Introduction

This article focuses on calculating returns from a Zecco account; Zecco is a discount broker that once offer free trades.  This policy changed in early 2011.  The company allows clients to published their annualized return.  The company has provided details of how this return is calculated, but it likely follows industry norms for mutual funds.  This article focuses on how to determine a return for yourself.  This article complements an earlier submission on the Dividend Discount Model.  After this article, you'll find an Excel-based description on how to calculate the annual return on your portfolio using cash flows.

Calculating an Interest Payment

Interest is a concept with which everyone is pretty familiar.  Checking accounts used to—and many still do—pay interest: periodic payments of cash the size of which depends on the balance in your account.  So, for example, if an account pays a paltry 1% a year (interest is normally expressed in terms of an annual interest rate) and you invest $100, at the end of that year, you’ll have $101 in your account.  That $100 consists of the original $100 that you deposited plus $1, which is the interest on that deposit.  The $100 is often referred to as the principal; the $1 is the interest.

In answer to another question, a percentage—like 1%—is how information is presented.  When doing math, it’s represented as 0.01.

$100 + $100 x 0.01 = $101
$100 x (1 + 0.1) = $101, so
$100 x 1.01 = $101

Simple and Compound Interest 

There are two types of interest.  The interest can be simple, which means that each interest payment is a percentage of your original investment.  In the example just discussed, each annual payment is 1% of the principal or $1.  Even if the rate of interest changes and, say, doubles to 2% each year, simple interest is paid on the principal.  In the year that the rate of interest increases, the interest payment becomes $2.  (We’ll assume that the interest rate changes right at the start of a year.

A more common scenario is that the interest is compound, which means that interest is paid on everything in your account.  So, for example, at the end of year one, when an interest payment is made, you would have $101 in the account.  At the end of year two, interest is calculated on the balance of $101, so you’d have $1.201 in your account as opposed to someone with $1.200 in an account that pays simple interest.  Big deal.
Here’s an example of how the difference grows over time.  The first example which follows shows an account that receives simple interest of 10% on an investment of $1,000.  Each year, interest of $100 is deposited into the account; this interest is calculated on the original investment of $1,000.
In both tables, the first column is the date; the second is the account balance on that date; the third column is the amount of interest earned and the final column is the resulting balance.

Account paying simple interest

Note the payment of $100 each year and that's it.

Account paying compound interest

Here the payment grows since interest is being paid on top of interest: the interest is compounding.

The column that’s shaded in blue is the balance on which the interest is paid.  The last column in each table is the resulting balance after the interest is paid.  At the end of ten years, the difference is quite large, $594, showing how rapidly interest paid on interest can grow the balance of an account.
A regular dividend payment is something similar interest in a savings account.  If take the cash dividend and don’t reinvest it in the same stock, it’s more similar to simple interest; reinvesting it in the same stock makes it somewhat akin to compound interest.  It’s not exactly the same, since the price of the stock changes too, but you get the picture.

Interest and Valuing Bonds 

A bond is a different instrument, but a discussion is useful if we’re to touch on the topic of “discounting cash flows.”  A bond is a security, just like a stock, but it’s a debt security, whereas a stock is an equity security.  A bond represents an agreement between two parties: one lends the other money and the party that receives the funds promises to pay regular interest for a specified period of time and to make a final payment at the expiration of the contract.  This final payment is paid when the bond “matures,” at which bond the contract between the two parties ceases to exist and no further interest payments are forthcoming.

Some terms required to understand bonds 

There are some terms that will be used that need to be explained.  Bonds are purchased to receive regular payments and redeemed for a fixed payment when the bond reaches the end of its life.  The first is the face value of the bond, which is the value at which the bond can be redeemed; in the last paragraph the final fixed payment received by the lender when the bond matures is the face value of the bond.  In the examples that follow, each bond will have a face value of $1,000.  This face value concept is a little confusing to some people, but think of the following: every postage stamp has a face value—it’s worth 5¢, 10¢, 44¢.  If you found a stamp that was 100 years old and put it on an envelope, the post office would see the face value and demand more stamps be added.  A collector, on the other hand, might pay you thousands of dollars for a rare stamp.  The face value doesn’t necessarily equal the real value.

