferring consumption. E.g. Long-bond yields have historically been equal to 2% plus inflation. The ‘2%’ is the real yield. It represents the risk premium for term risk – ignoring the compensation for inflation.

E.g. GDP is the yearly production of a country measured using the market value of items. Its year-to-year 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.

Nominal returns refer to the commonly quoted full return (measured by simple interest or compound interest). E.g. If historical inflation was 3%, and real long-bond yields were 2%, then nominal yields were quoted as 5%.

Instead of simple subtraction, you sometimes see the calculation of the ‘real’ return as:

((1+return) / (1+inflation)) – 1.

E.g. ((1+5%)/(1+3%)) – 1 = 1.9%

This is the technically correct math but the simple subtraction is good enough.

See also the discussion below on “Nominal vs. Effective rates”.**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’s resulting large number is used to impress you. 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 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 rates of other investments. When comparing investments it is the most logical, because their terms may differ.

You may hear the cumulative return referred to as a Total 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.

The term ‘total return’ is also used when referring to what is called ‘cumulative return’ above. **AVERAGE returns (arithmetic mean)** :You know how to calculate an average. But the question is: “Do investors WANT to find an average return of a multi-year period?” Consider the example

- Start with $100
- Earn 100% return the first year. ($100*100%=$100)
- Lose 50% the second year. ($200*50%=$100)
- End with the same original $100

The average return of the two years would be (100-50)/2 = 25%. But over the whole period there was no 25% gain. There was 0% true return (“geometric mean”). Using averages means that any losses will be undervalued because they are calculated on the higher amount at the year’s start. The arithmetic mean (average) will always be larger than the geometric mean.

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%.

Fortunately, when you hear the term ‘average’ used by mutual funds or others in the finance industry, it almost always refers to the true return (geometric mean) that you DO want to use to compare investments. They use the term because everyone understands the concept of ‘average’. **REALIZED profits vs. PAPER profits **:Realized profits are profits that have been proven by a transaction. E.g. dividend dollars have been received, or an asset has been sold. Paper profits have had no transaction to prove their value. E.g. dividends were paid in additional shares, not cash; or increases in market value have been calculated but the assets not yet sold. This distinction does not affect the method chosen to measure the returns. **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 mid-month 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 6-months, 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)**: Some investments have interest that compounds. E.g. a mortgage’s interest compounds. It means that any unpaid interest that is due, but not paid, is added to the balance of the principal … so the subsequent interest is calculated on the now-bigger balance. Of course if the mortgage payment is received, nothing compounds.

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 monthly-pay mortgage will compound monthly and a weekly-pay 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 being applied is not the yearly rate. It is the rate for only the length of the compounding period.

**REGULAR vs. COMPOUND interest (adjective)** :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 (regular), and
- products that retain the interest and reinvest it (compound).

**NOMINAL vs. EFFECTIVE rates** :The use of these terms is virtually synonymous with “simple vs. compound interest (name)” below. 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”. **SIMPLE vs. COMPOUND INTEREST (methodology)** :There are two ways to measure yearly interest, just like a distance can be measured in English or Metric. 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)**: Simple interest methodology ignores the time-value-of-money (TVM means a dollar today is worth more than a dollar tomorrow). If a $100 investment pays 12% interest in a year, simple interest methodology treats all the $12 as if paid at the year end, even if some is actually paid earlier in the year. It would make no difference if all the $12 was paid after the second day. The return would still be measured as 12%.

To calculate the rate of return, all the interest paid in a year is added together. The ratio of that total to the investment $$ at the beginning of the year equals the rate of return. E.g.

- Bonds pay interest twice yearly. If they are quoted to pay 12%, then 6% is paid each 6 months.
- GIC’s 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.

Because this method of measurement ignores the time-value-of-money its results do not measure the ‘true’ economic rate of return. But for many situations it is ‘close enough’. **COMPOUND INTEREST (methodology)** :Compound interest DOES take into account the time-value-of-money. It recognizes that being paid interest before the end of the year allows you to reinvest it to earn additional income. E.g.** 12% simple interest **that pays 6% twice a year would be quoted as 12.36% compound interest.

- $100 investment, beginning of year
- + $12 primary interest earned for year ($100 * 12%)
- + $0.36 secondary interest earned on primary interest paid after 6 months ($6 * 6%)
- = $112.36 investments, end of year,
- or alternate calculation: Value at year end = $100*(1+6%)(1+6%)= $112.36
**12.36% compound interest**.

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 total-return 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.**GOVERNMENT T-BILL rates** :Their interest rate is measured with a specific convention. Where ‘P’ stands for the price paid, the interest rate = (100 – P) divided by P. Instead of normalizing it to a year, the interest is prorated by ‘t’ the number of days: (365 / t) * 100.

#### Buy a financial calculator

Buying 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 back-to-school sales in September. At other times of the year they can be found in university book stores.

There are websites that have calculators, but all try to dummy-down the inputs by making them specific to one use. Excel has these as functions, but they are very cumbersome for one-off calculations. There are equations you can memorize: but not for all variables: not for the variables you will be solving for most frequently. If you refuse to buy a calculator here are tables with the basic terms/rates.

#### Time Value of Money

The three time-lines 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 pay-day-loan.
- 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 lump-sum 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 lump-sum 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