# 4.2 Calculating the Purchase Price of a Bond on an Interest Payment Date

## Learning Objectives

• Calculate the purchase price of a bond on an interest payment date

## Formula & Symbol Hub

#### Symbols Used

• $BAL$ = Principal balance
• $BV$ = Book value
• $FV$ = Face value of bond
• $PMT$ = Annuity payment amount
• $N$ = Number of annuity payments
• $I/Y$ = Nominal interest rate
• $P/Y$ = Number of payments per year or payment frequency
• $C/Y$ = Number of compounds per year or compounding frequency

#### Formulas Used

• ##### Formula 4.1 – Bond Payment

$\text{Payment}=\mbox{Face Value} \times \mbox{periodic bond rate}$

• ##### Formula 4.2 – Purchase Price on an Interest Payment Date

$\text{Purchase Price}=\mbox{Present Value of Face Value}+\mbox{Present Value of Bond Payments}$

• ##### Formula 4.3 – Premium (Bonds)

$\text{Premium}=\mbox{Purchase Price}-\mbox{Face Value}$

• ##### Formula 4.4 – Discount (Bonds)

$\text{Discount}=\mbox{Face Value}-\mbox{Purchase Price}$

## Introduction

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?

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’s purchase price.

On the bondâ€™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.

Because the bond pays interest semi-annually, two days of the year are defined as the interest payment dates. For now, we are interested in finding out the purchase price of a bond if the bond is purchased on one of these two interest payment dates. Because the bond sells on its interest payment date, the most recent interest payment date is the selling date.

When you purchase a bond, you are buying two promises: to be paid the face value of the bond at maturity and to be paid the periodic bond payments. The purchase price takes both of these promises into account. The purchase price of the bond on an interest payment date is the sum of the present value of all of the remaining payments on the bond on this date and the present value of the face value of the bond on this date, using the yield rate to calculate the present values.

### $\boxed{4.2}$ Purchase Price on an Interest Payment Date

$\Large{\color{red}{\text{Purchase Price}}}={\color{blue}{PV \text{ of Payments}}}+{\color{green}{PV \text{ of Face Value}}}$

${\color{red}{\text{Purchase Price:}}}$ is the calculated purchase price on the interest payment date.

${\color{blue}{PV \text{ of Payments:}}}$ is the sum of the present value of the remaining payments.

${\color{green}{PV \text{ of Face Value:}}}$ is the present value of the bond’s face value as of the payment date.

### Key Takeaways

The bond payment issued on the date of purchase (which in this case is an interest payment date) goes to the seller of the bond, and not the purchaser. The purchaser only starts to receive payments on the first interest payment date after the purchase of the bond.

The price the seller sells the bond for equals the price the purchaser pays for the bond.

Similar to T-bills and commercial papers, the time for the purchase price calculation is based on the remaining time to maturity (i.e. the time from the interest payment date to the maturity date) at the time of purchase.

Recall from the previous section that a bond can be bought at a premium, discount, or a par.

• Bonds purchased at par. The purchase price of the bond equals its face value. This happens when the coupon rate is the same as the yield rate.
• Bonds purchased at premium. The purchase price of the bond is higher than its face value. The difference between the purchase price and the face value is called the premium.

### $\boxed{4.3}$ Premium (Bonds)

$\Large\mbox{Premium}=\mbox{Purchase Price}-\mbox{Face Value}$

• Bonds purchased at discount. The purchase price of the bond is lower than its face value. The difference between the face value and the purchase price is called the discount.

### $\boxed{4.4}$ Discount (Bonds)

$\Large\mbox{Discount}=\mbox{Face Value}-\mbox{Purchase Price}$

You can use the financial calculator to find the purchase price of the bond. After all, the purchase price is just the sum of two present value calculations. You can calculate each present value separately and then add the results together. Alternatively, the calculator can do both calculations simultaneously and output the purchase price.

## Using the TI BAII Plus Calculator to Find the Purchase Price of a Bond on an Interest Payment Date

• Set the calculator to the END payment setting.
• Enter values for the known variables ($FV$, $PMT$, $N$, $I/Y$, $P/Y$ and $C/Y$), paying close attention to the cash flow sign convention for $PMT$ and $FV$.
• Because you are purchasing the bond, you will become the bond holder and receive the payments and the face value. So $PMT$ (the bond payment) and $FV$ (the face value) are both entered with a positive cash flow sign.
• After all of the known quantities are loaded into the calculator, press $CPT$ and then $PV$ to solve for the purchase price.

### Things to Watch Out For

There are two interest rates associated with a bond: the yield rate and the coupon rate. They are NOT interchangeable.