The second term is the coupon rate, which is the amount of interest paid on the bond; the coupon rate is expressed as a percentage of the face value of the bond.  Bonds are traded and while the face value of the bond remains the same, the price paid for a bond with a face value of, say, $1,000 may be higher or lower than that amount.  The reason will become clear in a moment.

Example of payments from a bond 

So, for example, imagine there’s a bond with a face value of $1,000, a coupon rate of 8% and a maturity date in twenty years.  The issuer of the bond will pay the owner 8% of the face value a year, that’s $80, and bond will be redeemed for $1,000 when it matures.  So imagine that this bound was purchased on January 1, 1991, then it will mature twenty years later on December 31, 2010.  The schedule of payments from the issuer is as follows:

Where the final payment on December 31, 2010 of $1,080 represents the redemption value of the bond (remember it has a face value of $1,000) along with the final interest payment of $80.  So far, so good: you lend someone $1,000 and you generate $1,600 in interest payments over twenty years and you get your $1,000 back again.

Working backwards to check the coupon rate 

So let’s say it’s December 31, 1990 and the potential buyer of the bond, who’s happy with an 8% return, wants to be certain that he’s getting the return he expects.  He realizes the issuer of the bond is asking for $1,000, so he decides to look at each of the twenty payments separately and decided if each of them is giving him 8% annually.

He starts off with the first payment of $80 that he’s going to get on December 31 of 1991.  He asks himself the following question: how much would I have to invest on January 1, 1991 at a rate of 8% a year to end up with $80 at the end of the year?  In other words, this $80 is eight per cent bigger than what amount of money.

The calculation isn’t that difficult.  We saw above that if the rate of interest is 1% a year, you multiply the principle ($100 in the example) by one plus the interest rate (expressed as a decimal) to get the value twelve months later.  So, if you know the value twelve months later, dividing it by one plus the rate of interest should give you the principal.

The value turns out to be $74.  To convince yourself further, increase $74 by 8%, or 0.08, and you get $80.

Next, the payment at the end of 1992 is examined.  This determination is a little trickier.  Now the person is asking himself: what do I have to invest today that will grow at 8% during 1991 and another 8% in 1992.  The first calculation that was just done helps with this second one, because you know that $74 on January 1 the next year, 1992, becomes $80 on December 31, 1992.  (It’s just been demonstrated that over the course of one year, if $74 grows at 8% it becomes $80.  So the $80 at the end of year 2 is worth $74 at the start of the year.)  The big question is what is $74 at the end of year one (the end of year one is the start of year two) worth on January 1, 1991?  That calculation is easy: instead of putting $80 in the equation above, put $74.  Divided $74 by 1.08 and you get $69.  (Note, I have rounded to the nearest dollar, but it’s no big deal.)

So, you’ve now demonstrated that to someone who wants an 8% return, $80 paid on December 31, 1991 is worth $74 on January 1, 1991.  Further, the next payment of $80 on December 31, 1992 (the following year) is worth $69 on January 1, 1991: the day on which the person is considering buying the bond.

To get to that figure of $69, you started off with $80 (the payment in two years) and divided it by 1.08 (to get the value at the start of the second year), then you divided that number by 1.08 (to get the value at the start of the first year).  You’ve done the following:

Fear not if you don’t like math.  What’s presented above is what’s discussed in the previous paragraphs.  The last component is the most important, since this figure is the number of years in the future that you receive the payment.  For example, if you get $80 in year 3 and you want to know what you’d have to invest today at an annual rate of 8% to get $80 three years in the future, just divide $80 by 1.08 three times.  If you round to a whole number you get $64.

The person might do this calculation for each of the twenty payments, figuring out what he’d have to invest today to get, say, $80 in fifteen years: the answer is $25.

The final payment is $1,080 which would be due in twenty years, can be analyzed in a similar manner; if a person knows he or she can invest at 8% a year, then $232 invested at

The result is that the value on January 1, 1991 of each of the future twenty payments can be calculated and summed up to get the total current value of the bond.  Guess what: it’s $1,000: the result is tabulated below.

The column on the right had side is simply what someone would have to invest at 8% a year to get to value in the middle column on the given date.  So, if you invested $20 on January 1, 1991 at 8% a year, you’d reach $80 (the payment from the bond) on December 31, 2008.

