4.3 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
Formula & Symbol Hub
Symbols Used
- [latex]BAL[/latex] = Principal balance
- [latex]BV[/latex] = Book value
- [latex]FV[/latex] = Face value of bond
- [latex]PMT[/latex] = Annuity payment amount
- [latex]N[/latex] = Number of annuity payments
- [latex]I/Y[/latex] = Nominal interest rate
- [latex]P/Y[/latex] = Number of payments per year or payment frequency
- [latex]C/Y[/latex] = Number of compounds per year or compounding frequency
Formulas Used
-
Formula 4.1 – Bond Payment
[latex]\text{Payment}=\mbox{Face Value} \times \mbox{periodic bond rate}[/latex]
-
Formula 4.2 – Purchase Price on an Interest Payment Date
[latex]\text{Purchase Price}=\mbox{Present Value of Face Value}+\mbox{Present Value of Bond Payments}[/latex]
-
Formula 4.3 – Premium (Bonds)
[latex]\text{Premium}=\mbox{Purchase Price}-\mbox{Face Value}[/latex]
-
Formula 4.4 – Discount (Bonds)
[latex]\text{Discount}=\mbox{Face Value}-\mbox{Purchase Price}[/latex]
Introduction
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 [latex]5.5\%[/latex] and you hold onto the bond until maturity, you will realize a yield to maturity of [latex]5.5\%[/latex].
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 ([latex]PV[/latex], [latex]FV[/latex], [latex]PMT[/latex], [latex]N[/latex], [latex]P/Y[/latex] and [latex]C/Y[/latex]), paying close attention to the cash flow sign convention for [latex]PV[/latex], [latex]PMT[/latex] and [latex]FV[/latex].
- Because you are paying out the purchase price, [latex]PV[/latex] (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 [latex]PMT[/latex] (the bond payment) and [latex]FV[/latex] (the face value) are both entered with a positive cash flow sign.
- After all of the known quantities are loaded into the calculator, press [latex]CPT[/latex] and then [latex]I/Y[/latex] to solve for the yield rate.
Key Takeaways
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.
To calculate a yield rate, the purchase date must be on an interest payment date.
Example 4.3.1
Suppose a [latex]\$10,000[/latex] face value bond is purchased for [latex]\$7,688.52[/latex] with [latex]20[/latex] years until maturity and has a coupon rate of [latex]4\%[/latex] semi-annually. What yield will the bondholder realize if she holds onto it until maturity?
Solution
Step 1: The given information is
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.
[latex]\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*}[/latex]
Step 2: Calculate the bond payment.
[latex]\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 10,000 \times\frac{0.04}{2}\\ & = & \$200\end{eqnarray*}[/latex]
Step 3: Calculate the yield rate.
| PMT Setting | END |
| [latex]N[/latex] | [latex]2 \times 20=40[/latex] |
| [latex]PV[/latex] | [latex]-7,688.52[/latex] |
| [latex]FV[/latex] | [latex]10,000[/latex] |
| [latex]PMT[/latex] | [latex]200[/latex] |
| [latex]I/Y[/latex] | [latex]?[/latex] |
| [latex]P/Y[/latex] | [latex]2[/latex] |
| [latex]C/Y[/latex] | [latex]2[/latex] |
[latex]I/Y=6\%[/latex]
Step 4: Write as a statement.
The yield rate is [latex]6\%[/latex] 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 [latex]97.5[/latex], it means that the bond’s purchase price is [latex]97.5\%[/latex] of its face value. If the face value of the bond is [latex]\$5,000[/latex], the purchase price is [latex]0.975\times 5,000 = $4,875[/latex].
[latex]\displaystyle{\mbox{Purchase Price}=\mbox{Face Value} \times \mbox{Market Quotation}}[/latex]
Things to Watch Out For
Although market quotations are percentages, they generally do not include a percent sign when written. It is common practice to see phrases like “The bond is purchased at [latex]102.3[/latex].” or “The bond has a market quotation of [latex]102.3[/latex].” This does NOT mean that the bond is purchased for [latex]\$102.30[/latex]. In this type of phrasing, it is understood that the the “[latex]102.3[/latex]” is a percent and that the purchase price of the bond is [latex]102.3\%[/latex] of the face value of the bond.
Example 4.3.2
A [latex]\$5,000[/latex] bond has a [latex]7\%[/latex] coupon. The bond is purchased at [latex]103.7[/latex] when there are [latex]10[/latex] years to maturity. What is the yield rate on the bond?
