Alexander Núñez Torres, PhD

Alexander Núñez Torres, PhD
School of Natural and Social Sciences Department of Economics and Business, Lehman College Chapter 5 – Bonds Alexander Núñez Torres, PhD Assistant Professor, Department of Economics and Business Unless otherwise noted, licensed under an Attribution-NonCommercial-ShareAlike 4.0 International.

Schedule Bond Concepts Zero-Coupon Bonds
Yield to Maturity of a Zero-Coupon Bond Risk-Free Interest Rates Coupon Bonds The Cash Flows of a Coupon Bond or Note Yield to Maturity of a Coupon Bond Coupon Bond Price Quotes Why Bond Prices Change Bond Prices and Interest Rates Bond Prices, maturity and risks Corporate Bonds Risks associated with corporate bonds Corporate yield curves, credit spreads and bond prices

Schedule Transforming coupon and zero-coupon bonds
Valuing a coupon bond with zero-coupon prices Using zero-coupon yields to value a coupon bond

Bond Concepts Bond Bond Certificate Terms of the bond
Amounts and dates of all payments to be made Payments Maturity date For good decisions, the benefits exceed the costs

Bond Concepts Term Face value (aka par value or principal amount)
Notional amount used to compute interest payments Usually standard increments, such as $1,000 Typically repaid at maturity Coupons

Bond Concepts Coupon rate
Set by the issuer and stated on the bond certificate By convention, expressed as an APR, so the amount of each coupon payment, CPN, is 𝐶𝑃𝑁= 𝐶𝑜𝑢𝑝𝑜𝑛 𝑅𝑎𝑡𝑒 𝑥 𝐹𝑎𝑐𝑒 𝑉𝑎𝑙𝑢𝑒 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐶𝑜𝑢𝑝𝑜𝑛 𝑃𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑝𝑒𝑟 𝑌𝑒𝑎𝑟

Bond Concepts: Table 5.1 Bond Certificate
States the terms of a bond as well as the amount, and dates of all payments to be made Coupons The promised interest payments of a bond. They are determined by the coupon rate, stated on the bond certificate. The amount to be paid is equal to: 𝐶𝑃𝑁= 𝐶𝑜𝑢𝑝𝑜𝑛 𝑅𝑎𝑡𝑒 𝑥 𝐹𝑎𝑐𝑒 𝑉𝑎𝑙𝑢𝑒 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐶𝑜𝑢𝑝𝑜𝑛 𝑃𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑝𝑒𝑟 𝑌𝑒𝑎𝑟 Maturity Date Final repayment date of the bond. Payment continue until this date. Face Value The notional amount used to compute the interest payment and that is usually repaid on the maturity date Term The time remaining until the repayment date.

Zero-Coupon Bonds Zero-coupon bonds
Only two cash flows: The bond’s market price at the time of purchase and the bond’s face value at maturity Treasury bills are zero-coupon U.S. government bonds with maturity of up to one year FV 1 Price

Zero-Coupon Bonds Yield to Maturity of a Zero-Coupon Bond
The discount rate that sets the present value of the promised bond payments equal to the current market price of the bond Yield to Maturity of an n-Year Zero-Coupon Bond: 1+ 𝑌𝑇𝑀 𝑛 = 𝐹𝑎𝑐𝑒 𝑉𝑎𝑙𝑢𝑒 𝑃𝑟𝑖𝑐𝑒 𝑛

Zero Coupon Bonds Example 5.1: Zero-coupon bonds for different maturities, 2018 Bellow the prices of treasury bills zero-coupon bonds that were trading in July Determine the corresponding yield to maturity for each bond. The table gives the prices and number of years to maturity and the face value is $1,000 per bond. Maturity 1 year 2 years 3 years 4 years 5 years Price $976.18 948.66 921.30 893.68 867.23