• The coupon rate is only used to calculate the bond payment. The coupon rate is NEVER used as the value for $I/Y$.
• The yield rate is the prevailing interest rate in the market. In the purchase price calculation, the yield rate is the interest rate entered for $I/Y$.

### Example 4.2.1

A $\50,000$ bond with a $10.15\%$ coupon is purchased when there are $10.5$ years remaining to maturity. If the yield to maturity at the time of purchase is $4.31\%$, what is the purchase price? What is the amount of premium or discount?

Solution

Step 1: The given information is

Because no other information is given, the frequency of the payments and the compounding frequencies (for the coupon rate and the yield rate) are assumed to be semi-annual.

$\begin{eqnarray*} FV & = & \50,000 \\ P/Y & = & 2 \\ I/Y & = & 4.31\% \\ C/Y & = & 2 \\ t & = & 10.5 \mbox{ years} \\ \mbox{Coupon Rate} & = & 10.15\% \end{eqnarray*}$

Step 2: Calculate the bond payment.

$\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 50,000\times \frac{0.1015}{2}\\ & = & \2,537.50\end{eqnarray*}$

Step 3: Calculate the purchase price.

 PMT Setting END $N$ $2 \times 10.5=21$ $PV$ $?$ $FV$ $50,000$ $PMT$ $2,537.50$ $I/Y$ $4.31$ $P/Y$ $2$ $C/Y$ $2$

$PV=\74,452.86$

Step 4: Calculate the premium or discount. Because the purchase price is greater than the face value, this is a premium bond.

$\begin{eqnarray*} \mbox{Premium} & = & \mbox{Purchase Price}-\mbox{Face Value} \\ & = & 74,452.86-50,000 \\ & = & \24,452.86 \end{eqnarray*}$

Step 5: Write as a statement.

The purchase price of the bond is $\74,452.86$ and the premium is $\24,452.86$.

### Example 4.2.2

A $\25,000$ bond with a $8.92\%$ coupon is purchased when there are $22.5$ years remaining to maturity. If the yield to maturity at the time of purchase is $9.46\%$, what is the purchase price? What is the amount of premium or discount?

Solution

Step 1: The given information is

Because no other information is given, the frequency of the payments and the compounding frequencies (for the coupon rate and the yield rate) are assumed to be semi-annual.

$\begin{eqnarray*} FV & = & \25,000 \\ P/Y & = & 2 \\ I/Y & = & 9.46\% \\ C/Y & = & 2 \\ t & = & 22.5 \mbox{ years} \\ \mbox{Coupon Rate} & = & 8.92\% \end{eqnarray*}$

Step 2: Calculate the bond payment.

$\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 25,000\times \frac{0.0892}{2}\\ & = & \1,115\end{eqnarray*}$

Step 3: Calculate the purchase price.

 PMT Setting END $N$ $2 \times 22.5=45$ $PV$ $?$ $FV$ $25,000$ $PMT$ $1,115$ $I/Y$ $9.46$ $P/Y$ $2$ $C/Y$ $2$

$PV=\23,751.28$

Step 4: Calculate the premium or discount.

Because the purchase price is less than the face value, this is a discount bond.

$\begin{eqnarray*} \mbox{Discount} & = & \mbox{Face Value}-\mbox{Purchase Price} \\ & = & 25,000-23,751.28 \\ & = & \1,248.72 \end{eqnarray*}$

Step 5: Write as a statement.

The purchase price of the bond is $\23,751.28$ and the discount is $\1,248.72$.

### Try It

1) A $\7,000$ bond with a coupon rate of $5\%$ is redeemable in ten years. Calculate the purchase price and the premium or discount if the yield rate is $7\%$.

Solution
 PMT Setting END $N$ $2 \times 10=20$ $PV$ ? $FV$ $7,000$ $PMT$ $175$ $I/Y$ $7$ $P/Y$ $2$ $C/Y$ $2$

$\begin{eqnarray*} PV&=&\6,005.13\\\\\mbox{Discount} & = & 7,000-6,005.13 \\ & = & \994.87 \end{eqnarray*}$

### Example 4.2.3

An investor purchased a $\10,000$ bond with a coupon rate of $6.5\%$ when there was $15$ years to maturity and the yield rate was $4\%$. Four years later, the investor decided to sell the bond when the yield to maturity was $5.2\%$.

1. At what price did the investor purchase the bond?
2. At what price did the investor sell the bond?
3. What was the investor’s gain or loss on the sale of the bond?
Solution

Step 1: Calculate the investor’s purchase price.

