Lecture 14 Term Structure of Interest Rates Investment Analysis and Portfolio Management Spring 2009 CAU Business School

Yield Curve 만기와 YTM 간의 관계 Yield curve 그 원인에 대해서는 여러 가지 설명이 제시 Yield curve 수평축에 만기(까지 남은 시간), 수직 축에 YTM을 표시한 곡선

Yield Curve의 예

How to Get Yield Curve? Zero coupon bond의 가격과 TYM 액면가 F, 만기까지 남은 시간이 t인 할인채권의 현재시장가격이 Pt인 경우 만기수익률 YTMt는 다음에서 계산 가능 액면가가 동일하고 만기가 다른 채권의 YTM을 구하여 이를 만기와 match: yield curve

Example

Yield Curve Under Certainty 미래의 모든 “연 이자율(short rate)”이 알려져 있다고 가정 만기수익률 곡선의 기울기가 의미하는 것 Flat: short rate이 모든 시기에 동일 Upward sloping: short rate이 시간의 경과에 따라 상승할 것으로 예상 Downward sloping: short rate이 시간의 경과에 따라 상승할 것으로 예상

Yield Curve Under Certainty Two strategies Buy a ZCB with two years to maturity and hold till maturity 액면가 $1,000, 현재가격 $890 Buy a ZCB with one year to maturity and then buy another ZCB with one year maturity with the proceeds from the first year investment 현재 구입하는 ZCB의 액면가 $934.5, 현재가격 $890 2년 후 두 전략의 투자 결과는 동일해야 할 것임!

Yield Curve Under Certainty 7.01%

Spot rate vs. Short rate Spot rate: yt Short rate: rt Rate the prevails today for a tome period corresponding to the zero’s maturity Example: two-year spot rate is 6% in our example. Short rate: rt The interest rate for a certain interval (usually, a year) available at different time points

Example Again 1원을 2년 만기 ZCB에 투자하든 1년 만기 ZCB에 투자한 후 이 투자수입을 다시 1년 만기 ZCB에 투자하더라도 2년 후 투자 성과는 동일 기하평균

Spot rate vs. Short rate

Generalization Note that 따라서 만기가 서로 다른 spot rate에서 미래의 short rate 개산 가능

Forward Rate Forward rate fn Forward rate의 해석 계약시점: 현재 계약 내용: 미래의 시점 (n-1)부터 시점 n까지 기간 동안 차입/대출 Forward rate: 위의 계약에 적용되는 이자율 Forward rate의 해석 Future short rate

Example: Forward Rate 4 yr rate = 8.00% 3yr rate = 7.00% f4 = ? (1.08)4 = (1.07)3 (1+fn) fn = .1106 or 11.06%

Example Zero-Coupon Rates Bond Maturity 12% 1 11.75% 2 11.25% 3 12% 1 11.75% 2 11.25% 3 10.00% 4 9.25% 5

Forward Rate Year 1 [(1.1175)2 / 1.12] - 1 = 0.115006

Interest Rate Uncertainty What can we say when future interest rates are not known today? Suppose that today’s rate is 5% and the expected short rate for the following year is E(r2) = 6% then: The rate of return on the 2-year bond is risky for if next year’s interest rate turns out to be above expectations, the price will lower and vice versa.

Interest Rate Uncertainty Investors require a risk premium to hold a longer-term bond. This liquidity premium compensates short-term investors for the uncertainty about future prices.

이자율 기간 구조: 이론 기대가설(expectations hypothesis) 유동성 선호가설(liquidity preference hypothesis)

Expectations Hypothesis Observed long-term rate is a function of today’s short-term rate and expected future short-term rates. Long-term and short-term securities are perfect substitutes. Forward rates that are calculated from the yield on long-term securities are market consensus expected future short-term rates.

Liquidity Premium Long-term bonds are more risky. Investors will demand a premium for the risk associated with long-term bonds. The yield curve has an upward bias built into the long-term rates because of the risk premium. Forward rates contain a liquidity premium and are not equal to expected future short-term rates.

이자율 기간구조의 해석 Higher expectations for forward rates or If the yield curve is to rise as one moves to longer maturities,,, A longer maturity results in the inclusion of a new forward rate that is higher than the average of the previously observed rates, Reason: Higher expectations for forward rates or Liquidity premium