14.1: Determining the Value of a Bond
 Page ID
 22152
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{\!\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\ #1 \}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\ #1 \}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{\!\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{\!\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left#1\right}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exactly what is your marketable bond worth? When you inspect the financial section of your local newspaper, you pay particular attention to the quotes on bonds, into which you invested a portion of your RRSP portfolio. Ten years ago, when prevailing bond rates were 5.5%, you purchased 10 Government of Canada $5,000 face value bonds with a 5% coupon and 20 years remaining to maturity. At the time you paid $4,699.02 each for them. Today's prevailing bond rates have dropped to 3.35% but are expected to rise in the near future. Should you hold on to those bonds? Should you cash them in by selling them in the bond market?
To help answer those questions, it is critical to know the value, or selling price, of your bonds today. This section explains the concept of a marketable bond along with its important characteristics and terminology. You calculate the bond's selling price first on interest payment dates and then on all other dates, which present a more complex situation.
Bond Basics
A marketable bond, as illustrated in the photo, is a debt that is secured by a specific corporate asset and that establishes the issuer’s responsibility toward a creditor for paying interest at regular intervals and repaying the principal at a fixed later date. A debenture is the same as a marketable bond, except that the debt is not secured by any specific corporate asset. Mathematically, the calculations are identical for these two financial tools, which this textbook refers to as bonds for simplicity. A typical bond timeline looks like the one below. Key terms are discussed as well.
Bond Issue Date
The bond issue date is the date that the bond is issued and available for purchase by creditors. Interest accrues from this date.
Bond Face Value
Also called the par value or denomination of the bond, the bond face value is the principal amount of the debt. It is what the investor lent to the bondissuing corporation. The amount, usually a multiple of $100, is found in small denominations up to $10,000 for individual investors and larger denominations up to $50,000 or more for corporate investors.
Bond Coupon Rate
Also known as the bond rate or nominal rate, the bond coupon rate is the nominal interest rate paid on the face value of the bond. The coupon rate is fixed for the life of the bond. Most commonly the interest is calculated semiannually and payable at the end of every sixmonth period over the entire life of the bond, starting from the issue date. All coupon rates used in this textbook can be assumed to be semiannually compounded unless stated otherwise.
Bond Market Rate
The bond market rate is the prevailing nominal rate of interest in the open bond market. Since bonds are actively traded, this rate fluctuates based on economic and financial conditions. On the issue date, the market rate determines the coupon rate that is tied to the bond. Market rates are usually compounded semiannually, as will be assumed in this textbook unless otherwise stated. Therefore, marketable bonds form ordinary simple annuities, since the interest payments and the market rate are both compounded semiannually, and the payments occur at the end of the interval.
Bond Redemption Price
Also called the redemption value or maturity value, the bond redemption price is the amount the bond issuer will pay to the bondholder upon maturity of the bond. The redemption price normally equals the face value of the bond, in which case the bond is said to be “redeemable at par” because interest on the bond has already been paid in full periodically throughout the term, leaving only the principal in the account. In some instances a bond issuer may in fact redeem the bond at a premium, which is a price greater than the face value. The redemption price is then stated as a percentage of the face value, such as 103%. For introductory purposes, this text sticks to the most common situation, where the redemption price equals the face value.
Bond Maturity Date
Also known as the redemption date or due date, the bond maturity date is the day upon which the redemption price will be paid to the bondholder (along with the final interest payment), thereby extinguishing the debt.
Bond Selling Date
The date that a bond is actively traded and sold to another investor through the bond market is known as the bond selling date. In the timeline, the selling date can appear anywhere on the timeline between the issue date and maturity date, and it may occur more than once as the bond is sold by one investor after another.