Solution
Step 1: The given information is
Because no other information is given, the frequency of the payments and the compounding frequencies are assumed to be semi-annual.
[latex]\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*}[/latex]
Step 2: Calculate the bond payment.
[latex]\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 5,000\times \frac{0.07}{2}\\ & = & \$175\end{eqnarray*}[/latex]
Step 3: Calculate the purchase price (PV).
[latex]\begin{eqnarray*} PV & = & FV \times \mbox{market quotation} \\ & = & 5,000\times 1.037\\ & = & \$5,185\end{eqnarray*}[/latex]
Step 4: Calculate the yield rate.
| PMT Setting | END |
| [latex]N[/latex] | [latex]2 \times 10=20[/latex] |
| [latex]PV[/latex] | [latex]-5,185[/latex] |
| [latex]FV[/latex] | [latex]5,000[/latex] |
| [latex]PMT[/latex] | [latex]175[/latex] |
| [latex]I/Y[/latex] | [latex]?[/latex] |
| [latex]P/Y[/latex] | [latex]2[/latex] |
| [latex]C/Y[/latex] | [latex]2[/latex] |
[latex]I/Y=6.49\%[/latex]
Step 5: Write as a statement.
The yield rate is [latex]6.49\%[/latex] compounded semi-annually.
Try It
1) A [latex]\$13,000[/latex] bond with a coupon rate of [latex]4.5\%[/latex] is redeemable in [latex]15[/latex] years. The bond is purchased today at [latex]95.4[/latex]. Calculate the yield rate on the bond.
Solution
| PMT Setting | END |
| [latex]N[/latex] | [latex]2 \times 15=30[/latex] |
| [latex]PV[/latex] | [latex]-12,402[/latex] |
| [latex]FV[/latex] | [latex]13000[/latex] |
| [latex]PMT[/latex] | [latex]292.50[/latex] |
| [latex]I/Y[/latex] | [latex]?[/latex] |
| [latex]P/Y[/latex] | [latex]2[/latex] |
| [latex]C/Y[/latex] | [latex]2[/latex] |
[latex]I/Y=4.94\%[/latex]
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 ([latex]I/Y[/latex]) 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 ([latex]PV[/latex], [latex]FV[/latex], [latex]PMT[/latex], [latex]N[/latex], [latex]P/Y[/latex] and [latex]C/Y[/latex]), paying close attention to the cash flow sign convention for [latex]PV[/latex], [latex]PMT[/latex] and [latex]FV[/latex].
- Because you paid out the purchase price, [latex]PV[/latex] (the purchase price) is entered with a negative cash flow sign.
- Because you received the bond payments as the bond holder, [latex]PMT[/latex] (the bond payment) is entered with a positive cash flow sign.
- Because you are selling the bond, you will receive the selling price. So [latex]FV[/latex] (the selling price) is entered with a positive cash flow sign.
- After all of the known quantities are loaded into the calculator, press [latex]CPT[/latex] and then [latex]I/Y[/latex] to solve for the investor’s yield.
Key Takeaways
- To calculate the investor’s yield, the purchase date and the selling date must be on an interest payment dates.
- 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.
- 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 4.3.3
A [latex]\$1,000[/latex] face value bond with a [latex]7\%[/latex] coupon and [latex]12[/latex] years to maturity was purchased for [latex]\$1,084.68[/latex]. It sold seven years later for [latex]\$920.87[/latex]. What rate of return did the bond holder realize on the sale?
Solution
Step 1: The given information, related to the sale, is
Because no other information is given, the frequency of the payments and the compounding frequencies are assumed to be semi-annual.
[latex]\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*}[/latex]
Step 2: Calculate the bond payment.
[latex]\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 1,000 \times\frac{0.07}{2}\\ & = & \$35\end{eqnarray*}[/latex]
Step 3: Calculate the bond holder’s rate of return on the sale.
| PMT Setting | END |
| [latex]N[/latex] | [latex]2 \times 7=14[/latex] |
| [latex]PV[/latex] | [latex]-1,084.68[/latex] |
| [latex]FV[/latex] | [latex]920.87[/latex] |
| [latex]PMT[/latex] | [latex]35[/latex] |
| [latex]I/Y[/latex] | [latex]?[/latex] |
| [latex]P/Y[/latex] | [latex]2[/latex] |
| [latex]C/Y[/latex] | [latex]2[/latex] |
[latex]I/Y=4.6\%[/latex]
Step 4: Write as a statement.
