# 7.3 Calculating the Yield Rate

## LEARNING OBJECTIVES

• Calculate the yield to maturity of a bond on an interest payment date.
• Calculate the investor’s yield when a bond is sold before maturity.

The yield to maturity is a bond’s overall rate of return when purchased at a market price and held until maturity. It includes both the semi-annual interest that the bondholders earn on their investment along with the gain or loss resulting from the difference between the market price on the selling date and the redemption price. This yield to maturity is exactly equal to the market rate of return on the date of purchase. For example, if you purchase a bond today with a yield rate of 5.5% and you hold onto the bond until maturity, you will realize a yield to maturity of 5.5%.

You can use the financial calculator to find the yield rate for a bond.  To calculate the yield rate, you need to know the purchase price of the bond, along with the other information about the bond.

## USING THE TI BAII PLUS CALCULATOR TO FIND THE YIELD RATE FOR A BOND

• Set the calculator to the END payment setting.
• Enter values for the known variables (PV, FV, PMT, N, P/Y and C/Y), paying close attention to the cash flow sign convention for PV, PMT and FV.
• Because you are paying out the purchase price, PV (the purchase price) is entered with a negative cash flow sign.
• Because you are the bond holder, you 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 I/Y to solve for the yield rate.

### NOTES

1. The calculated yield rate is the market rate of return that the bond holder earns if the bond is purchased at the given purchase price and held until maturity. It represents the rate of return the bond holder will receive on their investment, provided they hold the bond until maturity.  If the bond holder decides to sell the bond at any point before the maturity date, they will realize a different rate of return on the bond.
2. To calculate a yield rate, the purchase date must be on an interest payment date.

## EXAMPLE

Suppose a $10,000 face value bond is purchased for$7,688.52 with 20 years until maturity and has a coupon rate of 4% semi-annually. What yield will the bondholder realize if she holds onto it until maturity?

Solution:

Step 1:  Because no other information is given, the frequency of the payments and the compounding frequency for the yield rate are assumed to be semi-annual.  The given information is

$\begin{eqnarray*} PV & = & \7,688.52 \\ FV & = & \10,000 \\ P/Y & = & 2 \\ C/Y & = & 2 \\ t & = & 20 \mbox{ years} \\ \mbox{Coupon Rate} & = & 4\% \end{eqnarray*}$

Step 2:  Calculate the bond payment.

$\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 10,000 \times\frac{0.04}{2}\\ & = & \200\end{eqnarray*}$

Step 3:  Calculate the yield rate.

 PMT Setting END N $2 \times 20=40$ PV $-7,688.52$ FV $10,000$ PMT $200$ I/Y ? P/Y $2$ C/Y $2$

$I/Y=6\%$

The yield rate is 6% compounded semi-annually.

In order to calculate the yield rate, you need to know the purchase price of the bond.  As in the above example, you could be told the purchase price directly.  An alternative way to be given information about the purchase price is through a market quotation.

The market quotation of a bond is the purchase price expressed as a percentage of the face value.  For example, if a bond has a market quotation of 97.5, it means that the bond’s purchase price is 97.5% of its face value.  If the face value of the bond is $5,000, the purchase price is 0.975×5,000 =$4,875.

$\displaystyle{\mbox{Purchase Price}=\mbox{Face Value} \times \mbox{Market Quotation}}$

Although market quotations are percents, they generally do not include a percent sign when written.  It is common practice to see phrases like “The bond is purchased at 102.3.” or “The bond has a market quotation of 102.3.”  This does NOT mean that the bond is purchased for $102.30. In this type of phrasing, it is understood that the the “102.3” is a percent and that the purchase price of the bond is 102.3% of the face value of the bond. ## EXAMPLE A$5,000 bond has a 7% coupon.  The bond is purchased at 103.7 when there are 10 years to maturity.  What is the yield rate on the bond?

Solution:

Step 1:  Because no other information is given, the frequency of the payments and the compounding frequencies are assumed to be semi-annual.  The given information is

$\begin{eqnarray*} FV & = & \5,000 \\ P/Y & = & 2 \\ C/Y & = & 2 \\ t & = & 10 \mbox{ years} \\ \mbox{Coupon Rate} & = & 7\%\\ \mbox{Market Quotation} & = & 103.7\% \end{eqnarray*}$

Step 2:  Calculate the bond payment.

$\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 5,000\times \frac{0.07}{2}\\ & = & \175\end{eqnarray*}$

Step 3:  Calculate the purchase price (PV).

