Duration

Bond Valuation and Analysis in Python

Joshua Mayhew

Options Trader

The context behind duration

Prices and yields move inversely
We don't know how sensitive a bond is to interest rates
Duration measures interest rate sensitivity

Motivating example

Take a 5 year & 10 year bond, both with a 5% coupon
At a 5% yield they both have a price of USD 100:

-npf.pv(rate=0.05, nper=5, pmt=5, fv=100)
-npf.pv(rate=0.05, nper=10, pmt=5, fv=100)

100.00
100.00

Motivating example

If interest rates move up to 6%:

-npf.pv(rate=0.06, nper=5, pmt=5, fv=100)
-npf.pv(rate=0.06, nper=10, pmt=5, fv=100)

95.79
92.64

The 5 year bond lost 4.21% of its value, while the 10 year bond lost 7.36%
The 10 year bond was more sensitive to interest rate changes

What is duration?

Duration is the % price change for a 1% change in yields (interest rates).
Higher duration = higher interest rate risk
Typically used to:
- Measure interest rate risk
- Hedge interest rate risk
- Predict profit & loss as interest rates change

Calculating duration

We will use a simplified formula for duration:

${\large Duration = \frac{P_{down}\ -\ P_{up} }{2\ \times\ P\ \times\ \Delta y}}$

$P_{down}$ = Bond price at 1% lower yield
$P_{up}$ = Bond price at 1% higher yield
$P$ = Bond price at current yield
$\Delta y$ = Change in yield (we will use 1%)

Duration example

10 year bond, 5% annual coupon, 4% yield to maturity, what is its duration?

${ Duration = \frac{P_{down}\ -\ P_{up} }{2\ \times\ P\ \times\ \Delta y}}$

price = -npf.pv(rate=0.05, nper=10, pmt=5, fv=100)

price_up = -npf.pv(rate=0.06, nper=10, pmt=5, fv=100)
price_down = -npf.pv(rate=0.04, nper=10, pmt=5, fv=100)

duration = (price_down - price_up) / (2 * price * 0.01)
print(duration)

7.74

A 1% move in interest rates causes a 7.74% change in the bond price.

Summary

Bonds can behave differently for the same change in yields
Duration is the % price change of a bond for a 1% change in yields
Duration measures interest rate sensitivity

Let's practice!

Bond Valuation and Analysis in Python