Calculating the Bond Price on an Interest Payment Date
Marketable bonds and debentures are nonredeemable, which means the only way to cash these bonds in before the maturity date is to sell them to another investor. Therefore, the key mathematical calculation is what to pay for the bond. The selling date, maturity date, coupon rate, redemption price, and market rate together determine the bond price. On the bond’s issue date, the market rate determines the coupon rate, so these two rates are identical. As a result, the price of the bond equals its face value. After the bond is issued, interest starts to accrue on it, and the market rate begins to fluctuate based on market conditions. This changes the price of the bond.
The Formula
Because the bond pays interest semiannually, two days of the year are defined as the interest payment dates. To determine a bond’s selling price on these two days, you must use the formulas for present value of an ordinary annuity. Once you understand how to perform these basic calculations we will move on to the more complex formulas and techniques required to determine the selling price on the other 363 days of the year. Regardless of the selling date, Formula 14.1 expresses how to determine the price of any bond.
To determine the selling price of the bond, you must know the amount of the semiannual interest payment to the bondholder. You use Formula 14.2 to calculate this amount.
The market price of a bond on its selling date is the present value of all the future cash flows, as illustrated in the figure below. For the bond purchaser, this is a combination of the remaining coupon annuity payments plus the redemption price at maturity (which in this textbook always equals the face value). Formula 14.3 summarizes this calculation, which combines Formulas 9.3 and 11.4 together and simplifies the resulting expression.
The price of a bond fluctuates with the market rate over time. If the bond sells for a price higher than its face value, the difference is known as a bond premium. If the bond sells for a price lower than its face value, the difference is known as a bond discount. The amount of the premium or discount excludes any accrued interest on the bond. Why does the selling price change like this? Remember that the interest paid by the bond is a fixed rate (the coupon rate) determined at the time of issue.
• Assume a coupon rate of 5%. If the market rate has increased to 6%, it means that investors can buy bonds paying 6%. If you are trying to sell your 5% bond, no one wants to buy it unless you “put it on sale” in an amount that compensates for the 1% difference. Hence, you discount your bond.
• Alternatively, if the market rate decreases to 4%, it means that investors can buy bonds paying 4%. If you are trying to sell your 5% bond, it is very attractive to investors, so you add some extra margin, raising the price by an amount not exceeding the 1% difference. Hence, you sell at a premium price.
The figure after Formula 14.3 illustrates the relationship between the market rate, coupon rate, and the selling price of the bond. Notice that when the coupon rate is higher than the market rate, the selling price rises above its face value. Alternatively, when the coupon rate is lower than the market rate, the selling price falls below its face value. Apply Formula 14.4 to calculate the amount of the premium or discount on a bond.
How It Works
Follow these steps to calculate the price of a bond on its interest payment date:
Step 1: Draw a timeline extending from the selling date to the maturity date. Identify all known variables.
Step 2: Using Formula 14.2, calculate the amount of the regular bond interest payment. For future calculations do not round this number.
Step 3: Using
Formula 14.3, calculate the date price of the bond. On an interest payment date, the date price is equal to both the market price and cash price. Use the market rate for Formula 9.1 (Periodic Interest Rate).
 Typically, to calculate N you need both Formula 9.2 (Number of Compound Periods for Single Payments) for the redemption value and Formula 11.1 (Number of Annuity Payments) for the annuity. However, since compound periods and annuity payments are both semiannual, then each formula would produce the same value of \(N\). Therefore, you can use just Formula 11.1, recognizing that it represents both the number of compound periods as well as the number of annuity payments.
 The date price from Formula 14.3 equals the market price. Since there is no accrued interest, the cash price is the same as the date price.
Step 4: If required, use Formula 14.4 to calculate any bond premium or discount.
Important Notes
The BOND Function on the BAII Plus Calculator. On an interest payment date, you can solve any bond problem using the regular time value of money buttons on your calculator since bonds represent ordinary simple annuities. However, a builtin function for bonds on the BAII Plus calculator greatly simplifies bond price calculations, particularly when the bond is being sold on a date other than an interest payment date. The BOND function is located on the second shelf above the number 9 key and is accessed by pressing 2nd BOND.
