# Term Structure

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$Y(t) = \left(\frac{1}{P(t)} \right)^{\frac{1}{t}} -1$

## The Term Structure of Interest RatesEdit

Bonds are issued with different times to maturity and can be group into either short term or long term bonds. The time to maturity for short-term bonds is usually less than a year and these bonds are therefore considered highly liquid. Bonds usually pay a rate of interest (Coupon Bond), but there are also bonds that don't pay any interest until the maturity date (Zero-coupon Bond). Bonds can be traded on capital markets. The price of the bond is inversly related to its interest rate and will reflect a liquidity, as well as a risk premium. [1] Longer term bonds (10, 20, 30, 50 years) also pay a real rate of interest plus a liquidity premium. Bonds issued by the US treasury are considered risk free, bonds issued by other countries or corporations however usually pay a risk premium. The liquidity premium for short-term bonds is lower than for long term bonds. The yield curve is therefore usually upward sloping, i.e. increases with time to maturity. The subjective difference between a 1 and 2 year bond at the present time, is usually a magnitude greater than the difference between the subjective difference in a 30 year and a 31 year bond.

However, it stands to reason that a long-term bond could be sold as a series of shorter term bonds (i.e. What is the difference between 1 30-year bond and 30 one-year bonds?). There must also be some expectation of rate changes that is factored into the yield curve. This is an assumption that can be made about treasury securities, that they are essentially the same instrument (in the secondary market) except for the term to maturity of the issued bond. The yield curve can then be plotted and some assumptions can be made, including that the short run bonds will pay a lower nominal interest rate than the long-term bonds. As a bond nears maturity it should, according to theory and at the limit, trade at par with currency of the same denomination.

Problems arise when the underlying fundamentals of the economy change.

### Flat CurveEdit

Short-term interest rates can be bid up to similar levels as long-term rates. This means that the short-term is valued relatively higher than "normal". The curve then reflects constant yields across different times to maturity. This present preference (with respect to liquidity premium) indicates a increased demand for liquidity, or a willingness to wait a few periods before making longer-term bets. ==> uncertainty about the correct long-run bets that should be made in the economy.

### Inverted CurvesEdit

An inverted yiel curve indicates that long-term rates have been overvalued, as well as a general suspicion of the long-term fundamentals of the economy. This theoretically would signal an eminent contraction, i.e. long-term investments will prove less profitable than previously thought (excess selling in the long-term markets depresses the interest rate in the long-term such that short-term rates pay higher returns than long-term rates).

### Steep CurveEdit

A rise in entrepreneurial spirit would, theoretically, be reflected in active long-term investment. An imminent boom would be approaching as long-term investment rises out-of proportion with the intermediate term-to-maturity. This results in a sharp rise along the curve (low short-term, moderate intermediate-term, and high long-term).

## Calculating YieldEdit

• Face Value = The value at which the bond can be sold on the date of maturity. Also called the par value.
• Market Value = The value the bond will trade at at any given point in time. This is typically less than the face value although it converges to face value as the term to maturity shortens.
$Yield\equiv \frac{Face~Value}{Market~Value }$

such that If I paid 950 for a bond that will pay 1000 a year from today:

$1.05263158\equiv \frac{1000}{950}$
• Forward rate: The nominal rate on a 2-year bond is 5.5% and the rate on a 1-year bond is 5%. Therefore, the nominal forward rate on the 2nd year is 6%  :
$\frac{1.055^{2}}{1.05^{1}} - 1 = 6.00\%$

### Changes in YieldEdit

Theory suggests LR=SR. If LR>SR, we expect both to go up, because the SR is the first year of the LR, so the LR must be even higher than the current SR in future years, and since we expect SR=LR, the SR must be higher as well. If SR>LR, we expect both to go down, flip the reasoning.

## Problems with the theoryEdit

• Nominal interest rates mean that inflation will have effects on the yields
• Preffered Habitat: major source of speculation -- Institutional buying follows a path dependant norm.
• Market segmentation: The long-term assets and the short-term assets have different markets. Might explain a consistantly lower short-term yield.
• Markets go illiquid at about a year (justification for the segment as well as a short-term vs. long-term distinction).
• Transactions cost of participating in the financial markets are not zero. The knowledge, or at least the access to brokers is non-zero, large numbers of people do not participate.
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