The bond holder’s rate of return on the sale is [latex]4.6\%[/latex] compounded semi-annually.
Example 4.3.4
Sara purchased a [latex]\$25,000[/latex] bond paying [latex]2.8\%[/latex] when the yield rate was [latex]3.7\%[/latex] and there was [latex]10[/latex] years to maturity. After three years, Sara decides to sell the bond when the yield rate was [latex]4.5\%[/latex].
- What price did Sara pay to purchase the bond?
- At what price did Sara sell the bond?
- What rate of return did Sara realize on sale of the bond?
Solution
Step 1: Calculate the Sara’s purchase price.
[latex]\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 25,000 \times \frac{0.028}{2}\\ & = & \$350\end{eqnarray*}[/latex]
| PMT Setting | END |
| [latex]N[/latex] | [latex]2 \times 10=20[/latex] |
| [latex]PV[/latex] | [latex]?[/latex] |
| [latex]FV[/latex] | [latex]25,000[/latex] |
| [latex]PMT[/latex] | [latex]350[/latex] |
| [latex]I/Y[/latex] | [latex]3.7[/latex] |
| [latex]P/Y[/latex] | [latex]2[/latex] |
| [latex]C/Y[/latex] | [latex]2[/latex] |
[latex]PV=\$23,133.56[/latex]
Sara paid [latex]\$23,133.56[/latex] 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 [latex]10[/latex] years to maturity and then sold the bond three years later. So at the time of the sale there was [latex]7[/latex] ([latex]10-3[/latex]) years to maturity.
| PMT Setting | END |
| [latex]N[/latex] | [latex]2 \times 7=14[/latex] |
| [latex]PV[/latex] | [latex]?[/latex] |
| [latex]FV[/latex] | [latex]25,000[/latex] |
| [latex]PMT[/latex] | [latex]350[/latex] |
| [latex]I/Y[/latex] | [latex]4.5[/latex] |
| [latex]P/Y[/latex] | [latex]2[/latex] |
| [latex]C/Y[/latex] | [latex]2[/latex] |
[latex]PV=\$22,472.11[/latex]
Sara sold the bond for [latex]\$22,472.11[/latex].
Step 3: Calculate Sara’s rate of return on the sale.
| PMT Setting | END |
| [latex]N[/latex] | [latex]2 \times 3=6[/latex] |
| [latex]PV[/latex] | [latex]-23,133.56[/latex] |
| [latex]FV[/latex] | [latex]22,472.11[/latex] |
| [latex]PMT[/latex] | [latex]350[/latex] |
| [latex]I/Y[/latex] | [latex]?[/latex] |
| [latex]P/Y[/latex] | [latex]2[/latex] |
| [latex]C/Y[/latex] | [latex]2[/latex] |
[latex]I/Y=2.10\%[/latex]
Step 4: Write as a statement.
Sara’s rate of return on the sale is [latex]2.10\%[/latex] compounded semi-annually.
Try It
2) John purchased a [latex]\$12,000[/latex] bond with a [latex]4.7\%[/latex] coupon for [latex]\$13,750.09[/latex] when there was [latex]17[/latex] years to maturity and the yield rate was [latex]3.55\%[/latex]. After [latex]11[/latex] years, John sold the bond when the yield rate was [latex]5.3\%[/latex].
- At what price did John sell the bond?
- What rate of return did John realize on the sale of the bond?
Solution
a. Calculate the sale price of the bond.
| PMT Setting | END |
| [latex]N[/latex] | [latex]2 \times 6=12[/latex] |
| [latex]PV[/latex] | [latex]?[/latex] |
| [latex]FV[/latex] | [latex]12,000[/latex] |
| [latex]PMT[/latex] | [latex]282[/latex] |
| [latex]I/Y[/latex] | [latex]5.3[/latex] |
| [latex]P/Y[/latex] | [latex]2[/latex] |
| [latex]C/Y[/latex] | [latex]2[/latex] |
[latex]PV=\$11,634.05[/latex]
b. Calculate the rate of return on the sale.
| PMT Setting | END |
| [latex]N[/latex] | [latex]2 \times 11=22[/latex] |
| [latex]PV[/latex] | [latex]-13,750.09[/latex] |
| [latex]FV[/latex] | [latex]11,634.05[/latex] |
| [latex]PMT[/latex] | [latex]282[/latex] |
| [latex]I/Y[/latex] | [latex]?[/latex] |
| [latex]P/Y[/latex] | [latex]2[/latex] |
| [latex]C/Y[/latex] | [latex]2[/latex] |
[latex]I/Y=2.9\%[/latex]
Section 4.3 Exercises
- A [latex]\$1,000[/latex] bond with a [latex]6\%[/latex] coupon is purchased for [latex]\$1,546.16[/latex] when there was [latex]25[/latex] year to maturity. Calculate the yield to maturity.