The process that we just went through is an example of discounting a stream of future cash flows.  You’ve looked at a future value, $80 on December 31, 2008, and discounted it back to a value on January 1, 1991; that value was $20.  That value of $20 is the present value of a future cash flow.  When you add up the present value of each of the twenty future cash flows you come up with a current value of $1,000 for the bond—which turns out to be the face value for this debt security.

Pricing a bond if the required rate and coupon rate are not the same 

Let’s make the problem trickier.  Let’s say once again that the issuer has a bond with a maturity date twenty years into the future, a coupon rate of 8% and face value of $1,000.  Let’s say this time we have an investor who has happily earned 12% annual from her investments.  If she is offered this bond, she knows it’s too pricey for her tastes: we’ve just demonstrated above that the bond is worth $1,000 to someone who’s happy with 8% a year.  If someone is used to getting 12% annually, how would she value the bond?

Forget about the face value of the bond for a moment, that’s just a number on a stamp: as we discussed earlier, a stamp could be worth more or less than its face value; a bond could too.

The investor is interested in figuring out the current value to her of the future stream of twenty cash flows: twenty points in time when she will receive payments.  Those payments are $80 at the end of each of the first nineteen years and $1,080 at the end of year twenty.  (Even if the price at which the bond is sold isn’t $1,000, the contract says the buyer gets the face value when the instrument matures; it doesn’t matter what she paid to own or be the beneficiary of that contract.)

To calculate the value of that bond to an investor today (the term is net present value), we’d conduct the exact same exercise as described above.  We’d look at each of the twenty cash flows and ask the question: what would I have to invest today to get that cash flow in year 1 or 2, or 14 or 19—whatever the year in question might be.  The difference is that the investor knows that she can invest those funds elsewhere and earn 12%, so she wants to know what she has to invest today at the rate of 12% to get the same cash flow—the same amount of each of the twenty payments.  The coupon rate to her means nothing: it’s only how the payment is calculated not what the bond is worth.  She determines the worth of the bond, the cash flows, based on her expectation of a rate of return.

So, this time, the exercise would be repeated; however, instead of using 8%, we’d use 12%.  To get to $80 at the end of year one, you’d have to invest $71 today, January 1, 1991 at a rate of 12%.  You repeat the exercise for each of the other nineteen payments, accounting for the fact that each of them is further away in the future; add them all up and viola, you have the value to you of those twenty payments:

Just as you might have anticipated, the investor with higher expectations of a rate of return would offer a lower price for the bond.  If that’s the best price that the issuer can get, the coupon rate is 8%, but the effective rate is 12%.

You could convince yourself, by going through a similar exercise, that if there was an investor who was happy to get a 5% return, then shown the cash flows calculated from a bond with a coupon rate of 8%, that person would place a value of $1,374 on the instrument.

Introduction to discounted cash flows and net present value 

As discussed, this process is called discounting a set of future cash flows to determine a net present value.  The math can be pretty daunting for some people, but the reasons it’s done are clear to most: the question asked in another thread: would you prefer $100 today or $1,000 in a year.  If the only meaningful factor today is the size of the cash flow, then a rational person would likely defer receiving payment and settle for $1,000 in a year.  If we lived in a period of hyper inflation—hundreds of percent a year—a rational person might grab the $100 today and run.  Most people know intuitively that if they won that $380 million lottery they won’t get a lump sum payment of the “face value” of the prize.  The lottery pays the prize in equal installments over many years (twenty or something like that), but if you want a lump sum cash payment, it will be “discounted back” to what a stream of equal payments totaling $380 million is worth today.  That lump sum would come to around $240 million.

You have to decide for yourself the rate at which you discount those future cash flows.  If you’re too aggressive, you will value the opportunity at a very low price, which, if it’s much lower than the market price, means you won’t consider buying that asset.  Discounting cash flows was what was done in the blog that discussed the Dividend Discount Model.

In summary

Discounting a stream of future cash flows is at the heart of many valuation approaches, so it’s worth getting your arms around the issue.  If you appreciate why people place more value on receiving $1 today than a $1 a year from now, you’ve done the heavy lifting behind the concepts of net present value and discounting future cash flows!