Zero Coupon Bonds 𝑌𝑇𝑀 1 = $1, $ −1=0.0244=2.44% 𝑌𝑇𝑀 2 = $1, $ −1=0.0267=2.67% 𝑌𝑇𝑀 3 = $1, $ −1=0.0277=2.77% 𝑌𝑇𝑀 4 = $1, $ −1=0.0285=2.85% 𝑌𝑇𝑀 5 = $1, $ −1=0.0289=2.89% Example 5.1: Zero-Coupon bonds for different maturities, 2018 Using the previous equation, we have:

Zero-Coupon Bonds Risk-Free Interest Rates
Since the treasury bills bear virtually no risk, the law of one price guarantees that the risk-free interest rate equals the yield to maturity of the T-bill zero coupon bond. In other words, the yield curve of the T-bill represents the risk-free interest rate for different maturities. Hence, any risk-bearing security must yield a higher return. We often refer to this as the risk-free interest rate for that period (n)

Zero Coupon Bonds Example 5.3: Computing the Price of a Zero-Coupon Bond What is the price of a 4-year risk-free zero-coupon bond with a face value of $1,000? Plan: Using the zero-coupon bond yield of the T-bill, the price of the bond can be calculated. From Example 5.2 (not shown here), in July 2019, the 4-year yield to maturity was 1.84%

Zero Coupon Bonds 𝑃= $1,000 1.0184 4 =$929.66
Example 5.3: Computing the Price of a Zero-Coupon Bond Execute: 𝑃= $1, =$929.66

Coupon Bonds Pay face value at maturity
Also make regular coupon interest payments Coupon bonds: Two types of U.S. Treasury coupon securities are currently traded in financial markets: Treasury notes: Original maturities from one to ten years. Treasury bonds: Original maturities of more than ten years

Coupon Bonds Existing U.S. Treasury Securities Treasury Security
Original maturity Type Bills 4, 13, 26, and 52 weeks Discount Notes 2, 3, 5, and 10 years Coupon Bonds 20 and 30 years

Coupon Bonds Return on a coupon bond comes from:
The difference between the purchase price and principal value Periodic coupon payments In order to compute the yield to maturity of a coupon bond, it is necessary to compute the interest payments, and when they are paid

Coupon Bonds Example 5.4. The Cash Flows of a Coupon Bond
Assume that in May 15, 2019 the U.S. Treasury issued securities with May 2024 maturity, $1,000 face value and a 4.3% coupon rate with semiannual coupons. Since the original maturity is only 5 years, these would be called “notes” as opposed to “bonds”. The first coupon payment will be paid on November 15, What cash flows would you receive if you hold this note until maturity?

Coupon Bonds Example 5.4. The Cash Flows of a Coupon Bond
The description of the note should be sufficient to determine all of its cash flows. The phrase “May 2024 maturity, $1000 par value” tells us that this is a note with a face value of $1000 and five years to maturity. The phrase “4.3% coupon rate and semiannual coupons” tells us that the note pays a total of 4.3% of its face value each year in two equal semiannual installments.

Coupon Bonds Example 5.4, The Cash Flows of a Coupon Bond
The face value of this note is $1000. Because this note pays coupons semiannually, you will receive a coupon payment every six months 𝐶𝑜𝑢𝑝𝑜𝑛𝑠 (𝐶𝑃𝑁, 𝑃𝑀𝑇)= 𝐶𝑜𝑢𝑝𝑜𝑛 𝑅𝑎𝑡𝑒 𝑥 𝐹𝑎𝑐𝑒 𝑉𝑎𝑙𝑢𝑒 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐶𝑜𝑢𝑝𝑜𝑛 𝑃𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑝𝑒𝑟 𝑌𝑒𝑎𝑟 𝐶𝑜𝑢𝑝𝑜𝑛𝑠 𝐶𝑃𝑁, 𝑃𝑀𝑇 = 4.3% 𝑥 $1, =$21.5 Here is the timeline based on a six-month period and there are a total of 10 cash flows:

Coupon Bonds Example 5.4, The Cash Flows of a Coupon Bond or Note
1 $21.5 $1,021.5 2 Price 3 4 10 Example 5.4, The Cash Flows of a Coupon Bond or Note Note that the last payment occurs five years (ten six-month periods) from now and is composed of both a coupon payment of $21.5 and the face value payment of $1000.

