Chapter3. Fixed-Income Securities || Part2

3.4 Yield

Yield

• Interest rate implied by the payment structure
  수익률 : 지불 구조에 따른 이자율

• Interest rate at which the present value of the stream of payments (including coupon payments) is exactly equal to the current price

• Yields are always quoted on an annual basis

• YTM(Yield to maturity) is the IRR of the bond at the current price

 

Bond Price

• Suppose that a bond with face value F makes m coupon payments of c/m each year (total of c each year) and there are n periods remaining. If the current price is P, the YTM is the value of 𝛌 such that

 

Price-Yield Relationship

• General interest rate environment urges the yield of every bond to conform to that of other bonds
  이것이 바로 시장의 원리
• Yield of a bond can only change when bond’s price changesàAs yields move, prices move correspondingly
• Relationship between price and yield is shown by a price-yield curve

Price-Yield Curve

• Negative slope -> Price and yield have an inverse relationship

• If yield goes up, price goes down

• Larger coupons result in steeper curves

• What is the bond price when yield is zero?
  • When YTM = 0, bond is priced as if it offered no interest => Money in the future is not discounted so the present value is equal to the sum of all payments
• What is the bond price when yield and coupon rate are identical?
  • Value of the bond is equal to the par value (par bonds = Face Value, 신기하다!!)

• As maturity is increased, the price-yield curve becomes steeper => Longer maturities imply greater sensitivity of price to yield

• Suppose you purchase a 10% bond at par. It is likely that all bonds of maturity approximately 30 years would have yields of 10%. Then 10% would represent the market rate for such bonds.

  • Now suppose market conditions change and the yield increases to 11%. The price will drop to 91.28

  • With a 3-year 10% par bond, the price will drop to 97.50èInterest rate risk is lower

Yield Risk

• If yields change, bond prices also change

• Affects the near-term value (an immediate risk) => Price is governed by the price-yield curve

• Not affected if the bond is not sold
  • Continue to receive the promised coupon payments and the face value at maturity
  • Remains a fixed-income

3.5 Duration

Price and Maturity

• Bonds with long maturities have steeper price-yield curves than bonds with short maturities

• But maturity itself is not a complete quantitative measure of interest rate sensitivity

• Duration does give a direct measure of sensitivity
• '기울기가 크다'만으로 민감도를 측정하기가 어려움 => Duration 나옴

Duration

• Weighted average of the times that payments (cash flows) are made

• Weighting coefficients are the present values of the individual cash flows
• Suppose cash flows are received at times t0, t1, ...., tn the duration is

where PV(tk) is the present value of the cash flow that occurs at time tk

 

• Duration is the sensitivity of price with respect to changes in interest rate
  => Therefore, only cash flows defined from the bond is considered

같은 말 다른 표현

• where PV(tk) is the present value of the cash flow that occurs at time tk and PV is the total present value,

• Duration is a weighted average of cash flow times => Has units of time(대부분 year)

• When cash flows are all nonnegative, then t0 ≤ D ≤ tn
  => Duration is a time intermediate between the first and the last cash flows
• Consider a bond as multiple individual cash flows each at its maturity
  => Duration is an average of the maturities of all individual payments

• What is the duration of a zero-coupon bond?
  • Zero-coupon bond has a duration equal to its maturity
• What about nonzero-coupon bonds?
  • Nonzero-coupon bonds have durations strictly less than their maturity dates

Macaulay Duration

• Macaulay duration : use yield for computing duration

• Suppose m payments are made each year where the payment in period k is ck and there are n periods remaining. Then, the Macaulay duration is

 

• 이렇게 공식으로 쓰기도 한다. 

• Duration is useful because it directly measures the sensitivity of price to changes in yield

• Modified duration represents the percentage change in price of a bond per 1% point change in yield
• If a bond has a modified duration of 3 years
  • This means that a 1% increase (0.01 increase) in interest rates will reduce the bond price by about 3%

Note the difference between Macaulay duration and modified duration

▫ Macaulay duration calculates the weighted average of cash flow periods, whereas modified duration derives the bond price sensitivity