This spreadsheet has nine lines, which you scroll through using the ↓ and ↑ arrows. The first seven lines are considered the data entry lines, while the last two are the output lines. Upon opening the window, you should use the 2nd CLR Work function to erase previously loaded data. The data entry lines are as follows:
 SDT is the selling date. It is entered in the standard date format of MM.DDYY where MM is the month number (one or two digits), DD is the twodigit day number, and YY is the last two digits of the year. You must press the ENTER key to store this information. If the question does not involve specific dates, use January 1, 2000, or 1.0100 so as to determine the redemption date more easily.
 CPN is the nominal coupon rate. It is formatted as a percentage but without the percent sign; thus 5.5% is keyed in as 5.5. You must press the ENTER key to store this information.
 RDT is the redemption date or maturity date. It must be entered in the standard date format, and you must press ENTER to store this information. If the question does not involve specific dates, key in the appropriate date based on a selling date of January 1, 2000.
 RV is the redemption value or redemption price expressed as a percentage of the face value. Since the redemption price equals the face value, use the default setting of 100. This textbook never requires you to alter this number.
 ACT/360 is a toggle that you change by pressing 2nd SET. ACT counts the actual number of days in the transaction, while 360 treats every month as having 30 days. In Canada, ACT is the standard, so you should leave the calculator on this setting.
 2/Y OR 1/Y is a toggle that you change by pressing 2nd SET. 2/Y indicates a semiannual compound for both the market rate and coupon rate, while 1/Y indicates an annual compound. This textbook always uses the 2/Y setting.
 YLD is the nominal market rate for bonds at the time of sale. It follows the same format as the CPN, that is, a percentage but without the percent sign. You must press the ENTER key to store the information. The output lines are as follows:
 PRI is the market price of the bond. After scrolling to this line, press the CPT button to calculate this output, which is not computed automatically. The output is a percentage of the redemption price (which is the same as the face value). Thus, if you have a $1,000 face value bond, you need to take the output divided by 100 multiplied by the face value to arrive at the market price.
 AI is the accrued interest of the bond. This output is automatically calculated when you press the CPT button on the PRI line. If you are on an interest payment date, it has a value of zero. If you are in between payments, the output is, just like the PRI, a percentage of the redemption price, so you need to convert it to a value in the same manner as the PRI.
If you are interested in the cash price of the bond, you must add together the values associated with the PRI and AI outputs, as in Formula 14.1. When you are finished with the BOND function, press 2nd QUIT to leave the window.
Things To Watch Out For
When you calculate the price of a bond on the interest payment date, the date price is in fact calculating the market price. Recall that the cash price of the bond is always determined by Formula 14.1, where the market price and accrued interest must be totalled to arrive at the cash price. On interest payment dates, there is no accrued interest, so it always has a value of zero. When working with bonds, get in the habit now of thinking in the manner of Formula 14.1. Later on, when the bond is sold on a noninterest payment date and accrued interest is involved, this habit is handy for figuring out bond prices. With respect to the BAII Plus calculator, always add together the outputs of the PRI and AI windows to arrive at the selling price (cash price) of the bond.
Paths To Success
Earlier in this textbook Canada Savings Bonds were discussed. Note that these bonds are fully redeemable at any point, in that you can cash them in at any point with any financial institution before maturity. Therefore, Canada Saving Bonds are not considered marketable bonds and do not operate according to the principles discussed in this section.
 Explain the relationship between changes in the bond market rate and the price of the bond.
 What five variables are used in determining the price of a bond?
 True or false: If a $1,000 face value bond has a cash price of $1,125 and a market price of $1,100, it is selling at a bond premium of $100.
 What three variables determine the coupon payment amount?
 Answer

 As the market rate rises the bond price decreases, and vice versa.