Coupon Bonds Example 5.4, The Cash Flows of a Coupon Bond or Note
Evaluate: Since a note is just a package of cash flows, we need to know those cash flows in order to value the note. That’s why the description of the note contains all of the information we would need to construct its cash flow timeline.

Coupon Bonds Yield to Maturity of a Coupon Bond:
1 CPN CPN+FV 2 -Price 3 4 N Yield to Maturity of a Coupon Bond: Cash Flows shown in the timeline below: Coupon bonds have many cash flows, complicating the yield to maturity calculation The coupon payments are an annuity, and the face value or par value is a cash flow in the future. The price can be computed as:

𝑃=𝐶𝑃𝑁 𝑥 1 𝑦 1− 1 1+𝑦 𝑁 + 𝐹𝑉 1+𝑦 𝑁 Coupon Bonds
Yield to Maturity of a Coupon Bond: 𝑃=𝐶𝑃𝑁 𝑥 1 𝑦 1− 𝑦 𝑁 + 𝐹𝑉 1+𝑦 𝑁

Coupon Bonds Example 5.5: Computing Yield to Maturity of a Coupon Bond
Problem: In July 2019, Apple Corporation issued $300 Million note with semiannual coupons and a coupon rate of 2.60%. Imagine you buy a $1,000 face value note at a price of $956. The note matures in 7 years. Since the original maturity is only 7 years, these would be called “notes” as opposed to “bonds”. What cash flows would you receive if you hold this note until maturity?

Plan: We worked out bond’s cash flows in Example 5.4. From the cash flow timeline, we can see that the bond consists of an annuity of 14 payments of $13, paid every 6 months, and one lump-sum payment of $1000 in 7 years (14 6-month periods). 1 $13 $1,013 2 -$956 3 4 14

1 $13 $1,013 2 $956 3 4 14 Example 5.5: Computing Yield to Maturity of a Coupon Bond Plan: Compute using the equation to solve for the yield to maturity. However, we must use 6-month intervals consistently throughout the equation.

Execute: Because the bond has ten remaining coupon payments, we compute its yield y by: 𝑃=𝐶𝑃𝑁 𝑥 1 𝑌𝑇𝑀 1− 𝑌𝑇𝑀 𝑁 + 𝐹𝑉 1+𝑌𝑇𝑀 𝑁 956=$13 𝑥 1 𝑌𝑇𝑀 1− 𝑌𝑇𝑀 , 𝑌𝑇𝑀 14

Execute: We can solve it by trial-and-error, financial calculator, or a spreadsheet. To use a financial calculator, we enter the price we pay as a negative number for the PV (it is a cash outflow), the coupon payments as the PMT, and the bond’s par value as its FV. Finally, we enter the number of coupon payments remaining (14) as N.

Execute:

Execute: Therefore, y = 1.65%. Because the bond pays coupons semiannually, this yield is for a six-month period. We convert it to an APR by multiplying by the number of coupon payments per year. Thus the bond has a yield to maturity equal to a 3.30% APR with semiannual compounding.

Coupon Bonds Example 5.6: Computing the Bond Price of a Coupon Bond
Problem: Consider a five-year, $1,000 bond with a 2.4% coupon rate and semiannual coupons. Suppose the bond’s yield to maturity is 2% (as an APR with semiannual compounding). What is the price of the bond?

Plan: Given the yield, the coupon rate and face value of the bond, we can use the equation to compute the price of the bond. Because this bond pays semiannual coupons, the 6-month yield of the bond is 1%.

