Introduction to bonds
A bond is a fixed-interest security that entitles the holder to regular interest payments (coupon) and repayment of the invested capital (nominal value). If a payment is not made, the debtor defaults, which can result in a loan default or even insolvency. Both the price of the bond and the coupon are expressed as a percentage of the nominal value.
Both companies and governments can issue bonds. In Europe, the coupon is usually paid annually, whereas in the United States it is paid semi-annually. Bonds without a coupon are known as zero-coupon bonds.
The bond market comprises numerous types of bonds, which are differentiated according to their characteristics and conditions.
Here are some examples:
- T-bills (US): Short-term bonds with maturities of up to one year.
- T-Notes (US): Medium-term bonds with maturities of 2 to 10 years.
- T-Bonds (US): Long-term bonds with terms of between 10 and 30 years.
- Callable bonds: Can be redeemed early by the issuer.
- Convertible bonds: Allow conversion into shares of the issuer.
- Puttable bonds: Allow the holder to redeem or extend the bond early.
- Floating rate bonds: The coupon is linked to a benchmark interest rate, such as the interest rate on a Treasury bill.
- TIPS (Treasury Inflation-Protected Securities): Inflation-protected bonds that secure real value.
International bonds and currency components
Common types of international bonds include
- Yankee bonds: Bonds issued by foreign issuers that are sold in the U.S.
- Samurai bonds: Bonds issued in yen in Japan by foreign issuers.
- Bulldog bonds: Sterling-denominated bonds issued by foreign issuers in the UK.
- Eurodollar bonds: Bonds denominated in US dollars that are sold outside the US.
- Euroyen and Eurosterling bonds: Bonds denominated in yen or sterling and sold outside Japan and the UK respectively.
Valuation of bonds using the DCF method
The value of a bond is derived from the discounted cash flow received by the buyer. First, the bond is purchased at the current price P, then the regular coupon payments (CPN) are received and at the end of the term the nominal value (FV). The present value (PV) of all future payments can be calculated as follows:
PV= CPN/(1+r) + CPN/ (1+r)^2 +... +CPN+FV/(1+r)^N
This is the maximum fundamental value of the bond that a buyer should be prepared to pay. Since many coupons are constant, the calculation is simplified to:
PV=CPN- [1/r - 1/r(1+r)^N] + FV/(1+r)^N
Here r corresponds to the yield of comparable investments and thus reflects the risk of the bond.
Sample calculation
In October 2022, Mr. MĂŒller purchases a French government bond with a nominal value of EUR 100, a coupon of 4.25 % and a term until October 2026. With a yield of comparable bonds of 0.15 %, this results in a bond value of:
PV=4.25/1.0015 + 4.25/(1.0015)^2 + 4.25/(1.0015)^3 + 104.25/(1.0015)^4=116.34
This value is expressed as a percentage of the nominal value as 116.34%. This means that the bond value is higher than the nominal value, as the market yield is lower than the coupon.
The yield you make is then your previous YTM or, in the case of an early payment, can also be calculated as follows = profit x 100 / capital invested = relative yield
Relationship between bond price and interest rate
If the bond price rises, the yield falls, and vice versa. As a general rule, if interest rates for similar bonds rise, the price falls, as higher yields can only be achieved through lower entry prices.
Yield curve and maturities
The yield curve shows the relationship between the term and the interest rate and is derived from the yields on government bonds. Long-term bonds usually offer higher interest rates (normal yield curve) in order to compensate for long-term risks and inflation uncertainties.
Bonds and the default risk
The level of the bond yield is significantly influenced by the issuer's default risk. A higher risk means a higher yield. Rating agencies classify bonds into rating classes according to their risk, which help investors to make a decision.
Quiz
A 10-year bond is issued with a face value of 1,000 US dollars and pays 60 US dollars interest annually. What happens to the nominal interest rate (i.e. the coupon) of the bond if the YTMs of comparable bonds rise shortly after issue?
a) no change
b) rises
c) falls
d) depends on the rating of the bond
A bond with a term of 10 years is issued at a nominal value of 1,000 US dollars and pays interest of 60 US dollars annually. What happens to the YTM of the bond if the YTMs of comparable bonds rise shortly after issue?
a) no change
b) rises
c) falls
d) depends on the rating of the bond
A bond with a term of 10 years is issued at a nominal value of 1,000 US dollars and pays interest of 60 US dollars per year. What happens to the price of the bond if the YTMs of comparable bonds rise shortly after issue?
a) no change
b) rises
c) falls
d) depends on the rating of the bond
Summary
- The price of a bond is determined using the DCF formula.
- If the general interest rate level rises, the bond price falls and vice versa.
- Long-term bonds generally offer higher interest rates as the interest rate and inflation risk increases.
- The default risk of a bond is assessed by rating agencies and divided into rating classes.
Source: Brealey, R., Myers, S., Allen, F., Edmans, A. (2022): Principles of Corporate Finance, 14th edition, McGraw Hill, ISBN 1260013901 and lecture slides.
