Advanced Concepts: Continuous Compounding
Understanding continuously compounded yields and their applications in JGB analysis
Why Euler's Number (e) Appears in Finance
You may have noticed e-rt in the continuous compounding formula. Euler's number (e ≈ 2.71828) is fundamental to finance because it represents the mathematical limit of compounding.
Simple Example:
If you invest ¥100 at 10% annual interest, what happens as you compound more frequently?
- Annual compounding: ¥100 × (1 + 0.10)1 = ¥110.00
- Semi-annual: ¥100 × (1 + 0.05)2 = ¥110.25
- Quarterly: ¥100 × (1 + 0.025)4 = ¥110.38
- Daily: ¥100 × (1 + 0.10/365)365 = ¥110.516
- Continuous: ¥100 × e0.10 = ¥110.517...
As compounding frequency approaches infinity, the result converges to ert. This is why derivatives pricing (which assumes continuous trading) uses e extensively. For JGBs with discrete semi-annual coupons, we use the discrete formula, but understanding e is essential for advanced bond math and derivatives.