Execute: 𝐶𝑜𝑢𝑝𝑜𝑛𝑠 (𝐶𝑃𝑁, 𝑃𝑀𝑇)= 2.4% 𝑥 $1,000 2 𝐶𝑜𝑢𝑝𝑜𝑛𝑠 𝐶𝑃𝑁, 𝑃𝑀𝑇 = 2.4% 𝑥 $1, =$12.0 1 $12 $1,012 2 Price 3 4 10

1 $12 $1,012 2 Price 3 4 10 Example 5.6: Computing the Bond Price of a Coupon Bond Execute: 𝑃=𝐶𝑃𝑁 𝑥 1 𝑌𝑇𝑀 1− 𝑌𝑇𝑀 𝑁 + 𝐹𝑉 1+𝑌𝑇𝑀 𝑁 𝑃𝑟𝑖𝑐𝑒=$12 𝑥 − , 𝑃𝑟𝑖𝑐𝑒=$1,018.94

Coupon Bonds Example 5.6: Computing a Bond Price from its YTM Execute:
We can also use a financial calculator: The effective annual yield corresponding to 1.0% every six months is (1+0.01) 2 −1=0.0201, 𝑜𝑟 2.01%

Coupon Bonds Coupon Bond Price Quotes
Prices and yields are often used interchangeably Bond traders usually quote yields rather than prices One advantage is that the yield is independent of the face value of the bond When prices are quoted in the bond market, they are conventionally quoted per $100 face value

Why Bond Prices Change Zero-coupon bonds always trade for a discount. Why? Coupon bonds may trade at a discount or at a premium. Why? Most issuers of coupon bonds choose a coupon rate so that the bonds will initially trade at, or very close to, par After the issue date, the market price of a bond changes over time.

Lets read the Bond Pricing Document
Why Bond Prices Change Interest Rate Changes and Bond Prices If a bond sells at par the only return investors will earn is from the coupons that the bond pays Therefore, the bond’s coupon rate will exactly equal its yield to maturity As interest rates in the economy fluctuate, the yields that investors demand will also change Lets read the Bond Pricing Document

Bond Prices Immediately after coupon payment
When It means The bond is considered 𝑃𝑟𝑖𝑐𝑒 > 𝐹𝑎𝑐𝑒 𝑉𝑎𝑙𝑢𝑒 𝐶𝑜𝑢𝑝𝑜𝑛 𝑅𝑎𝑡𝑒 > 𝑌𝑇𝑀 𝐴𝑏𝑜𝑣𝑒 𝑝𝑎𝑟, 𝑎𝑡 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 𝑃𝑟𝑖𝑐𝑒 = 𝐹𝑎𝑐𝑒 𝑉𝑎𝑙𝑢𝑒 𝐶𝑜𝑢𝑝𝑜𝑛 𝑅𝑎𝑡𝑒 = 𝑌𝑇𝑀 𝐴𝑡 𝑝𝑎𝑟 𝑃𝑟𝑖𝑐𝑒 < 𝐹𝑎𝑐𝑒 𝑉𝑎𝑙𝑢𝑒 𝐶𝑜𝑢𝑝𝑜𝑛 𝑅𝑎𝑡𝑒 < 𝑌𝑇𝑀 𝐵𝑒𝑙𝑜𝑤 𝑝𝑎𝑟, 𝑎𝑡 𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡

Why Bond Prices Change Example 5.7: Coupon Bonds, Discount, par or Premium Problem: Consider three ten-year, $1,000 bonds with a 5%, 3% and 1% coupon rates respectively. Suppose the bond’s yield to maturity is 3% for all bonds. What is the price of each of the bonds?

Why Bond Prices Change Example 5.7: Coupon Bonds, Discount, par or Premium Plan: From the description of the bonds, we can determine their cash flows. Each bond has 10 years to maturity and pays its coupons annually. Therefore, each bond has an annuity of coupon payments, paid annually for 10 years, and then the face value paid as a lump sum in 10 years.

