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BFC5280 Institutional asset and liability management
创建日期：2024-08-24 15:57:46
所属分类：金融Finance
标签： BFC5280
Institutional asset and liability management
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BFC5280
Institutional asset and liability management
Interest rate risk – Part III: Limitations of the Duration
model & Convexity

Key Learning objectives
 Understand the relative strengths and weaknesses of the duration model
 Understand the economic meaning of convexity & its properties
 Consider adjustments to the duration model to incorporate convexity
 Understand the impact of term structures and default risk

 Understand the relative strengths and weaknesses of the duration model
 Understand the economic meaning of convexity & its properties
 Consider adjustments to the duration model to incorporate convexity
 Understand the impact of term structures and default risk

 Duration model and immunization strategies are useful tools for regulators
◼ The model used by APRA and BIS to monitor bank interest rate risk-taking is
heavily based on this model
 Critics, however, claim that it is difficult to apply in real-world situations

Limitations of the duration model
 Duration matching can be costly and time-consuming
◼ The impact of the growing purchased funds markets, asset securitization, loan sales markets
on balance sheet restructuring (speed and cost)?
◼ Taking hedging positions in derivative products?
 Immunization is a dynamic problem
◼ Duration changes as the bond approaches maturity
◼ There is a trade-off between being perfectly immunized and the transaction costs
◼ So, most managers rebalance at discrete intervals (e.g., quarterly)

Limitations of the duration model
 Large interest rate changes and convexity
◼ Duration measures the price sensitivity of fixed income securities for small (
in the order of 1 basis point) changes in interest rates
◼ When interest rate shocks are much larger, duration becomes a less accurate
predictor of interest rate sensitivity
◼ When R changes, D changes. Hence, D is not a constant.

Limitations of the duration model
 Large interest rate changes and convexity
◼ Duration model predicts symmetric effects for rate increases and decreases on bond prices
◼ For large interest rate increases, duration model over-predicts the fall in bond prices while
for large interest rate decreases it under-predicts the increase in bond prices

Key Learning objectives
 Understand the relative strengths and weaknesses of the duration model
 Understand the economic meaning of convexity & its properties
 Consider adjustments to the duration model to incorporate convexity
 Understand the impact of term structures and default risk

Characteristics of Convexity
 Convexity is desirable
◼ The greater the convexity of a security or a portfolio of securities, the more
insurance or interest rate protection an FI manager has against interest rate
increases and the greater the potential gains after interest rate decreases.
 Convexity and duration
◼ The larger the interest rate changes and the more convex a fixed-income security
or portfolio, the greater the error the FI manager faces in using just duration to
immunize exposure to interest rate shocks.
 All fixed-income securities are convex

Economic meaning of convexity
Point (A)
FV = $1,000
N = 6 (annual)
CR = R = 8%
P = $1,000
D = 4.993
Point (C)
FV = $1,000
N = 6 (annual)
CR = 8%
R = 10%
P = $912.895
Point (E)
FV = $1,000
N = 6 (annual)
CR = 8%
R = 6%
P = $1,098.347
Point (B)
Point (D)

Properties of convexity
❑ As the bond’s maturity (N)
increases, so does the convexity
(CX)
❑ Coupon bonds of the same maturity
(N) have less convexity (CX) than do
zero-coupon bonds

Properties of convexity
❑ For coupon bonds and discount or zero-coupon bonds of the same duration, part 3 of the table
shows that the coupon bond has more convexity.

Properties of convexity
❑ Assets are more convex than liabilities

Key Learning objectives
 Understand the relative strengths and weaknesses of the duration model
 Understand the economic meaning of convexity & its properties
 Consider adjustments to the duration model to incorporate convexity
 Understand the impact of term structures and default risk

Basics of continuous compounding
 Annual compounding
 More frequent compounding
 Daily Compounding
 Continuous compounding

Basics of continuous compounding (CC)
 When compounding frequency approaches infinity, m ≈ ∞
 With CC, an amount A invested for n years at rate R grows to:
where e = 2.71828
 So, when, A = 100, n = 1, and R = 0.1, future value of A is:
 This is (to two decimal places) the same as the value with daily compounding:

Bond price and yield under CC
 The relationship between the price P and its yield y is:
 From this, it follows that…
 The term in parenthesis above is the ratio of the present value of the Cashflow
(Ci) at time ti to the bond price

Convexity of a bond
 The convexity of a bond is:
where, CX is the convexity measure, y is the bond’s yield measured with
continuous compounding, C is the cashflow and P is the market price of the bond
 Please, note: The exact calculation of CX won’t be required in the assessments –
you will be given information related to CX if needed for calculating price
changes or returns.
=
σ=0

2−

A Graphic Illustration of Duration Approximation

Duration and Convexity
 Duration (D) is the slope of the price–yield curve, and
◼ Essentially, the first-order effect (dP/dR) of an interest rate change on the bond’s
price is the price-yield curve slope effect—measured by Duration (D)
 The second order effect (dP2/d2R) measures the change in the slope of the
price-yield curve
◼ This is the curvature, or convexity (CX), effect
 There are other higher order effects ( from Taylor expansion series)
 Can be ignored for practical purposes

Convexity of a bond and IR risk
 Thus, incorporating convexity gives…
◼ The first term is the simple duration model
◼ The second term is the second-order effect/convexity/curvature adjustment
 In modified duration term
 The parameter CX reflects the degree of curvature in the price–yield curve at the
current yield level
◼ The degree to which the capital gain effect exceeds the capital loss effect for an equal
change in yields up or down.

Convexity of a bond and IR risk
 Even the above equation is not complete, because CX also changes with interest rate
(just as D does).
 Higher orders (up to infinity) of derivatives and interest rate changes should be added,
which should remind you of Taylor expansion in your advanced mathematics course.
 However, we typically ignore higher orders of approximation errors because they are
typically trivial → how big could be (∆) be if n >= 3?

Key Learning objectives
 Understand the relative strengths and weaknesses of the duration model
 Understand the economic meaning of convexity & it’s properties
 Consider adjustments to the duration model to incorporate convexity
 Understand the impact of term structures and default risk
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