Dollar convexity and bond price prediction

Bond Valuation and Analysis in Python

Joshua Mayhew

Options Trader

Dollar convexity

Convexity = % change in duration for 1% change in yields
Dollar convexity = $ change in duration for 1% change in yields:

$ \text{Dollar Convexity} = \text{Convexity} \times \text{Bond Price} \times 0.01^2$

Dollar convexity example

10 year bond with 3% coupon, 5% yield and USD 100 face value, what is its dollar convexity?

price = -npf.pv(rate=0.05, nper=10, pmt=3, fv=100)
price_up = -npf.pv(rate=0.06, nper=10, pmt=3, fv=100)
price_down = -npf.pv(rate=0.04, nper=10, pmt=3, fv=100)
convexity = (price_down + price_up - 2 * price) / (price * 0.01 ** 2)
dollar_convexity = convexity * price * 0.01 ** 2
print("Dollar Convexity: ", dollar_convexity)

Dollar Convexity:  0.69

The convexity adjustment

Convexity can be used to improve bond price prediction
Convexity adjustment = how much bond prices change due to convexity

$ \text{Convexity Adjustment} = 0.5 \times \text{Dollar Convexity} \times 100^2 \times (\Delta y)^2$

Convexity adjustment example

10 year bond with 3% coupon, 5% yield and USD 100 face value
What is its convexity adjustment?

price = -npf.pv(rate=0.05, nper=10, pmt=3, fv=100)
price_up = -npf.pv(rate=0.06, nper=10, pmt=3, fv=100)
price_down = -npf.pv(rate=0.04, nper=10, pmt=3, fv=100)
convexity = (price_down + price_up - 2 * price) / (price * 0.01 ** 2)

dollar_convexity = convexity * price * 0.01 ** 2

convexity_adjustment = 0.5 * dollar_convexity * 100 ** 2 * 0.01 ** 2
print("Convexity Adjustment: ", convexity_adjustment)

Convexity Adjustment:  0.35

Combining duration and convexity

Predicting price changes from duration alone:

$ \text{Price Change} = -100 \times \text{Dollar Duration} \times \Delta y$

Predicting price changes from both duration and convexity:

$ \text{Price Change} = -100 \times \text{Dollar Duration} \times \Delta y \ + \ \text{Convexity Adjustment}$

$ = -100 \times \text{Dollar Duration} \times \Delta y \ + \ 0.5 \times \text{Dollar Convexity} \times 100^2 \times (\Delta y)^2$

Combining duration and convexity improves price estimation

Duration and convexity example

10 year bond, 3% coupon, 5% yield, USD 100 face value:

price = -npf.pv(rate=0.05, nper=10, pmt=3, fv=100)
price_up = -npf.pv(rate=0.06, nper=10, pmt=3, fv=100)
price_down = -npf.pv(rate=0.04, nper=10, pmt=3, fv=100)

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

convexity = (price_down + price_up - 2 * price) / (price * 0.01 ** 2)
dollar_convexity = convexity * price * 0.01
convexity_adjustment = dollar_convexity * 100 ** 2 * 0.01 ** 2

combined_prediction = -100 * dollar_duration * 0.01 + convexity_adjustment
print("Predicted Price Change: ", combined_prediction)

Predicted Price Change:  -6.64

Let's practice!

Bond Valuation and Analysis in Python