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The yield to maturity of zero coupon bonds is called yield curve. The yield to maturity also called the spot rate.

Distinguish the difference between the short interest rate and the yield to maturity.

The yield to maturity is the average short interest rate on each period. Is not an arithmetic average, but a geometric one.

Coupon Bonds.

The yield to maturity differs for coupon bonds of the same maturity if their coupon rates differ.

Yield to maturity of coupon bonds is a bit less than yield to maturity of zero coupon bond.

Coupon bonds’ yield to maturity is weighted average spot rates of each period of maturity. The yield on the last payment dominates. 

The pure yield curve refers to the yield to maturity and time to maturity for zero coupon bonds.

The yields to maturity of different bonds over 1-year holding period are equal with the interest rate of certainty.

Forward interest rate/ forward rate:

E.g. Invest in 3 year zero coupon bond

       Invest in 2 year zero coupon bond , after 2 years reinvest the proceeds in a 1-year bond

Forward rate prevail in the second year to make the long- and short-term investments equally attractive.

Forward rate embody a premium compared with the expected future short interest rate.

The risk averse investors will be willing to hold the long term bond unless the expected short term interest E(r2) is less than the break even value f2. Because the lower the expectation of r2, the higher the anticipated return on the long-term period.

The liquidity premium compensates short term investors for the uncertainty about the price at which they will be able to sell their long term bonds at the end of the year. f2-E(r2)

The yield to maturity is different for coupon bonds with different rates.

A single yield curve relating yield with the time to maturity can not be appropriate for all bonds.

Theories of the term structure

Expectation hypothesis

The forward rate equals to the market consensus expectation of the future short interest rate. F2= E(r2), the liquidity premium is zero.

Relate the yield of long term bonds to expectations of future short interest rate. (1+y2)2 = (1+r1)(1+f2) since f2=E(r2), (1+y2)2 = (1+r1)[1+ E(r2)].

The yield to maturity can be determined by current and expected future short interest rates.

Liquidity preference

The short term investors will be unwilling to hold the short term bond unless the f2 > E(r2), the long term investors will be unwilling to hold the long term bonds unless the f2 < E(r2).

Liquidity preference theory of term structure believes that short term investors dominate the market, so the forward rate exceeds the expected short rate. f2>E(r2).

The liquidity premium is predicted to be positive.

(1 + yn)n = (1+r) (1+f1)(1+f2)…(1+fn)

Long term investments are more risky.

Investors with long term bonds demand a premium associated with risk.

The yield curve has an upward bias built into long term rates because of risk premium.

The liquidity premium may be overlaid on the expected spot rates to determine the forward interest rate.

An upward yield curve is associated with a forward rate higher than the spot or current yield to maturity.

Forward rates contain a liquidity premium and not equal to expected future short rate.

The yield curve continually rises, starting at level of 10% for 1-year bonds, but eventually approaching 11% for long term bonds and more forward rates at 11% are averaged into the yields to maturity.

Term Premiums: the spread between yields on long term and short term bonds, generally positive, then anticipated declines in rates could account for a downward-sloping yield curve.