Why Bond Prices Change Example 5.7: Coupon Bonds, Discount, par or Premium Plan: They are all priced so that their yield to maturity is 3%, meaning that 3% is the discount rate that equates the present value of the cash flows to the price of the bond. Therefore, we can compute the price of each bond as the PV of its cash flows, discounted at 3%.

Why Bond Prices Change Example 5.7: Coupon Bonds, Discount, par or Premium Execute: For the 5% coupon bond, the annuity cash flows are $50 per year (5% of each $1,000 face value). Similarly, the annuity cash flows for the 3% and 1% bonds are $30 and $10 per year. We use a $1,000 face value for all of the bonds.

Why Bond Prices Change Example 5.7: Coupon Bonds, Discount, par or Premium The bond prices are:

Why Bond Prices Change Example 5.8: Interest rate, maturity, and sensitivity of bonds Problem: Consider a 10-year and a 20-year bond, with 10% annual coupons. What would be the price of the bonds if the yield to maturity is 5%. What would be the price if the yield to maturity is 6%?

Why Bond Prices Change Example 5.8: Interest rate, maturity, and sensitivity of bonds The bond prices are: 10 year, 10% Annual Coupon 20 year, 10% Annual Coupon 5% $691.13 $501.51 6% $632.00 $426.50

Why Bond Prices Change Example 5.7: Coupon Bonds, Discount, par or Premium Evaluate: The prices reveal three important points previously discussed: First, all else equal, bonds with longer maturities have a lower price. Second, the bond prices have an inverse relationship with interest rates. Finally, long-term bonds are more sensitive on interest rate changes, that is, changes in interest rates will have greater effect on long-term bonds than short-term bonds

Why Bond Prices Change Interest Rates Risk and Bond Prices
Effect of time on bond prices is predictable, but unpredictable changes in rates also affect prices Bonds with different characteristics will respond differently to changes in interest rates Investors view long-term bonds to be riskier than short-term bonds

Corporate Bonds Credit Risk: U.S. Treasury securities are widely regarded to be risk-free. Credit risk is the risks associated with default. There is the probability that a company with outstanding debt defaults. Due to credit risk, corporate bonds have to pay higher coupons to attract buyers to their bonds. Corporate Bond Yields: Yield to maturity of a defaultable bond is not equal to the expected return of investing in the bond. A higher yield to maturity does not necessarily imply that a bond’s expected return is higher.

Corporate Bonds Bond Ratings: Several companies rate the creditworthiness of bonds: (Standard & Poor’s and Moody’s). These ratings help investors assess creditworthiness.

Corporate Bonds Corporate Yield Curves:
We can plot a yield curve for a corporate bonds just as we can for Treasuries The credit spread is the difference between the yields of corporate bonds and Treasuries

Corporate Bonds Example 5.09: Credit Spreads, corporate yields and Bond Prices Problem: Your firm has a credit rating of A. You notice that the credit spread for 10-year maturity debt is 80 basis points (0.80%). Your firm’s ten-year debt has a coupon rate of 5%. You see that new 10-year Treasury notes are being issued at par with a coupon rate of 4.5%. What should the price of your outstanding 10-year bonds be?

Corporate Bonds Example 5.09: Credit Spreads, corporate yields and Bond Prices Plan: If the credit spread is 80 basis points, then the yield to maturity (YTM) on your debt should be the YTM on similar treasuries plus 0.9%. The fact that new 10-year treasuries are being issued at par with coupons of 4.5% means that with a coupon rate of 4.5%, these notes are selling for $1,000 per $1,000 face value. Thus their YTM is 4.5% and your debt’s YTM should be 4.5% + 0.8% = 5.3%.

Corporate Bonds Example 5.09: Credit Spreads, corporate yields and Bond Prices Plan: The cash flows on your bonds are $5 per year for every $1,000 face value. Armed with this information, you can compute the price of your bonds.