 Selling date, maturity date, coupon rate, redemption price, and market rate
 True; the bond premium is the difference between the face value and the market price.
 Face value, nominal coupon rate, and coupon rate compounding frequency; see Formula 14.2
A Government of Canada $50,000 bond was issued on January 15, 1991, with a 25year maturity. The coupon rate was 10.15% compounded semiannually. What cash price did the bond have on July 15, 2005, when prevailing market rates were 4.31% compounded semiannually? What was the amount of the bond premium or discount?
Solution
This bond makes interest payments six months apart, on January 15 and July 15 each year. The bond is being sold on July 15, which is an interest payment date. On an interest payment date, solve for the date price, which is the same as the cash price. Also calculate the premium or discount.
What You Already Know
Step 1:
The timeline for the bond sale appears below.
Coupon Interest Payment: \(CPN\) = 10.15%, \(CY\) = 2, Face Value = $50,000
Bond: \(FV\) = $50,000, \(IY\) = 4.31%, \(CY\) = 2, \(PMT_{BOND}\) = Formula 14.2, \(PY\) = 2, Years Remaining = 10.5
How You Will Get There
Step 2:
Apply Formula 14.2 to determine the periodic bond interest payment.
Step 3:
Apply Formulas 9.1, 11.1, and 14.3 to determine the price of the bond on its interest payment date. The cash price in Formula 14.1 equals the date price.
Step 4:
Apply Formula 14.4 to determine the bond premium or discount.
Perform
Step 2:
\[PMT_{BOND}=\$ 50,000 \times \dfrac{0.1015}{2}=\$ 50,000 \times 0.05075=\$ 2,537.50 \nonumber \]
Step 3:
\(i=4.31 \% / 2=2.155 \% ; N=2 \times 10.5=21\) (compounds and payments)
\[\text { Date Price }=\dfrac{\$ 50,000}{(1+0.02155)^{21}}+\$ 2,537.50\left[\dfrac{1\left[\frac{1}{1+0.02155}\right]^{21}}{0.02155}\right]=\$ 74,452.86 \nonumber \]
Step 4:
Premium or Discount \(=\$ 74,452.86\$ 50,000.00=\$ 24,452.86\)
Calculator Instructions
SDT  CPN  RDT  RV  Days  Compound  YLD  PRI  AI 

7.1505  10.15  1.1516  100  ACT  2/Y  4.31  Answer: 148.905720  Answer: 0 
Transforming the output:
\[\dfrac{148.905720+0}{100} \times \$ 50,000=\$ 74,452.86012 \nonumber \]
The bond has a coupon rate that is substantially higher than the market rate, so it is sold at a premium of $24,452.86 for a cash price of $74,452.86. Note that because the sale occurs on the interest payment date, there is no accrued interest, so the market price and cash price are the same.
A $25,000 Government of Canada bond was issued with a 25year maturity and a coupon rate of 8.92% compounded semiannually. Twoandahalf years later the bond is being sold when market rates have increased to 9.46% compounded semiannually. Determine the selling price of the bond along with the amount of premium or discount.
Solution
Although there are no specific dates, the coupon is semiannual, making interest payments every six months. If the bond is being sold 2½ years after issue, this makes the sale date an interest payment date. On an interest payment date, solve for the date price, which is the same as the cash price. Also calculate the premium or discount.
What You Already Know
Step 1:
The timeline for the bond sale appears below.
Coupon Interest Payment: \(CPN\) = 8.92%, \(CY\) = 2, Face Value = $25,000
Bond: \(FV\) = $25,000, \(IY\) = 9.46%, \(CY\) = 2, \(PMT_{BOND}\) = Formula 14.2, \(PY\) = 2, Years Remaining = 22.5
How You Will Get There
Step 2:
Apply Formula 14.2 to determine the periodic bond interest payment.
Step 3:
Apply Formulas 9.1, 11.1, and 14.3 to determine the price of the bond on its interest payment date. The cash price in Formula 14.1 equals the date price.
