Much of the discussion in this series on bond basics has been about the inverse relationship between bond yield and bond price: when one goes up, the other goes down, and vice versa. My goal in this post is to help you begin to understand the mathematical formula that specifies bond price in terms of bond yield, since understanding this can facilitate a deeper understanding of bond fundamentals. We can start by considering something familiar: earning interest in a savings account. We can develop the simple formula that describes this, then with some elementary algebra, we can build on it to develop the formula that gives us bond price in terms of bond yield.

In Part 1 of this series, I explained that a bond is basically a loan from an investor to a corporation or government. Similarly, a savings account is essentially a loan from a saver to a bank (or credit union). The bank compensates the saver for the loan by paying interest, very much like the corporation or government compensates the bondholder by paying interest. The main difference is that the saver can take back all or part of the loan from the bank at any time by removing money from the savings account. In bond terms, the term to maturity of the savings account is zero years, so we can think of it as an extremely short-term bond.

If you deposit $100 in a savings account with an annual interest rate of 1%, you will earn $1 in interest in one year, because 1% of $100 is $1 (taking percentages of 100 is trivial: 1% of 100 is 1, 10% of 100 is 10, 27.4% of 100 is 27.4, etc.). Adding this $1 in interest to your original $100 gives you a total of $101 in your account after one year. From now on I'm going to drop the dollar sign for convenience; e.g., 100 grows to 101 at the end of one year. Also, all interest rates will be understood to be annual interest rates.

If we use FV to represent the Future Value of the savings account at the end of one year, we can write this equation to describe the future value in terms of the initial value of 100 and the interest rate of 1%:

FV = 100 + 100 * 1%

Here I'm using * for the multiplication operator, so 1% of 100 is written as 1% * 100, which also can be written as 100 * 1% (due to the commutative property of multiplication).

The right-hand side of the equation above can be evaluated directly in a spreadsheet formula or by Google (copy and paste it into a Google search box to verify). Some calculators have a % key that also would allow you to evaluate the expression as written. However, it is standard to write such expressions using decimal numbers rather percentages.

In decimal form, 1% is 0.01. If you don't remember this from elementary school arithmetic, then consider that % can be interpreted as the arithmetic operation "divide by 100". Using / as the division operator, 1% = 1 / 100 = 0.01. Dividing by 100 is the same as moving the decimal point two places to the left (adding one or more zeros if necessary). So 100% = 1, 10% = 0.1 and 1% = 0.01.

Rewriting the above equation using 0.01 instead of 1%:

FV = 100 + 100 * 0.01

Based on the distributive property, we can factor out the 100 in the expression on the right-hand side of this equation, and rewrite it as:

FV = 100 * (1 + 0.01)

If we want to be able to apply this formula to any starting amount, instead of just 100, we can designate the starting amount as PV, which stands for Present Value. Similarly, if we want to be able to apply the formula using any interest rate, instead of just 1%, we can designate the interest rate as

FV = PV * (1 + i)

This is the basic

We can use some simple algebra to rearrange the above equation to answer the question, "what is the present value that will result in a given future value at the end of one year?" For example, what value do I start with to end up with 101 at the end of one year at an interest rate of 1%? To derive the formula to answer questions like this, we simply divide both sides of the above equation by (1+i), and move each side of the equation to the other side, which gives us:

PV = FV / (1+i)

(Still using / as the division operator). In words, the present value is the future value divided by the sum of one and the interest rate.

Of course if we substitute 101 for FV and 0.01 for i in this equation, and solve it, we will get 100 as the result for PV. We know this because we've already determined that 101 is the Future Value after one year given the Present Value of 100 and an interest rate of 1%. Just as we used some simple algebra to rearrange the TVM equation, we can rearrange our words to say that 100 is the Present Value given a Future Value of 101 and an interest rate of 1%.

In this version of the one-year TVM formula, in which we solve for PV in terms of FV,

How does this relate to bond price and yield? Before answering this, we must introduce one more TVM term:

That's a mouthful, but read it carefully while looking at the equation for present value in terms of future value and an interest rate (or discount rate) for a one-year period:

PV = FV / (1+i)

Now that we're talking about bonds, we'll replace PV with P (for Price), and replace i with y (for yield), and write it this way:

P = FV / (1 + y)

We can improve this further by expressing FV in terms of known bond characteristics.

The Future Value when the bond matures after one year is the face value of the bond plus the interest earned in one year. We learned in Part 1 that face value also is referred to as par value, or simply par. To avoid confusion between the terms future value and face value, I'll use the term par value or par from now on.

Since bonds are priced as a percent of par value, we note the price of a bond valued at par is 100 (100% of par value). In our calculations, we will use 100 as par value, since this will give us a result consistent with the bond pricing convention. So 100 will be the principal portion of the cash flow we receive after one year.