Corporate Bonds Example 5.09: Credit Spreads, corporate yields and Bond Prices Execute: 50 𝑥 − =$977.17

Valuing a Coupon Bond with Zero-Coupon Prices
It is possible to replicate the cash flows of a coupon bond using zero-coupon bonds using the Law of One Price. For example, a four-year, $1000 bond that pays 10% annual coupons: 1 $100 $1,100 2 3 4 1-year zero 2-year zero 3-year zero 4-year zero

Valuing a Coupon Bond with Zero-Coupon Prices
We can calculate the cost of the zero-coupon bond portfolio that replicates the three-year coupon bond as: By the Law of One Price, the three-year coupon bond must trade for a price of $1,267.67 Zero-Coupon Bond Face Value Required Cost 1 Year $100 $97.62 2 Years $94.87 3 Years $92.13 4 Years $1,100 11 x $89.37 = $983.05 Total Cost: $1,267.67

Valuing a Coupon Bond with Zero-Coupon Yields
We can also use the zero-coupon yields to value a coupon bond. 𝑃=𝑃𝑉 𝐵𝑜𝑛𝑑 𝐶𝑎𝑠ℎ 𝐹𝑙𝑜𝑤𝑠 = 𝐶𝑃𝑁 1+ 𝑌𝑇𝑀 1 + 𝐶𝑃𝑁 1+ 𝑌𝑇𝑀 …+ 𝐶𝑃𝑁+𝐹𝑉 1+ 𝑌𝑇𝑀 𝑛 𝑛

For the three-year, $1,000 bond with 10% annual coupons considered earlier, we can use the equation to calculate its price using the zero-coupon yields: 𝑃= 100 ( ) , 𝑃=$1,267.67 The price is identical to the price computed earlier by replicating the bond.

Using a financial calculator: 𝑃=$1,267.67= 𝑦 𝑦 𝑦 , 𝑦 4

Coupon Bond Yields Therefore, the yield to maturity of the bond is 2.83%. We can check this directly: 𝑃= … , =$1,267.67 The yield to maturity is the weighted average of the yields of the zero-coupon bonds of equal and shorter maturities.

Coupon Bond Yields As the example shows, as the coupon increases, earlier cash flows become more important in the PV calculation. The shape of the yield curve keys us in on trends with the yield to maturity: If the yield curve is upward sloping, the yield to maturity decreases with the coupon rate of the bond. If the yield curve is downward sloping, the yield to maturity increases with the coupon rate of the bond.

References Berk, J.B., DeMarzo, P.M., & Harford, J.V.T. (2015). Fundamentals of Corporate Finance. (3rd ed.). Boston: Pearson Education. Board of Governors of the Federal Reserve System. (2019). Monetary policy. Retrieved from Financial Industry Regulatory Authority. (2017). What interest rate hike could do to your bond portfolio. Retrieved from Fitch Ratings, Inc. (2019). About. Retrieved from Moody's Investors Service, Inc. (2019). About us. Retrieved from Standards and Poor’s Financial Services LLC. (2019). S & P global ratings. Retrieved from U.S. Securities and Exchange Commission. (n.d.-a). About the SEC. Retrieved from U.S. Securities and Exchange Commission. (n.d.-b). Bonds. Retrieved from investing/basics/investment-products/bonds U.S. Securities and Exchange Commission. (n.d.-c). Interest rate risk - when interest rates go up, prices of fixed-rate bonds fall. Retrieved from

Alexander Núñez Torres, PhD
School of Natural and Social Sciences Department of Economics and Business, Lehman College Chapter 5 – Bonds Alexander Núñez Torres, PhD Assistant Professor, Department of Economics and Business

Alexander Núñez Torres, PhD

Similar presentations

Presentation on theme: "Alexander Núñez Torres, PhD"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Alexander Núñez Torres, PhD

Similar presentations

Presentation on theme: "Alexander Núñez Torres, PhD"— Presentation transcript:

Similar presentations

About project

Feedback