### Bond Theorem

Bond valuation is a technique for determining the theoretical fair value of a particular bond. Bond valuation includes calculating the present value of the bond’s future interest payments, also known as its cash flow, and the bond’s value upon maturity, also known as its face value or par value. Because a bond’s par value and interest payments are fixed, an investor uses bond valuation to determine what rate of return is required for a bond investment to be worthwhile.

#### Understanding Bond Valuation

A bond is a debt instrument that provides a steady income stream to the investor in the form of coupon payments. At the maturity date, the full face value of the bond is repaid to the bondholder. The characteristics of a regular bond include:

**Coupon rate:** Some bonds have an interest rate, also known as the coupon rate, which is paid to bondholders semi-annually. The coupon rate is the fixed return that an investor earns periodically until it matures.

**Maturity date:** All bonds have maturity dates, some short-term, others long-term. When the bond matures, the bond issuer repays the investor the full face value of the bond. For corporate bonds, the face value of a bond is usually $1,000 and for government bonds, face value is $10,000. The face value is not necessarily the invested principal or purchase price of the bond.

**Current Price:** Depending on the level of interest rate in the environment, the investor may purchase a bond at par, below par, or above par. For example, if interest rates increase, the value of a bond will decrease since the coupon rate will be lower than the interest rate in the economy. When this occurs, the bond will trade at a discount, that is, below par. However, the bondholder will be paid the full face value of the bond at maturity even though he purchased it for less than the par value.

### Term Structure of Interest Rates

The **term structure of interest rates** is the variation of the yield of bonds with similar risk profiles with the terms of those bonds. The **yield curve** is the relationship of the yield to maturity (YTM) of bonds to the time to maturity, or more accurately, to duration, which is sometimes referred to as the effective maturity. In most cases, bonds with longer maturities have higher yields. However, sometimes the yield curve becomes inverted, with short-term notes and bonds having higher yields than long-term bonds. Sometimes, the yield curve may even be flat, where the yield is the same regardless of the maturity. The actual shape of the yield curve depends on the supply and demand for specific bond terms, which, in turn, depends on economic conditions, fiscal policies, expected forward rates, inflation, foreign exchange rates, foreign capital inflows and outflows, credit ratings of the bonds, tax policies, and the current state of the economy. The yield curve changes because a component of the supply and demand for short-term, medium-term, and long-term bonds varies, to some extent, independently. For instance, when interest rates rise, the demand for short-term bonds increases faster than the demand for long-term bonds, which causes a flattening of the yield curve. Such was the case in 2006, when __T-bills__ were paying the same high rate as 30-year Treasury bonds.

The term structure of interest rates has 3 characteristics:

- The change in yields of different term bonds tends to move in the same direction.
- The yields on short-term bonds are more volatile than long-term bonds.
- The yields on long-term bonds tend to be higher than short-term bonds.

The expectations hypothesis has been advanced to explain the 1^{st} 2 characteristics and the premium liquidity theory have been advanced to explain the last characteristic.

### Market Segmentation Theory

Because bonds and other debt instruments have set maturities, buyers and sellers of debt usually have preferred maturities. Bond buyers want maturities that will coincide with their liabilities or when they want the money, while bond issuers want maturities that will coincide with expected income streams. **Market Segmentation Theory** (MST) posits that the yield curve is determined by supply and demand for debt instruments of different maturities. Generally, the debt market is divided into 3 major categories in regard to maturities: short-term, intermediate-term, and long-term. The difference in the supply and demand in each market segment causes the difference in bond prices, and therefore, yields. There are many different factors that would cause differences in the supply and demand for bonds of a certain maturity, but much of that difference will depend on current interest rates and expected future interest rates. If current interest rates are high, then future rates will be expected to decline, thus increasing the demand for long-term bonds by investors who want to lock in high rates while decreasing the supply, since bond issuers do not want to be locked into high rates. Therefore, long-term interest rates will be lower than short-term rates. On the other hand, if current interest rates are low, then bond buyers will tend to avoid long-term bonds so that they are not locked into low rates, especially since bond prices will decline when interest rates rise, which will generally happen if interest rates are already low. On the other hand, borrowers generally want to lock in low rates, so the supply for long-term bonds will increase. Hence, a lower demand and a higher supply will cause long-term bond prices to fall, thereby increasing their yield.

### Preferred Habitat Theory

**Preferred Habitat Theory** (PHT) is an extension of the market segmentation theory, in that it posits that lenders and borrowers will seek different maturities other than their preferred or usual maturities (their usual habitat) if the yield differential is favorable enough to them. For instance, if short-term rates are a lot lower than long-term rates, then bond issuers will issue more short-term bonds to take advantage of the lower rates even though they would prefer longer maturities to match their expected income streams; likewise, lenders will tend to buy long-term debt if the yield advantage is significant, even though carrying long-term debt has increased risks.

