Bond Convexity Calculator
From the price at an initial yield and at yields up/down, estimate the bond's convexity — the curvature of the price-yield curve.
輸入資料
計算結果
重點速覽：Convexity measures the curvature of the price-yield relation. Estimate: convexity = (P_down + P_up - 2 x P0) / (P0 x (Δy)^2), where P0 is the price at the initial yield and P_down/P_up are prices if the yield moves down/up by Δy. Higher convexity is good for holders: price rises more when yields fall and falls less when yields rise. It refines the duration (straight-line) estimate for small yield changes. WARNING: It is an approximation for small moves; it complements, not replaces, duration. Education only, not advice.
計算公式
凸性 = (P− + P+ − 2 × P0) ÷ (P0 × Δy²)。
其中 P0=初始價格、P−=殖利率下降後價格、P+=殖利率上升後價格、Δy=殖利率變動幅度 (小數)。
$$\text{Convexity} = \dfrac{P_- + P_+ - 2P_0}{P_0 \times (\Delta y)^2}$$$$\dfrac{\Delta P}{P} \approx -D \times \Delta y + \tfrac{1}{2}\,\text{Convexity} \times (\Delta y)^2$$使用說明
- Enter the bond price at the initial yield (P0).
- Enter the estimated prices if the yield moves down and up by the same amount.
- Enter the size of the yield change (Δy).
- View the convexity.
初始價格 P0 = HK$1,000、殖利率變動 ±0.5% 時，不同重估價格對應的凸性
| 殖利率下降後 (P−) | 殖利率上升後 (P+) | Δy | 凸性 | 解讀 |
|---|---|---|---|---|
| HK$1,015 | HK$986 | ±0.5% | 40 | 凸性偏低，價格反應接近直線 |
| HK$1,025 | HK$977 | ±0.5% | 80 | 中等凸性 |
| HK$1,032 | HK$971 | ±0.5% | 120 | 本工具預設值，凸性明顯 |
| HK$1,050 | HK$955 | ±0.5% | 200 | 高凸性，曲率大、價格保護強 |
理財情境案例
案例一：從重估價格算凸性
某債券初始價格 P0 = HK$1,000。當殖利率下降 0.5% 時價格升至 P− = HK$1,032；上升 0.5% 時跌至 P+ = HK$971，Δy = 0.005。
凸性 = (1,032 + 971 − 2 × 1,000) ÷ (1,000 × 0.005²) = 3 ÷ 0.025 = 120。此凸性可搭配存續期一起估算大幅利率變動下的價格反應。
案例二：凸性修正如何影響價格估算
延續上例，假設存續期 D = 6.1、凸性 = 120。若殖利率上升 1% (Δy = +0.01)，價格變動 ≈ −6.1 × 0.01 + 0.5 × 120 × 0.01² = −0.055，即約下跌 5.5%，比單看存續期的 6.1% 為佳。
若殖利率下降 1% (Δy = −0.01)，價格變動 ≈ 6.1 × 0.01 + 0.5 × 120 × 0.0001 = 0.067，即約上漲 6.7%。可見凸性修正讓下跌變小、上漲變大，這正是正凸性對持有人有利的體現。
常見問題
What does convexity mean simply?
Duration estimates price change as a straight line; convexity is the curvature. Positive convexity means the actual price falls less when yields rise and rises more when yields fall than duration alone predicts — good for holders. Most plain bonds have positive convexity.
How does convexity correct the duration estimate?
Duration alone over-estimates the fall when yields rise and under-estimates the rise when yields fall. Adding the convexity term (≈ 0.5 x convexity x (Δy)^2) corrects this. Example: duration 6.1, convexity 120, yield +1% → change ≈ -6.1 x 0.01 + 0.5 x 120 x 0.0001 = -5.5% (vs -6.1% by duration only); yield -1% → ≈ +6.7%.
Why do some bonds have negative convexity?
Bonds with embedded options, such as callable bonds (issuer can redeem early when rates fall) or MBS/prepayable bonds, can show negative convexity in part of the range — price gains are capped when yields fall. So convexity sign matters; check for call features.
Does convexity only matter for big yield moves?
The correction is small for tiny moves and grows with (Δy)^2. For a 0.1% move it is negligible; for a 1-2% move it is material. For highly curved bonds (long maturities, low coupons), even moderate moves need the convexity adjustment.
How does this relate to Hong Kong bonds?
Hong Kong dollar bonds (Exchange Fund Notes, bank issues, blue-chip bonds) are priced off HIBOR/US rates; longer maturities and lower coupons have higher convexity and bigger price swings. Use duration + convexity together to gauge interest-rate risk, and consult the HKMA and IFEC materials. This tool is educational, not advice.
相關工具
參考資料
內容審核：香港計算器財經團隊。計算邏輯與公式參考香港金融管理局（HKMA）及投資者及理財教育委員會（IFEC）之個人理財計算指引，結果僅供參考，實際以相關機構公佈為準。