Solution
[latex]2.91\%[/latex]
- A [latex]\$5,000[/latex] bond paying [latex]5.75\%[/latex] is purchased for [latex]76.4822[/latex] when there was [latex]18.5[/latex] years to maturity. Calculate the yield to maturity.
Solution
[latex]8.25\%[/latex]
- A [latex]\$25,000[/latex] bond has a [latex]7.25\%[/latex] coupon rate. The bond was purchased for [latex]\$23,123.77[/latex] and sold six years later for [latex]\$21,440.13[/latex]. Calculate the investor’s rate of return on the sale price.
Solution
[latex]6.8363\%[/latex]
- A [latex]\$100,000[/latex] bond with a [latex]8.65\%[/latex] coupon rate is purchased for [latex]\$106,590[/latex]. The bond is sold [latex]11.5[/latex] years later for [latex]\$109,687.75[/latex]. Calculate the investor’s rate of return on the sale price.
Solution
[latex]8.271\%[/latex]
- A [latex]\$500,000[/latex] bond has a [latex]9\%[/latex] coupon rate and is redeemable in [latex]15[/latex] years. The bond is purchased at [latex]105.72[/latex]. Calculate the yield rate.
Solution
[latex]8.33\%[/latex]
- A [latex]\$495,000[/latex] face value Province of Ontario bond issued with a coupon rate of [latex]7.5\%[/latex] is sold for [latex]\$714,557.14[/latex]. If [latex]20[/latex] years remain until maturity, what was the yield to maturity?
Solution
[latex]4.2\%[/latex]
- A [latex]\$100,000[/latex] face value bond is issued with a [latex]5\%[/latex] coupon and [latex]22[/latex] years until maturity. If it sells for [latex]\$76,566.88 9.5[/latex] years later, what was the yield to maturity?
Solution
[latex]8\%[/latex]
- A [latex]\$50,000[/latex] face value Government of Canada bond is purchased for [latex]\$48,336.48[/latex] and sold nine years later for [latex]\$51,223.23[/latex]. If the coupon rate is [latex]4.985\%[/latex], calculate the investor’s yield.
Solution
[latex]5.6742\%[/latex]
- Great-West Life (GWL) issued a [latex]\$100,000[/latex] face value bond carrying a coupon rate of [latex]6.14\%[/latex] and [latex]20[/latex] years to maturity. Eight years later, GWL decides to buy back some of its outstanding bonds when current market rates are [latex]7.29\%[/latex]. If the bond was held from the issue date until the selling date, what yield did the bondholder realize on her investment?
Solution
[latex]5.2091\%[/latex]
- A [latex]\$50,000[/latex] face value bond with a [latex]5.1\%[/latex] coupon is purchase [latex]8.5[/latex] years before maturity for [latex]\$53,336.24[/latex]. After [latex]1.5[/latex] years, the bond is sold for [latex]\$53,109.69[/latex].
- Calculate the yield to maturity on the purchase date.
- Calculate the yield to maturity on the selling date.
- Calculate the investor’s yield.
Solution
a. [latex]4.16\%[/latex]; b. [latex]4.07\%[/latex]; c. [latex]4.5041\%[/latex]
- A [latex]\$75,000[/latex] bond carrying a [latex]4.08\%[/latex] coupon is purchased [latex]19.5[/latex] years before maturity when the yield rate was [latex]4.2\%[/latex]. After [latex]3.5[/latex] years, the bond is sold when yield rate is [latex]4.05\%[/latex].
- Calculate the purchase price of the bond along with the amount of the premium or discount.
- Calculate the selling price of the bond along with the amount of the premium or discount.
- Calculate the rate of return when the bond was sold.
Solution
a. Purchase Price=[latex]\$73,809.92[/latex], Discount=[latex]\$1,190.08[/latex]; b. Purchase Price=[latex]\$75,263.06[/latex], Premium=[latex]\$263.06[/latex]; c. [latex]4.6701\%[/latex]
Attribution
“7.3 Calculating the Yield Rate” from Business and Financial Mathematics by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.
“14.2: Calculating a Bond’s Yield” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.