$\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 10,000 \times \frac{0.065}{2}\\ & = & \325\end{eqnarray*}$

 PMT Setting END $N$ $2 \times 15=30$ $PV$ ? $FV$ $10,000$ $PMT$ $325$ $I/Y$ $4$ $P/Y$ $2$ $C/Y$ $2$

$PV=\12,799.56$

The investor paid $\12,799.56$ for the bond.

Step 2: Calculate the selling price of the bond.

The selling price of the bond for the investor equals the purchase price of the bond for the person the investor is selling the bond to. The purchase price is based on the time to maturity and the yield rate at the time the bond is sold. The investor purchased the bond when there was $15$ years to maturity and then sold the bond four years later. So at the time of the sale there was $11$ ($15-4$) years to maturity.

 PMT Setting END $N$ $2 \times 11=22$ $PV$ $?$ $FV$ $10,000$ $PMT$ $325$ $I/Y$ $5.2$ $P/Y$ $2$ $C/Y$ $2$

$PV=\11,078.66$

The investor sold the bond for $\11,078.66$.

Step 3: Calculate the gain or loss on the sale.

Because the investor sold the bond for less than they paid for it, the investor realized a loss on the sale.

$\begin{eqnarray*} \mbox{Loss} & = & 12,799.56-11,078.66 \\ & = & \1,720.90 \end{eqnarray*}$

Step 4: Write as a statement.

The investor’s lost $\1,720.90$ on the sale of the bond.

### Try It

2) You purchased a $\3,000$ bond paying a $2.5\%$ coupon rate when there was eight years to maturity and the yield rate was $5.7\%$. After two years, you decided to sell the bond when the yield to maturity was $4.9\%$.

1. What price did you pay for the bond?
2. What price did you sell the bond for?
3. What was your gain or loss on the sale of the bond?
Solution

a. Calculate the purchase price of the bond.

 PMT Setting END $N$ $2 \times 8=16$ $PV$ $?$ $FV$ $3,000$ $PMT$ $37.50$ $I/Y$ $5.7$ $P/Y$ $2$ $C/Y$ $2$

$PV=\2,390.09$

b. Calculate the selling price of the bond.

 PMT Setting END $N$ $2 \times 6=12$ $PV$ $?$ $FV$ $3,000$ $PMT$ $37.50$ $I/Y$ $4.9$ $P/Y$ $2$ $C/Y$ $2$

$PV=\2,629.60$

c. Calculate the gain on the sale of the bond.

$\begin{eqnarray*} \mbox{Gain} & = & 2,629.60-2,390.09 \\ & = & \239.51 \end{eqnarray*}$

## Section 4.2 Exercises

1. A $\1,000$ bond with a $7.16\%$ coupon is redeemable in $27$ years.
1. Calculate the purchase price if the yield to maturity is $6.3\%$.
2. Calculate the premium or discount.
Solution

a. $\1,110.93$; b. Premium=$\110.93$

2. A $\5,000$ bond with a $7.64\%$ coupon is redeemable in $6.5$ years.
1. Calculate the purchase price if the yield to maturity is $3.96\%$.
2. Calculate the premium or discount.
Solution

a. $\6,045.43$; b. Premium=$\1,045.53$

3. With $17$ years until maturity, Julio purchased an $\8,000$ Government of Saskatchewan bond with a coupon rate of $5.55\%$. The market yield was $8.65\%$.
1. Calculate the purchase price of the bond.
2. Calculate the premium or discount.
Solution

a. $\5,777.23$; b. Discount=$\2,222.77$

4. A $\5,000$ with $14$ years to maturity has a coupon rate of $4.42\%$ and a yield of $4.07\%$.
1. Calculate the purchase price of the bond.
2. Calculate the premium or discount.
Solution

a. $\5,185.37$; b. Premium=$\185.37$

5. A $\25,000$ bond has a coupon rate of $5.12\%$. James purchased the bond when there was seven years to maturity and the yield rate was $4.18\%$. Five years later, James sold the bond when the yield to maturity was $3.87\%$.
1. Calculate the price James paid for the bond.
2. Calculate the price James sold the bond for.
3. Calculate James’ gain or loss on the sale of the bond.
Solution

a. $\26,413.52$; b. $\25,595.90$; c. Loss=$\817.62$

6. William purchased a $\18,000$ bond with a coupon rate of $11.25\%$, twelve years to maturity and a yield rate of $8.3\%$. After holding the bond for seven years, William sold the bond when the yield rate was $5.7\%$.
1. At what price did William purchase the bond?
2. At what price did William sell the bond?
3. What was William’s gain or loss on the sale of the bond?
Solution

a. $\21,986.61$; b. $\22,293.64$; c. Gain=$\307.03$