T HE SHORTAGES OF THE DURATION MODELS

Although the duration model is adopted by many financial institutions, it is hard to apply in the real world situation. The criticisms of the duration model approximately contain that the duration matching is costly; the immunization is a dynamic problem and the large interest rate changes and convexity.

In the principle of duration model, when we calculate the DA and DL, if there is the duration gap between them, the manager should change the DA or DL to make the company immunize from the interest rate risk. In the other hand, the interest rate change in the real world is not like our assumption which changes instantaneously. To make sure the duration of portfolio investment exactly matches the investment horizon, the manager need to rebalance the portfolio continuously. However, restructure the balance sheet is really time-consuming and costly. But this problem can be settled by using the financial derivatives such as the swaps, futures and forwards. This we will mention in part 6 of our paper.

Here we concentrate on the large interest rate changes and convexity. Because as we know, the interest rate drops dramatically since 1996, considering the convexity is close to the situation of China.

As Saunders (2006) pointing out, the duration accurately measure the price sensitivity of the bond for small change of interest rate, but when the interest rate shocks larger, the duration measure will be less accurate. With large interest rate increases, the duration model will over estimate the fall of the bond price; with the large interest rate decreases, it will under estimate the rise of the bond price. The effect shows like

Figure 1: Duration versus True Relationship

△P/P

△R/R True

Duration model Error

-D Error

Source: Saunders (2006) p 239.

The duration model predicts symmetric effects for interest rate increases or decreases on bond price. But in reality, for interest rate increases, the capital loss effect tends to be smaller than the capital gain effect is for the interest rate decreases. This is because the bond price-yield relationship is much more convex than linear, as assumed by the basic duration model. Thus we know that convexity is a volatility measure for bonds used in conjunction with modified duration in order to measure how the bond's price will change as interest rates change. It is equal to the negative of the second derivative of the bond's price relative to its yield, divided by its price. For example, since a non-callable bond's duration usually increases as interest rates decrease, its convexity is positive²⁶. For any manager of a financial institution, the convexity is a necessary

26 http://www.investorwords.com/5480/convexity.html

feature he should capture in the portfolio of assets.

By the definition of convexity (CX) parameter equals:

CX = Scaling factor

[

]

(negative effect) (positive effect) rise in yield fall in yield

a one-basic-point + from a one-basic-point Capital loss from Capital gain

The sum of the two terms in the brackets reflects the degree to which the capital gain effect exceeds the capital loss effect for a small one-basic-point interest rate change down and up. The scaling factor normalizes this measure to account for a larger 1 percent change in interest rate. A commonly used scaling factor is 108 so that:

^⎥⎦

⎢⎣ ⎤

⎡ Δ +

− +

= Δ

P P P

CX 10⁸ P

The value for CX can be inserted into the bond price prediction equation with the convexity adjustment

( )

2

1CX R R

P MD

P =− ×Δ + Δ Δ

(

)

duration D Modified

MD= − = +

1

This equation can reduce the error between predicted value and true value. (Sunders 2006)

Here are the three characteristics of convexity:

1. Convexity is desirable. The greater the convexity of a security or a portfolio of securities, the more protection against interest rate increases and the greater the potential gains after interest rate falls.

2. Convexity and duration. The larger the interest rate changes and the more convex a

fixed-income security or portfolio, the greater the error in using the duration to immunize exposure to interest rate shocks.

3. All fixed-income securities are convex. As the yield goes to infinity, the bond price falls asymptotically toward zero, but by definition a bond’s price can never be negative. Therefore, zero must be the minimum bond price (see Figure 2).

Saunders (2006)

Figure 2: The Natural Convexity of Bonds

Price-yield curve convexity

∞ Yield (R) Price

0

Source: Saunders (2006) p 249.

Commonly, the manager seeks to attain greater convexity in the asset portfolio than in the liability portfolio, as shown in figure. Because if in this situation, no matter the interest rate rises or falls, the company may get a benefit on the net worth.

However, the above calculations assume that the cash flows are not affected by the fluctuation of interest rates and the cash flows are discounted at a single rate.

Furthermore, the modified duration can not accurately measure the interest rate sensitivity of bonds with embedded options, such as mortgage backed securities and callable corporate bonds.

Figure 3: Assets are more convex than liabilities

Equity Asset

Liability Asset,

Liability, Equity Value ($)

R-2% R% R+2% Interest rates

Source: Saunders (2006) P 252

The only duration formula that can measure the risk of bonds that have embedded options is called “Effective Duration,” or “Option Adjusted Duration”. Effective Duration adjusts the risk of bonds by taking into account the relative sensitivity of bonds with different coupon rates and terms to prevailing interest rates.²⁷ Here we give the calculation of effective duration:

R P

P D_eff P

Δ

= ⁻ − ⁺ 2

Where P = initial price of the security

P- = estimated value of the security if the yield decreases by △R P+ = estimated value of the security if the yield increases by △R △R = change of yield of a security

And the calculation of effective convexity is:

( )

P

P P CX_eff P

Δ

−

= ⁻ − ⁺

27 http://www.vcallc.com/mailings/2000/dur0800.htm