The most important aspects of bonds- Valuation of bonds
Bonds
A bond is a fixed-interest security with a contractual claim to interest payments (coupon, coupon) and the repayment of the capital amount provided (nominal value, principal value). If the payment is not made, the debtor is in default, which leads to a loan default or 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, although coupon payments in Europe are usually made annually are usually made annually in Europe, while in the United States semi-annually in the United States.
If a bond does not contain a coupon, it is called a zero-coupon bond referred to as a zero-coupon bond.
Although we often hear about the stock market, the bond market is much larger.
The bond market also has different bonds.
Here are a few explanations: T-Bills(US 1 year), T-Note(US 2-10 years), T-Bond(US 10-30 years), Callable bonds: Can be redeemed by the issuer before maturity, Convertible bondsConvertible bonds: Can be exchanged for common shares of the company, Puttable bonds: Give the holder the option to redeem or extend the bond, Floating rate bonds: Have an adjustable coupon (e.g., the interest rate of a Treasury bill), Treasury Inflation Protected Securities (TIPS) (Inflation Protected Securities) & .... .
In addition, there are such circumstances: Foreign bondsIssued by a borrower from a country other than the one in which the bond is sold, Yankee bonds: Foreign bonds, e.g. from a German issuer, sold in the U.S., Samurai bonds: Yen-denominated bonds sold in Japan by foreign issuers, Bulldog bonds: Bonds denominated in British pounds sold by foreigners in the UK, Eurodollars: Dollar-denominated bonds sold outside the United States (not just in Europe), Euroyen: Yen-denominated bonds sold outside Japan& Eurosterling: Sterling-denominated bonds sold outside the UK.
But how do you value bonds?
DCF valuation
The buyer of a bond acquires the following cash flow: first he must purchase the bond at a price P, then he regularly receives a coupon payment CPN and at the end of the term the repayment of the nominal value FV. The total value of all payments is therefore PV= CPN/(1+r) + CPN/ (1+r)^2 +... +CPN+FV/(1+r)^N. This corresponds to the fundamental value of the bond, a buyer should not pay more for it, since most coupons are constant, the bond price can also be calculated as: PV=CPN- [1/r - 1/r(1+r)^N] + FV/(1+r)^N , where r corresponds to the yield that a comparable investment would bring, thus taking into account the risk of the bond.
Here is an example to digest. In October 2022, Mr. Müller buys a French government bond worth EUR 100. The bond offers a coupon of 4.25% and matures in October 2026. What is the value of the bond if comparable French government bonds offer a yield of 0.15%? PV=4.25/1.0015 + 4.25/(1.0015)^2 + 4.25/(1.0015)^3 + 104.25/(1.0015)^4=116.34, Usually bond prices are quoted as a percentage of the nominal amount, i.e. 116.34%. The coupon yield is also known as the nominal yield and now the question arises: Why is the value of the bond (PV) greater than the nominal value? (more on this later)
Conversely, if we knew that the bond price was 116.34, we could calculate the yield y 116.34=4.25/1+y +4.25/(1+y)^2 + 4.25/(1+y)^3 + 104.25/(1+y)^4 This yield is known as the yield to maturity (YTM) for bonds and the YTM is calculated either by "trial and error" or with the help of a financial calculator or in Excel (function: IKV).In our example, the YTM is 0.15%, but now the question arises: Why is the YTM lower than the coupon? (More on this later).
Another example: In November 2022, you buy a US government bond with a term of exactly three years. The bond has a coupon interest rate of 4.25%, which is paid semi-annually, and has a nominal value of 1000 US dollars. If investors demand an annual yield of 0.965% for a comparable investment, what is the value of the US government bond in November 2022? PV=21.25/1.004825 + 21.25/(1.004825)^2 +...+1021.25/(1.004825)^6=1096.90
Bond prices and interest rates
The yield decreases when bond prices riseand vice versa. Why is this the case? Buying the payment stream of coupon and redemption payments becomes more expensive when the bond price is higher, which lowers the yield on the investment.
Also decreases price falls when interest rates for similar bonds rise and vice versa. Why is this the case?
If comparable bonds offer a higher yield, the bond in question must also do so, otherwise it would "disappear" from the market. With fixed coupon payments, a bond can only offer a higher yield if it can be bought more cheaply.
Simply put:Price down=YTM up and Price up=YTM falls (with the exception of bonds that change interest rates).
Interest rate sensitivity of bond prices
Bonds with long maturities react ceteris paribus (c.p.) more sensitively to changes in interest rates, as the discount effect is stronger with longer payment series. Bonds with a low coupon also react c.p. more sensitively to changes in interest rates if the coupon is low, as the final repayment of the nominal value accounts for a larger proportion of the bond's payment series. If the nominal value is not paid until the end of the term, the discount effect is particularly strong here.
Quiz
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 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 10-year term is issued at a nominal value of USD 1,000 and pays interest of USD 60 per year. 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 annually. 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
Yield curve
So far we have used a constant (i.e. maturity-independent) discount rate to calculate the value of a bond. However, a more precise calculation is to discount different interest rates for payments at different times. These interest rates are taken from the yield curve and it provides the annual interest rates (spot interest rates) at each maturity. The yield curve is derived from the yields of government bonds.
Please note Normal yield curve:
Short-term interest rates < long-term interest rates
Inverse yield curve: short-dated interest rates > long-dated interest rates
So why do long-dated bonds "normally" offer higher interest rates?
1.ExpectationAs an alternative, instead of investing in a long-dated bond, one could also invest in a series of short-dated bonds.
However, in market equilibrium, both investments should deliver the same return (what if they didn't...?) Only if this is fulfilled will investors be willing to hold all bonds. If investors expect interest rates to rise in the future, the investments with longer maturities must already offer higher interest rates today and the inverse interest rate structure then results from the expectation of falling interest rates
2.RiskThe longer the term of a bond, the greater its interest rate sensitivity; this uncertainty must be offset by a higher interest rate (otherwise investors would not be prepared to bear the higher risks compared to short-dated investments).
3.InflationThe longer the term of an investment, the less certain it is what "real value" the nominal repayment amount will still have in the future (i.e. how many goods and services can be paid for with it). An interest rate that rises with the term to maturity is therefore a risk premium for the
assumption of inflation risks.
Bonds and the default risk
The yield on a bond is largely determined by the default risk. The higher the default risk, the higher the yield (YTM) demanded by investors on the market.
The default risk is assessed by rating agencies and divided into classes (see Google=> What bond ratings are there?).
The most important thing today
The bond price is determined using the DCF formula
If the general interest rate level rises, then the bond price falls c.p. (and vice versa)
The interest rate is usually dependent on the term. Interest rates generally rise with the term (normal yield curve)
The default risk is also expressed in the bond yield
Rating agencies measure the default risk using different rating classes
Sneak peak: Equities
SourceBrealey, R., S. Myers, F. Allen, A. Edmans (2022): Principles of Corporate Finance, 14th edition, McGraw Hill, ISBN 1260013901 and lecture slides
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