### Expectations Hypothesis

There are several versions of the expectations hypothesis, but essentially, the **expectations hypothesis** (aka **Pure Expectation Theory**, **Unbiased Expectations Theory**) states that different term bonds can be viewed as a series of 1-period bonds, with yields of each period bond equal to the **expected short-term interest rate** for that period. For example, compare buying a 2-year bond with buying 2 1-year bonds sequentially. If the interest rate for the 1^{st} year is 4% and the ** expected** interest rate, which is often referred to as the

**forward rate**, for the 2

^{nd}year is 6%, then one can be either buy a 1-year bond that yields 4%, then buy another bond yielding 6% after the 1

^{st}one matures for an average interest rate of 5% over the 2 years, or one can buy a 2-year bond yielding 5%—both options are equivalent: (4%+6%) / 2 = 5%. Hence, the sequential 1-year bonds are equivalent to the 2-year bond. (Actually, the geometric mean gives a slightly more accurate result, but the average is simpler to calculate and the argument is the same.)

Note that this relationship must hold in general, for if the sequential 1-year bonds yielded more or less than the equivalent long-term bond, then bond buyers would buy either one or the other, and there would be no market for the lesser yielding alternative. For instance, suppose the 2-year bond paid only 4.5% with the expected interest rates remaining the same. In the 1^{st} year, the buyer of the 2-year bond would make more money than the 1^{st}year bond, but he would lose more money in the 2^{nd} year—earning only 4.5% in the 2^{nd}year instead of 6% that he could have earned if he didn’t tie up his money in the 2-year bond. Additionally, the price of the 2-year bond would decline in the secondary market, since bond prices move opposite to interest rates, so selling the bond before maturity would only decrease the bond’s return.

Note, however, that expected future interest rates are just that – expected. There is no reason to believe that they will be the actual rates, especially for extended forecasts, but, nonetheless, the expected rates still influence present rates.

According to the expectations hypothesis, if future interest rates are expected to rise, then the yield curve slopes upward, with longer term bonds paying higher yields. However, if future interest rates are expected to decline, then this will cause long term bonds to have lower yields than short-term bonds, resulting in an **inverted yield curve**. The inverted yield curve often results when short-term interest rates are higher than historical averages, since there is a greater expectation that rates will decline, so long term bond issuers would be reluctant to issue bonds with higher rates when the expectation is that lower rates will prevail in the near future.

The expectations hypothesis helps to explain 2 of the 3 characteristics of the term structure of interest rates:

- The yield of bonds of different terms tend to move together.
- Short-term yields are more volatile than long-term yields.

However, the expectations hypothesis does not explain why the yields on long-term bonds are usually higher than short-term bonds. This could only be explained by the expectations hypothesis if the future interest rate was expected to continually rise, which isn’t plausible nor has it been observed, except in certain brief periods.

### Liquidity Premium Theory

The liquidity premium theory has been advanced to explain the 3^{rd} characteristic of the term structure of interest rates: that bonds with longer maturities tend to have higher yields. Although illiquidity is a risk itself, subsumed under the liquidity premium theory are the other risks associated with long-term bonds: notably interest rate risk and inflation risk. Naturally, increased risks will lower demand for those bonds, thus increasing their yield. This increase in yield is the risk premium to compensate buyers of long-term bonds for their increased risk.

**Liquidity** is defined in terms of its **marketability** — the easier it is to sell a bond at its value in the secondary marketplace, the more liquid it will be, thus reducing liquidity risk. This explains why long-term Treasuries have such low yields, because they are the easiest to sell. Assets may be illiquid because they are riskier and/or because supply exceeds demand. Additionally, illiquid assets are more difficult to price, since previous sale prices may be stale or nonexistent.

A bond’s yield can theoretically be divided into a risk-free yield and the risk premium. The risk-free yield is simply the yield calculated by the formula for the expectation hypothesis. The risk premium is the **liquidity premium** that increases with the term of the bond. Hence, the yield curve slopes upward, even if future interest rates are expected to remain flat or even decline a little, and so the **liquidity premium theory of the term structure of interest rates** explains the generally upward sloping yield curve for bonds of different maturities.

Liquidity Premium = Illiquid Bond YTM – Liquid Bond YTM

Besides liquidity, there are 2 other risks with holding bonds that increases with the term of the bond: inflation risk and interest rate risk. Both the inflation rate and the interest rate become more difficult to predict farther into the future. **Inflation risk** reduces the real return of the bond. **Interest rate risk** is the risk that bond prices will drop if interest rates rise, since there is an inverse relationship between bond prices and interest rates. Of course, interest rate risk is only a real risk if the bondholder wants to sell before maturity, but it is also an opportunity cost, since the long-term bondholder forfeits the higher interest that could be earned if the bondholder’s money was not tied up in the bond. Therefore, a longer term bond must pay a higher **risk premium** to compensate the bondholder for the greater risk.

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