Step 4:
Apply Formula 14.4 to determine the bond premium or discount.
Perform
Step 2:
\[PMT_{BOND}=\$ 25,000 \times \dfrac{0.0892}{2}=\$ 25,000 \times 0.0446=\$ 1,115 \nonumber \]
Step 3:
\(i=9.46 \% / 2=4.73 \% ; N=2 \times 22.5=45\) (compounds and payments)
\[\text { Date Price }=\dfrac{\$ 25,000}{(1+0.0473)^{45}}+\$ 1,115\left[\dfrac{1\left[\dfrac{1}{1+0.0473}\right]^{45}}{0.0473}\right]=\$ 23,751.28 \nonumber \]
Step 4:
Premium or Discount \(=\$ 23,751.28\$ 25,000.00=\$ 1,248.72\)
Calculator Instructions
SDT  CPN  RDT  RV  Days  Compound  YLD  PRI  AI 

1.0100  8.92  7.0122  100  ACT  2/Y  9.46  Answer: 95.005105  Answer: 0 
Transforming the output:
\[\dfrac{95.005105+0}{100} \times \$ 25,000=\$ 23,751.27631 \nonumber \]
The bond has a coupon rate that is slightly lower than the market rate, so it is sold at a discount of $1,248.72 for a cash price of $23,751.28. Note that because the sale occurs on the interest payment date, there is no accrued interest, so the market price and cash price are the same.
Calculating the Bond Price on a Noninterest Payment Date
There are only two days of the year upon which the cash price and the market price of the bond are the same value. Those days are the interest payment dates, when you determine the bond’s value using
Formula 14.3. However, what happens if the bond is sold on one of the other 363 days of the year?On these other dates, the cash price and the market price are not equal. For each day that elapses after an interest payment date, interest for the next payment starts to accrue such that over the next six months, enough interest is available to make the next interest payment.
According to Formula 14.1, when an investor wants to purchase a bond in between interest payment dates, the buyer has to pay the seller a cash price equalling the market price of the bond plus the accrued interest. Why? Assume a bond makes semiannual interest payments of $50. When the buyer acquires the bond from the seller, two months have elapsed since the last interest payment date. Since the seller held the bond for two months of the sixmonth payment interval, it is fair and reasonable for the seller to receive the interest earned during that time frame. However, the bond will not make its next interest payment until four months later, at which time the buyer, who now owns the bond, will receive the full $50 interest payment for the full six months. Thus, at the time of buying the bond, the buyer has to pay the seller the bond’s market price plus the portion of the next interest payment that legally belongs to the seller. In this example, an interest amount representing two of the six months needs to be paid.
The Formula
Arriving at the bond’s price in between interest payment dates is a little complex because the cash price is increasing according to a compound interest formula, while in practice the accrued interest on the bond is increasing according to a simple interest formula. If this seems peculiar to you, you would be right, but that is just how bonds happen to work!
You require three important pieces of information: what to pay (the cash price), how much simple interest was included in the cash price (the accrued interest), and what the actual value of the bond was (the market price). To arrive at these three numbers, follow these steps:
 First calculate the cash price of the bond as shown in Formula 14.5.
 Calculate the accrued interest included in that price as shown in Formula 14.6.
 Determine the market price from Formula 14.1.
How It Works
Follow these steps to calculate the price of a bond in between interest payment dates:
Step 1: Draw a timeline like the one to the right, extending from the preceding interest payment date to the maturity date. Identify all known variables.
Step 2: Using
Formula 14.2, calculate the amount of the regular bond interest payment. For future calculations do not round this number.Step 3: Using Formula 14.3, calculate the date price of the bond on the interest payment date just preceding the selling date.
 Use the market rate for Formula 9.1 (Periodic Interest Rate).