$\begin{eqnarray*} PV & = & FV \times \mbox{market quotation} \\ & = & 5,000\times 1.037\\ & = & \5,185\end{eqnarray*}$

Step 4:  Calculate the yield rate.

 PMT Setting END N $2 \times 10=20$ PV $-5,185$ FV $5,000$ PMT $175$ I/Y ? P/Y $2$ C/Y $2$

$I/Y=6.49\%$

The yield rate is 6.49% compounded semi-annually.

## TRY IT

A $13,000 bond with a coupon rate of 4.5% is redeemable in 15 years. The bond is purchased today at 95.4. Calculate the yield rate on the bond. Click to see Solution  PMT Setting END N $2 \times 15=30$ PV $-12,402$ FV $13000$ PMT $292.50$ I/Y ? P/Y $2$ C/Y $2$ $I/Y=4.94\%$ ## The Investor’s Yield The yield-to-maturity calculation requires that the bond is held until its maturity date, at which point the future value redemption price is known and equal to its face value. However, this condition may not hold. The bond holder may sell the bond at any point before the maturity date. If the bond is sold, the future value is based on the prevailing bond rate at that time, and the price is generally not equal to its face value. What then is the bond’s yield for the investor? To find the investor’s yield, calculate the rate of return (I/Y) with the present value set to the purchase price, the future value set to the selling price, the payment set to the bond payment, and time based on the length of time the investor held the bond. You can use the financial calculator to find the investor’s yield. To calculate the investor’s yield, you need to know the purchase price of the bond, the selling price of the bond, and the other information about the bond. ## USING THE TI BAII PLUS CALCULATOR TO FIND THE INVESTOR’S YIELD • Set the calculator to the END payment setting. • Enter values for the known variables (PV, FV, PMT, N, P/Y and C/Y), paying close attention to the cash flow sign convention for PV, PMT and FV. • Because you paid out the purchase price, PV (the purchase price) is entered with a negative cash flow sign. • Because you received the bond payments as the bond holder, PMT (the bond payment) is entered with a positive cash flow sign. • Because you are selling the bond, you will receive the selling price. So FV (the selling price) is entered with a positive cash flow sign. • After all of the known quantities are loaded into the calculator, press CPT and then I/Y to solve for the investor’s yield. ### NOTES 1. To calculate the investor’s yield, the purchase date and the selling date must be on an interest payment dates. 2. For an investor’s yield calculation, the time is based on the length of time between the purchase date and the selling date. The time to maturity is not involved in an investor’s yield calculation. 3. In an investor’s yield calculation, the present value is the purchase price of the bond and the future value is the selling price of the bond. The face value is not involved in an investor’s yield calculation (except to find the bond payment). ## EXAMPLE A$1,000 face value bond with a 7% coupon and 12 years to maturity was purchased for $1,084.68. It sold seven years later for$920.87. What rate of return did the bond holder realize on the sale?

Solution:

Step 1:  Because no other information is given, the frequency of the payments and the compounding frequencies are assumed to be semi-annual.  The given information, related to the sale, is

$\begin{eqnarray*} PV & = & \1,084.68 \\ FV & = & \920.87 \\ P/Y & = & 2 \\ C/Y & = & 2 \\ t & = & 7 \mbox{ years} \\ \mbox{Coupon Rate} & = & 7\% \end{eqnarray*}$

Step 2:  Calculate the bond payment.

$\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 1,000 \times\frac{0.07}{2}\\ & = & \35\end{eqnarray*}$

Step 3:  Calculate the bond holder’s rate of return on the sale.

 PMT Setting END N $2 \times 7=14$ PV $-1,084.68$ FV $920.87$ PMT $35$ I/Y ? P/Y $2$ C/Y $2$

$I/Y=4.6\%$

The bond holder’s rate of return on the sale is 4.6% compounded semi-annually.

## EXAMPLE

Sara purchased a $25,000 bond paying 2.8% when the yield rate was 3.7% and there was 10 years to maturity. After three years, Sara decides to sell the bond when the yield rate was 4.5%. 1. What price did Sara pay to purchase the bond? 2. At what price did Sara sell the bond? 3. What rate of return did Sara realize on sale of the bond? Solution: Step 1: Calculate the Sara’s purchase price. $\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 25,000 \times \frac{0.028}{2}\\ & = & \350\end{eqnarray*}$  PMT Setting END N $2 \times 10=20$ PV ? FV $25,000$ PMT $350$ I/Y $3.7$ P/Y $2$ C/Y $2$ $PV=\23,133.56$ Sara paid$23,133.56 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 10 years to maturity and then sold the bond three years later.  So at the time of the sale there was 7 (10-3) years to maturity.

