D

=

∑

I

=

1

T

t

∗

C

(

1

+

r

)

t

+

T

∗

F

(

1

+

r

)

t

∑

I

=

1

T

C

(

1

+

r

)

t

+

F

(

1

+

r

)

t

where:

D

=

The MacAulay duration of the bond

T

=

the number of periods until expiration

I

=

the

I

t

h

time frame

C

=

periodic coupon payment

r

=

the periodic yield to maturity

F

=

par value at maturity

begin {aligned} & D = frac { sum_ {i = 1} ^ T { frac {t * C} { left (1 + r right) ^ t}} + frac {T * F} { left (1 + r right) ^ t}} { sum_ {i = 1} ^ T { frac {C} { left (1 + r right) ^ t}} + frac {F} { left (1 + r right) ^ t}} \ textbf {where:} \ & D = text {The MacAulay duration of the bond} \ & T = text {the number of periods until maturity} \ & i = text {the} i ^ {th} text {period of time} \ & C = text {periodic coupon payment} \ & r = text {periodic yield until maturity} \ & F = text {the par value at maturity} \ end {aligned}

where: D = ∑ I = 1 T ( 1 + r ) t C + ( 1 + r ) t F ∑ I = 1 T ( 1 + r ) t t ∗ C + ( 1 + r ) t T ∗ F D = The MacAulay duration of the bond T = the number of periods until expiration I = the I t h time frame C = periodic coupon payment r = the periodic yield to maturity F = par value at maturity

Duration in Fixed Income Management

Duration is essential for the management of fixed income portfolios, for the following reasons:

It is a simple summary statistic of the effective average maturity of a portfolio.
It is an essential tool to immunize portfolios from interest rate risk.
Estimate the sensitivity to interest rates of a portfolio.

The duration metric has the following properties:

The duration of a zero coupon bond equals the time to maturity.
By keeping maturity constant, the duration of a bond is shorter when the coupon rate is higher, due to the impact of higher coupon prepayments.
By keeping the coupon rate constant, the duration of a bond generally increases with time to maturity. But there are exceptions, such as with instruments such as deeply discounted bonds, where duration can fall with increases in maturities.
Holding other factors constant, the duration of coupon bonds is longer when bond yields to maturity are lower. However, for zero coupon bonds, the duration is equal to the time to maturity, regardless of the yield to maturity.
The duration of the perpetuity of the level is (1 + y) / year. For example, with a 10% return, the duration of the perpetuity that pays $ 100 per year will equal 1.10 / 0.10 = 11 years. However, with a return of 8%, it will be equal to 1.08 / 0.08 = 13.5 years. This principle makes it clear that maturity and duration can differ greatly. Case in point: the maturity of the perpetuity is infinite, while the duration of the instrument with a 10% yield is only 11 years. The cash flow weighted by present value at the beginning of the perpetuity life dominates the calculation of the duration.

Duration for gap management

Many banks have mismatches between the maturities of assets and liabilities. Bank liabilities, which are primarily deposits owed to customers, are generally short-term in nature, with low-duration statistics. In contrast, a bank’s assets primarily comprise outstanding commercial and consumer loans or mortgages. These assets tend to last longer and their values are more sensitive to fluctuations in interest rates. In periods when interest rates rise unexpectedly, banks can experience drastic falls in net worth if their assets fall more in value than their liabilities.

A technique called gap management is a widely used risk management tool, in which banks try to limit the “gap” between the duration of assets and liabilities. Gap management relies heavily on adjustable rate mortgages (ARMs) as key components in reducing the life of bank asset portfolios. Unlike conventional mortgages, ARM mortgages do not lose value when market rates rise, because the rates they pay are tied to the current interest rate.

On the other side of the balance sheet, the introduction of longer-term bank certificates of deposit (CDs) with fixed maturities serve to lengthen the duration of bank liabilities, also contributing to the reduction of the duration gap.

Understanding Gap Management

Banks employ gap management to equalize the duration of assets and liabilities, effectively immunizing their overall position from interest rate movements. In theory, the assets and liabilities of a bank are approximately equal in size. Therefore, if their durations are also the same, any change in interest rates will affect the value of assets and liabilities to the same degree and, consequently, changes in interest rates would have little or no ultimate effect on the net worth. Thus, immunizing net worth requires a portfolio duration, or gap, of zero.

Institutions with future fixed obligations, such as pension funds and insurance companies, differ from banks in that they operate with a view to future commitments. For example, pension funds are required to maintain sufficient funds to provide workers with an income stream upon retirement. As interest rates fluctuate, so do the value of assets held by the fund and the rate at which those assets generate income. Therefore, portfolio managers may wish to protect (immunize) the future equity of the fund at some target date against interest rate movements. In other words, immunization protects assets and liabilities of equivalent duration, so that a bank can meet its obligations, regardless of interest rate movements.

Convexity in fixed income management

Unfortunately, duration has limitations when used as a measure of interest rate sensitivity. While the statistic calculates a linear relationship between changes in price and bond yield, in reality, the relationship between changes in price and yield is convex.

In the image below, the curved line represents the change in prices, given a change in returns. The straight line, tangent to the curve, represents the estimated change in price, through the duration statistic. The shaded area reveals the difference between the duration estimate and the actual price movement. As indicated, the greater the change in interest rates, the greater the error in estimating the price change of the bond.

Image by Julie Bang © Investopedia 2019

Convexity, a measure of the curvature of changes in the price of a bond, relative to changes in interest rates, addresses this error by measuring the change in duration as interest rates fluctuate. The formula is as follows:

C

=

D

two

(

B

(

r

)

)

B

∗

D

∗

r

two

where:

C

=

convexity

B

=

the price of the bond

r

=

the interest rate

D

=

duration

begin {aligned} & C = frac {d ^ 2 left (B left (r right) right)} {B * d * r ^ 2} \ & textbf {where:} \ & C = text {convexity} \ & B = text {the price of the bond} \ & r = text {the interest rate} \ & d = text {duration} \ end {aligned}

C = B ∗ D ∗ r two D two ( B ( r ) ) where: C = convexity B = the price of the bond r = the interest rate D = duration

In general, the higher the coupon, the lower the convexity, because a 5% bond is more sensitive to changes in interest rates than a 10% bond. Due to the call function, the callable bonds will show negative convexity if the yields fall too low, which means that the duration will decrease when the yields decrease. Zero coupon bonds have the highest convexity, where the ratios are only valid when the bonds compared have the same duration and yields to maturity. On time: a high convex bond is more sensitive to changes in interest rates and, consequently, should see greater fluctuations in price when interest rates move.

The opposite occurs with low convexity bonds, whose prices do not fluctuate as much when interest rates change. When graphed on a two-dimensional graph, this relationship should generate a long sloping U shape (hence the term “convex”).

Low-coupon and zero-coupon bonds, which tend to have the lowest yields, show the highest interest rate volatility. In technical terms, this means that the modified duration of the bond requires a larger adjustment to keep up with the higher change in price after the interest rate moves. Lower coupon rates lead to lower returns and lower returns lead to higher degrees of convexity.

The bottom line

Constantly changing interest rates introduce uncertainty in fixed income investing. Duration and convexity allow investors to quantify this uncertainty, helping them manage their fixed income portfolios.

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