 Recall that you use only Formula 11.1 and recognize that it represents both the number of compound periods as well as the number of annuity payments.
Step 4: Calculate the cash price of the bond using
Formula 14.5. Calculate the time ratio by determining the exact number of days the seller held the bond as well as the exact number of days involved in the current payment interval.Step 5: Calculate the accrued interest on the bond using Formula 14.6. Use the time ratio from step 5.
Step 6: Calculate the market price of the bond using Formula 14.1.
Step 7: If required, use Formula 14.4 to calculate the bond premium or discount.
Important Notes
The DATE Function on the BAII Plus Calculator. To create the time ratio, determine the number of days since the last interest payment date as well as the total number of days in the current payment interval. You can compute this through the DATE function. For a full discussion of this function, recall the instructions at the end of Chapter 8. To arrive at the required numbers:
 Compute the number of days since the last interest payment date by entering the last interest payment date as DT1 and the selling date as DT2. Compute the DBD.
 Compute the total number of days in the current payment interval by entering the last interest payment date as DT1 and the next interest payment date as DT2. Compute the DBD.
Things To Watch Out For
In calculations of bond premiums and discounts on noninterestpayment dates, the most common mistake is to use the cash price instead of the market price. Remember that the cash price includes both the accrued interest and the market price. The accrued interest does not factor into the value of the bond, since it represents a proportioning of the next interest payment between the seller and the buyer. Therefore, the amount of the bond premium or discount should not include the accrued interest. Use only the market price to determine the premium or discount.
Paths To Success
On the BAII Plus, the \(PRI\) and \(AI\) outputs of the BOND worksheet must be summed together to arrive at the cash price. Both of these outputs represent a percentage of the face value (the same base) and can be summed before converting the percentage into a dollar amount. For example, if \(PRI\) = 98% and \(AI\) = 2.5%, you could take the total of 100.5% to figure out the cash price. Using the calculator efficiently, you would store the \(PRI\) into a memory cell, scroll down, and then add the recall of the memory cell to the \(AI\) to arrive at the total percentage of face value.
 Is the cash price or market price higher when a bond is sold in between interest payment dates?
 Why can you not use Formula 14.3 directly to obtain the market price of a bond selling in between interest payment dates?
 Answer

 Cash price; it includes the accrued interest.
 Accrued interest is calculated using simple interest, which results in slightly different numbers than if the bond price used compound interest alone.
A $20 million face value Bell Canada bond was issued on July 19, 1999, with a 30year maturity and a coupon rate of 6.55% compounded semiannually. At what cash price did the bond trade on November 10, 2010, when market yields were posted at 5.892% semiannually? What was the accrued interest? What was the market price? What premium or discount does this represent?
Solution
Notice that this bond makes interest payments six months apart, on January 19 and July 19 of each year. The bond sale date, November 10, is not an interest payment date. Calculate four variables: the cash price, accrued interest (\(AI\)), market price (\(PRI\)), and the bond premium or discount.
What You Already Know
Step 1:
The timeline of the bond sale appears below.
Coupon Interest Payment: \(CPN\) = 6.55%, \(CY\) = 2, Face Value = $20 million
Bond: \(FV\) = $20 million, \(IY\) = 5.892%, \(CY\) = 2, \(PMT_{BOND}\) = Formula 14.2, \(PY\) = 2, Years Remaining = 18.5 years to preceding interest payment date
How You Will Get There
Step 2:
Apply Formula 14.2 to determine the periodic bond interest payment.
Step 3:
Apply Formulas 9.1, 11.1, and 14.3 to determine the price of the bond on its preceding interest payment date.
Step 4:
Apply Formula 14.5 to determine the cash price of the bond.
Step 5:
Apply Formula 14.6 to determine the accrued interest.
Step 6:
Apply Formula 14.1 to determine the market price.
Step 7:
Apply Formula 14.4 to determine the bond premium or discount.
