Question

Analyze the following three-bond portfolio and answer the following questions:

Bonds	Percent of Portfolio
Treasuries	25%
Corporate AAA	50%
Mortgage-backed	25%

a. If the portfolio duration is 11.5, what is the percent contribution to spread duration of the Treasury-sector bonds?

b. Evaluate the various convexity calculations and how those would affect the duration of the individual bonds and this portfolio of bonds (without needing to make calculations). Also analyze the advantages and disadvantages of using key rate and effective duration.

Answer 1

a. If the portfolio duration is 11.5, what is the percent contribution to spread duration of the Treasury-sector bonds?

To answer above need some more Information.

b. Evaluate the various convexity calculations and how those would affect the duration of the individual bonds and this portfolio of bonds (without needing to make calculations). Also analyze the advantages and disadvantages of using key rate and effective duration.

A bonds convexity refers to the sensetivity of the bond's modified duration to changes in yields.

At low yields the modified duration changes quickly as yield changes , therfore the convexity is high.

At high yields the modified duration changes slowly as yield changes, therefore the convexity is low

C=

Effective duration and convexity

Effective duration is defined as follows:

The modified duration and convexity measures that they are based on assumptions that all promised cash flow to a bond will actually be made. But in reality, with an embedded call option, the bond can be repurchased prior to maturity by the issuer. This will result in the last few years’ worth of cash flows not being received by the bond owner. As a result, the modified duration and convexity measures may not accurately reflect the risk of this bond.

n order to correct for this problem, two alternative measures were developed: effective duration and effective convexity

Effective duration is defined as follows:

where:

D_E = effective duration

P₀ = the initial price of a bond

P_– = the price of the bond after the yield curve shifts down by Δy basis points

P₊ = the price of the bond after the yield curve shifts up by Δy basis points

Dy = the change in yield

Also analyze the advantages and disadvantages of using key rate and effective duration.:

Key rate duration: Key rate duration measure in which it is assumed that yield changes do not change the expected cash flows (treasury bills)

  Effective duration, measure in which yield changes may change the expected cash flow (callable bonds)  

Key rate durations have several advantages over existing measures:

i. They can identify the price sensitivity of a bond to each segment of the spot yield curve. Effective duration is the total risk exposure, and the key rate durations are the component parts of the effective duration. As is the case with effective duration, key rate durations aggregate linearly, so that portfolio analysis is straightforward

ii. They recognize that the yield curve movement is driven by multiple market factors. The validity of the key rate durations does not depend on any equilibrium model of the yield curve movement. Key rate durations are applicable over a broad range of arbitrary yield curve movements.

iii. The use key rate durations to create a replicating portfolio of a bond with embedded options using zero-coupon bonds. Thus the cash flow of the replicating portfolio correctly represents the instantaneous expected cash flow of the option.

However, there are some disadvantage over key rate and modified duration:

i. The choice of the key rates is subjective. A natural election will be to employ as key rates the most common used treasury rates in the markets. whereas the portfolio manager can reduce (increase) this group according to the structure of his portfolio.

ii. The movements in the yield curve are not independent. e.g changes in the 8- year and 10-year rates in the same direction; but a change in the 9-year rate in the opposite way is possible in the KRD model, but it is unlikely that this happen in the reality.

iii. There is a lost of accuracy due to the fact that the KRD do not use the past yield curves changes to estimate the interest rate risk.