 PMT Setting END N $2 \times 7=14$ PV ? FV $25,000$ PMT $350$ I/Y $4.5$ P/Y $2$ C/Y $2$

$PV=\22,472.11$

Sara sold the bond for $22,472.11. Step 3: Calculate Sara’s rate of return on the sale.  PMT Setting END N $2 \times 3=6$ PV $-23,133.56$ FV $22,472.11$ PMT $350$ I/Y ? P/Y $2$ C/Y $2$ $I/Y=2.10\%$ Sara’s rate of return on the sale is 2.10% compounded semi-annually. ## TRY IT John purchased a$12,000 bond with a 4.7% coupon for $13,750.09 when there was 17 years to maturity and the yield rate was 3.55%. After 11 years, John sold the bond when the yield rate was 5.3%. 1. At what price did John sell the bond? 2. What rate of return did John realize on the sale of the bond? Click to see Solution 1. Calculate the sale price of the bond.  PMT Setting END N $2 \times 6=12$ PV ? FV $12,000$ PMT $282$ I/Y $5.3$ P/Y $2$ C/Y $2$ $PV=\11,634.05$ 2. Calculate the rate of return on the sale.  PMT Setting END N $2 \times 11=22$ PV $-13,750.09$ FV $11,634.05$ PMT $282$ I/Y ? P/Y $2$ C/Y $2$ $I/Y=2.9\%$ ## Exercises 1. A$1,000 bond with a 6% coupon is purchased for $1,546.16 when there was 25 year to maturity. Calculate the yield to maturity. Click to see Answer 2.91% 2. A$5,000 bond paying 5.75% is purchased for 76.4822 when there was 18.5 years to maturity.  Calculate the yield to maturity.

8.25%

3. A $25,000 bond has a 7.25% coupon rate. The bond was purchased for$23,123.77 and sold six years later for $21,440.13. Calculate the investor’s rate of return on the sale price. Click to see Answer 6.8363% 4. A$100,000 bond with a 8.65% coupon rate is purchased for $106,590. The bond is sold 11.5 years later for$109,687.75.  Calculate the investor’s rate of return on the sale price.

8.271%

5. A $500,000 bond has a 9% coupon rate and is redeemable in 15 years. The bond is purchased at 105.72. Calculate the yield rate. Click to see Answer 8.33% 6. A$495,000 face value Province of Ontario bond issued with a coupon rate of 7.5% is sold for $714,557.14. If 20 years remain until maturity, what was the yield to maturity? Click to see Answer 4.2% 7. A$100,000 face value bond is issued with a 5% coupon and 22 years until maturity. If it sells for $76,566.88 9.5 years later, what was the yield to maturity? Click to see Answer 8% 8. A$50,000 face value Government of Canada bond is purchased for $48,336.48 and sold nine years later for$51,223.23. If the coupon rate is 4.985%, calculate the investor’s yield.

5.6742%

9. Great-West Life (GWL) issued a $100,000 face value bond carrying a coupon rate of 6.14% and 20 years to maturity. Eight years later, GWL decides to buy back some of its outstanding bonds when current market rates are 7.29%. If the bond was held from the issue date until the selling date, what yield did the bondholder realize on her investment? Click to see Answer 5.2091% 10. A$50,000 face value bond with a 5.1% coupon is purchase 8.5 years before maturity for $53,336.24. After 1.5 years, the bond is sold for$53,109.69.
1. Calculate the yield to maturity on the purchase date.
2. Calculate the yield to maturity on the selling date.
3. Calculate the investor’s yield.

a. 4.16%; b. 4.07%; c. 4.5041%

11. A $75,000 bond carrying a 4.08% coupon is purchased 19.5 years before maturity when the yield rate was 4.2%. After 3.5 years, the bond is sold when yield rate is 4.05%. 1. Calculate the purchase price of the bond along with the amount of the premium or discount. 2. Calculate the selling price of the bond along with the amount of the premium or discount. 3. Calculate the rate of return when the bond was sold. Click to see Answer a. Purchase Price=$73,809.92, Discount=$1,190.08; b. Purchase Price=$75,263.06, Premium=\$263.06; c. 4.6701%