Perform
Step 2:
\[PMT_{BOND}=\$ 20,000,000 \times \dfrac{0.0655}{2}=\$ 655,000 \nonumber \]
Step 3:
\(i=5.892 \% / 2=2.946 \% ; N=2 \times 19=38\) (compounds and payments)
\[\text { Date Price }=\dfrac{\$ 20,000,000}{(1+0.02946)^{38}}+\$ 655,000\left[\dfrac{1\left[\dfrac{1}{1+0.02946}\right]^{38}}{0.02946}\right]=\$ 21,492,511.69 \nonumber \]
Step 4:
\[t=\dfrac{\text { July } 19,2010 \text { to November } 10,2010}{\text { July } 19,2010 \text { to January } 19,2011}=\dfrac{114}{184} \nonumber \]
\[\text { Cash Price }=(\$ 21,492,511.69)(1+0.02946)^{\frac{114}{184}}=\$ 21,882,632.40 \nonumber \]
Step 5:
\[(\$ 655,000) \dfrac{114}{184}=\$ 405,815.22 \nonumber \]
Step 6:
\[\$ 21,882,632.40=PRI+\$ 405,815.22 \quad \$ 21,476,817.18=PRI \nonumber \]
Step 7:
Premium or Discount \(=\$ 21,476,817.18\$ 20,000,000.00=\$ 1,476,817.18\)
Calculator Instructions
SDT  CPN  RDT  RV  Days  Compound  YLD  PRI  AI 

11.1010  6.55  7.1929  100  ACT  2/Y  5.592  Answer: 107.384085  Answer: 2.029076 
Transforming the output:
\[PRI=\dfrac{107.384085}{100} \times \$ 20,000,000=\$ 21,476,817.18\nonumber \]
\[AI=\dfrac{2.029076}{100} \times \$ 20,000,000=\$ 405,815.22 \nonumber \]
The bond had a coupon rate that is higher than the market rate, so it sold at a premium of $1,476,817.18 for a market price of $21,476,817.18. The buyer also owed the seller $405,815.22 of accrued interest; therefore, the cash price was $21,882,632.40.
A $50,000 Government of Ontario bond was issued on March 1, 1995, with a coupon rate of 9.5% compounded semiannually and 50 years until maturity. Harvey bought the bond on July 17, 1996, when market rates were 8.06% compounded semiannually, and sold the bond on December 12, 2008, with a semiannual yield of 3.45%. Based on the market price, what gain or loss did Harvey realize?
Solution
Notice that this bond makes interest payments six months apart, on March 1 and September 1 of each year. Since the bond is being bought on July 17 and sold on December 12, neither date represents an interest payment date. Calculate the market price (PRI) for both dates and then determine the difference.
What You Already Know
Step 1:
The timeline for the bond sale appears below. Coupon Interest Payment: \(CPN\) = 9.5%, \(CY\) = 2, Face Value = $50,000
Bond Purchase: \(FV\) = $50,000, \(IY\) = 8.06%, \(CY\) = 2, \(PMT_{BOND}\) = Formula 14.2, \(PY\) = 2, Years Remaining = 49 years to preceding interest payment date
Bond Sale: \(FV\) = $50,000, \(IY\) = 3.45%, \(CY\) = 2, \(PMT_{BOND}\) = Formula 14.2, \(PY\) = 2, Years Remaining = 36.5 years to preceding interest payment date
How You Will Get There
Step 2:
Apply Formula 14.2 to determine the periodic bond interest payment.
For each of the purchase and sale, perform steps 3 through 6:
Step 3:
Apply Formulas 9.1, 11.1, and 14.3 to determine the price of the bond on its preceding interest payment date.
Step 4:
Apply Formula 14.5 to determine the cash price of the bond.
Step 5:
Apply Formula 14.6 to determine the accrued interest.
Step 6:
Apply Formula 14.1 to determine the market price.
Step 7:
Determine the difference between the market prices (PRI) from the purchase to the sale.