The interest earned in one year is the coupon rate times the par value of the bond, as discussed in Part 1. So for a bond with a coupon rate of 1%, the interest portion of our future cash flow will be 1% * 100 = 1.

Putting this together, the future cash flow, FV in the equation, is:

FV = 100 + 100 * 0.01

Replacing 0.01 with the letter r, for coupon

FV = 100 + 100 * r

Which we can rewrite as:

FV = 100 * (1+r)

Substituting the right-hand side of the above equation for FV in our bond price formula:

P = FV / (1+y)

In Part 2 I used words and logic to explain that the yield equals the coupon rate for a bond priced at par. Now we can verify this with the one-year bond pricing formula we've developed. For a bond with a coupon rate of 1% and a yield of 1%:

P = 100 * 1

P = 100

In Part 2 I also used words and logic to explain that the price of a one-year bond with a yield of 2% and a coupon rate of 1% is about 99. Now we also can verify this with the one-year bond pricing formula we've developed. For a bond with a coupon rate of 1% and a yield of 2%:

P = 100 * 1.01/1.02

P = 100 * 0.99

P = 99

Also in Part 2, we saw that the price of a bond with a coupon rate of 1% and a yield of 0% is 101; let's check that with our formula too:

P = 100 * 1.01/1

P = 100 * 1.01

P = 101

In Part 3 I mentioned that the inverse relationship between bond price and yield is related to the formula for bond price in terms of bond yield, because yield is in the denominator of the right-hand side of the equation. Now we can see this clearly, and can see that when y increases in the denominator of the expression, the value of the expression decreases, and thus P decreases, and vice versa.

In the next part of this series on bond basics, I'll extend the simple one-year bond price formula to the general case of a bond with a term to maturity of any number of years.

In Part 1 of this series, I explained that a bond is basically a loan from an investor to a corporation or government. Similarly, a savings account is essentially a loan from a saver to a bank (or credit union). The bank compensates the saver for the loan by paying interest, very much like the corporation or government compensates the bondholder by paying interest. The main difference is that the saver can take back all or part of the loan from the bank at any time by removing money from the savings account. In bond terms, the term to maturity of the savings account is zero years, so we can think of it as an extremely short-term bond.

If you deposit $100 in a savings account with an annual interest rate of 1%, you will earn $1 in interest in one year, because 1% of $100 is $1 (taking percentages of 100 is trivial: 1% of 100 is 1, 10% of 100 is 10, 27.4% of 100 is 27.4, etc.). Adding this $1 in interest to your original $100 gives you a total of $101 in your account after one year. From now on I'm going to drop the dollar sign for convenience; e.g., 100 grows to 101 at the end of one year. Also, all interest rates will be understood to be annual interest rates.

If we use FV to represent the Future Value of the savings account at the end of one year, we can write this equation to describe the future value in terms of the initial value of 100 and the interest rate of 1%:

FV = 100 + 100 * 1%

Here I'm using * for the multiplication operator, so 1% of 100 is written as 1% * 100, which also can be written as 100 * 1% (due to the commutative property of multiplication).

The right-hand side of the equation above can be evaluated directly in a spreadsheet formula or by Google (copy and paste it into a Google search box to verify). Some calculators have a % key that also would allow you to evaluate the expression as written. However, it is standard to write such expressions using decimal numbers rather percentages.

In decimal form, 1% is 0.01. If you don't remember this from elementary school arithmetic, then consider that % can be interpreted as the arithmetic operation "divide by 100". Using / as the division operator, 1% = 1 / 100 = 0.01. Dividing by 100 is the same as moving the decimal point two places to the left (adding one or more zeros if necessary). So 100% = 1, 10% = 0.1 and 1% = 0.01.

Rewriting the above equation using 0.01 instead of 1%:

FV = 100 + 100 * 0.01

Based on the distributive property, we can factor out the 100 in the expression on the right-hand side of this equation, and rewrite it as:

FV = 100 * (1 + 0.01)

If we want to be able to apply this formula to any starting amount, instead of just 100, we can designate the starting amount as PV, which stands for Present Value. Similarly, if we want to be able to apply the formula using any interest rate, instead of just 1%, we can designate the interest rate as

*i*and rewrite the equation as:FV = PV * (1 + i)

This is the basic

*time value of money*(TVM) formula that allows you to calculate the future value (FV) at the end of one year given a present value (PV) and an interest rate (i). We can easily extend this formula to calculate the future value at the end of any number of years, but for now I'll just stick with a one-year time period.We can use some simple algebra to rearrange the above equation to answer the question, "what is the present value that will result in a given future value at the end of one year?" For example, what value do I start with to end up with 101 at the end of one year at an interest rate of 1%? To derive the formula to answer questions like this, we simply divide both sides of the above equation by (1+i), and move each side of the equation to the other side, which gives us:

PV = FV / (1+i)

(Still using / as the division operator). In words, the present value is the future value divided by the sum of one and the interest rate.