Perform
Step 2:
\[PMT_{BOND}=\$ 50,000 \times \dfrac{0.095}{2}=\$ 2,375 \nonumber \]
Bond Purchase (July 17, 1996)
Step 3:
\(i=8.06 \% / 2=4.03 \% ; N=2 \times 49=98\) (compounds and payments)
\[\begin{aligned} \text { Date Price }&=\dfrac{\$ 50,000}{(1+0.0403)^{98}}+ \$ 2,375\left[\dfrac{1\left[\dfrac{1}{1+0.0403}\right]^{98}}{0.0403}\right] \\=& \$ 58,747.02738 \end{aligned}\nonumber \]
Step 4:
\[t=\dfrac{\text { March } 1,1996 \text { to July } 17,1996}{\text { March } 1,1996 \text { to September } 1,1996}=\dfrac{138}{184} \nonumber \]
\[\begin{aligned} \text { Cash Price }&=(\$ 58,747.02738)(1+0.0403)^{\frac{138}{184}} \\&=\$ 60,513.86 \end{aligned} \nonumber \]
Step 5:
\[AI=(\$ 2,375) \dfrac{138}{184}=\$ 1,781.25 \nonumber \]
Step 6:
\[\begin{array}{l}{\$ 60,513.86=PRI+\$ 1,781.25} \\ {\$ 58,732.61=PRI}\end{array} \nonumber \]
Bond Sale (December 12, 2008)
Step 3:
\(i=3.45 \% / 2=1.725 \% ; N=2 \times 36.5=73\) (compounds and payments)
\[\begin{aligned} \text { Date Price }&=\dfrac{\$ 50,000}{(1+0.01725)^{73}}+ \$ 2,375\left[\dfrac{1\left[\dfrac{1}{1+0.01725}\right]^{73}}{0.01725}\right] \\=& \$ 112,522.6856 \end{aligned} \nonumber \]
Step 4:
\[t=\dfrac{\text { September } 1,2008 \text { to December } 12,2008}{\text { September } 1,2008 \text { to March } 1,2009}=\dfrac{102}{181} \nonumber \]
\[\begin{aligned} \text { Cash Price }&=(\$ 112,522.6856)(1+0.01725) ^{ \frac{102}{181}} \\&= \$ 113,612.43 \end{aligned} \nonumber \]
Step 5:
\[AI=(\$ 2,375) \dfrac{102}{181}=\$ 1,338.40 \nonumber \]
Step 6:
\[\begin{array}{l}{\$ 113,612.43=PRI+\$ 1,338.40} \\ {\$ 112,274.03=PRI}\end{array} \nonumber \]
Step 7:
\[\text { Gain }=\$ 112,274.03\$ 58,732.61=\$ 53,541.42 \nonumber \]
Calculator Instructions
Bond  SDT  CPN  RDT  RV  Days  Compound  YLD  PRI  AI 

Purchase  7.1796  9.5  3.0145  100  ACT  2/Y  8.06 
Answer: 117.465216 
Answer: 3.5625 
Sale  12.1208  \(\surd\)  \(\surd\)  \(\surd\)  \(\surd\)  \(\surd\)  3.45 
Answer: 224.548072 
Answer: 2.676795 
Bond Purchase (July 17, 1996)
Transforming the output:
\[PRI=\dfrac{117.465216}{100} \times \$ 50,000=\$ 58,732.60843 \nonumber \]
Bond Sale (December 12, 2008)
Transforming the output:
\[PRI=\dfrac{224.548072}{100} \times \$ 50,000=\$ 112,274.0362 \nonumber \]
(Difference of $0.01 is due to rounding.)
Harvey acquired the bond for a market price of $58,732.61 and sold the bond approximately 12.5 years later for $112,274.03 because of the very low market rates in the bond market. As a result, the gain on his bond amounts to $53,541.42.