Of course if we substitute 101 for FV and 0.01 for i in this equation, and solve it, we will get 100 as the result for PV. We know this because we've already determined that 101 is the Future Value after one year given the Present Value of 100 and an interest rate of 1%. Just as we used some simple algebra to rearrange the TVM equation, we can rearrange our words to say that 100 is the Present Value given a Future Value of 101 and an interest rate of 1%.

In this version of the one-year TVM formula, in which we solve for PV in terms of FV,

*i*typically is referred to as the*discount rate,*since we discount the future value using this rate to determine the present value. Thus we can say that the present value equals the future value received one year from now discounted by the discount rate.How does this relate to bond price and yield? Before answering this, we must introduce one more TVM term:

*cash flow*. In TVM lingo, FV in the equation above is referred to as a*cash flow*. In the simple example of our savings account, there is one future cash flow, which is the principal and interest payment we receive at the end of one year. Now we can define bond price in terms of bond yield using TVM terminology:**A bond's price is the present value of its future cash flows discounted at a rate equal to the bond's yield.**That's a mouthful, but read it carefully while looking at the equation for present value in terms of future value and an interest rate (or discount rate) for a one-year period:

PV = FV / (1+i)

For a one-year bond (term to maturity of one year), PV in the above equation is the bond's price, FV is the future cash flow (principal and interest), and i is the discount rate equal to the bond's yield. So the same formula we used to calculate the initial deposit into a savings account given the interest rate and the value after one year can be used to calculate the price of a bond given the yield and the principal and interest payment received at the end of one year.

Now that we're talking about bonds, we'll replace PV with P (for Price), and replace i with y (for yield), and write it this way:

P = FV / (1 + y)

We can improve this further by expressing FV in terms of known bond characteristics.

The Future Value when the bond matures after one year is the face value of the bond plus the interest earned in one year. We learned in Part 1 that face value also is referred to as par value, or simply par. To avoid confusion between the terms future value and face value, I'll use the term par value or par from now on.

Since bonds are priced as a percent of par value, we note the price of a bond valued at par is 100 (100% of par value). In our calculations, we will use 100 as par value, since this will give us a result consistent with the bond pricing convention. So 100 will be the principal portion of the cash flow we receive after one year.

The interest earned in one year is the coupon rate times the par value of the bond, as discussed in Part 1. So for a bond with a coupon rate of 1%, the interest portion of our future cash flow will be 1% * 100 = 1.

Putting this together, the future cash flow, FV in the equation, is:

FV = 100 + 100 * 0.01

Replacing 0.01 with the letter r, for coupon

**r**ate, we can write this as:FV = 100 + 100 * r

Which we can rewrite as:

FV = 100 * (1+r)

Substituting the right-hand side of the above equation for FV in our bond price formula:

P = FV / (1+y)

P = 100 * (1+r) / (1+y)

In Part 2 I used words and logic to explain that the yield equals the coupon rate for a bond priced at par. Now we can verify this with the one-year bond pricing formula we've developed. For a bond with a coupon rate of 1% and a yield of 1%:

P = 100 * (1+r) / (1+y)

P = 100 * (1 + 0.01) / (1 + 0.01)P = 100 * 1

P = 100

In Part 2 I also used words and logic to explain that the price of a one-year bond with a yield of 2% and a coupon rate of 1% is about 99. Now we also can verify this with the one-year bond pricing formula we've developed. For a bond with a coupon rate of 1% and a yield of 2%:

P = 100 * (1+r) / (1+y)

P = 100 * (1 + 0.01) / (1 + 0.02)P = 100 * 1.01/1.02

P = 100 * 0.99

P = 99

P = 100 * (1+r) / (1+y)

P = 100 * (1 + 0.01) / (1 + 0)P = 100 * 1.01/1

P = 100 * 1.01

P = 101

We now have the formula to calculate the price of a one-year bond, and have verified that it gives results consistent with those we achieved through reasoning about equalizing the one-year returns of bonds with the same coupon rates but different yields. The formula is more powerful than the reasoning we used, because we can expand on it to derive a formula for bonds with more than one year to maturity, for which it would be difficult to determine the price based on the reasoning used in Part 2.

In Part 3 I mentioned that the inverse relationship between bond price and yield is related to the formula for bond price in terms of bond yield, because yield is in the denominator of the right-hand side of the equation. Now we can see this clearly, and can see that when y increases in the denominator of the expression, the value of the expression decreases, and thus P decreases, and vice versa.

In the next part of this series on bond basics, I'll extend the simple one-year bond price formula to the general case of a bond with a term to maturity of any number of years.

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