APRとAPYの違い（APR vs APY）

単利と複利の違い。

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In DeFi and banking, understanding APR vs APY is crucial for calculating returns.

APR (Simple Interest): Does not account for reinvesting yields. If you have 100% APR, you earn the same amount each day based on the principal.
APY (Compound Interest): Assumes yields are reinvested periodically. 100% APY is achieved with a lower daily rate because the 'interest earns interest'.

Formula:
- APR = Periodic Rate × Number of Periods per Year
- APY = (1 + Periodic Rate)^Periods - 1

Borrowers prefer lower APR (paying less interest), while lenders/stakers prefer higher APY (earning more).

        graph LR
  Center["APRとAPYの違い（APR vs APY）"]:::main
  Rel_staking["staking"]:::related -.-> Center
  click Rel_staking "/terms/staking"
  Rel_yield_farming["yield-farming"]:::related -.-> Center
  click Rel_yield_farming "/terms/yield-farming"
  Rel_defi["defi"]:::related -.-> Center
  click Rel_defi "/terms/defi"
  classDef main fill:#7c3aed,stroke:#8b5cf6,stroke-width:2px,color:white,font-weight:bold,rx:5,ry:5;
  classDef pre fill:#0f172a,stroke:#3b82f6,color:#94a3b8,rx:5,ry:5;
  classDef child fill:#0f172a,stroke:#10b981,color:#94a3b8,rx:5,ry:5;
  classDef related fill:#0f172a,stroke:#8b5cf6,stroke-dasharray: 5 5,color:#94a3b8,rx:5,ry:5;
  linkStyle default stroke:#4b5563,stroke-width:2px;

🕸️ Open in Universe

🧒 5歳でもわかるように説明

APRは「そのままの金利」で、APYは「利息にさらに利息がつく」計算方法です。暗号資産の運用では、利益を再投資するAPYの方が最終的にもらえるお金が多くなります。

🤓 Expert Deep Dive

## Technical Analysis: APR vs. APY

This analysis provides a technically rigorous examination of Annual Percentage Rate (APR) and Annual Percentage Yield (APY), highlighting their mathematical underpinnings, practical implications, and advanced considerations, particularly within decentralized finance (DeFi) and traditional banking contexts.

### Missing Technical Nuances

Compounding Frequency Impact: The divergence between APR and APY is directly proportional to compounding frequency. A higher frequency (e.g., daily vs. monthly) amplifies the "interest on interest" effect. Mathematically, APY = (1 + Periodic Rate)^n - 1, where 'n' is the number of compounding periods per year. As 'n' increases, APY grows disproportionately faster than APR for a fixed periodic rate.
Mathematical Derivation of APY: The APY formula models a geometric progression where each period's earnings are added to the principal. Subsequent interest is calculated on this augmented principal, leading to exponential growth. This iterative reinvestment is the core of compound interest.
Real-World Applications & Edge Cases:
Variable Rates: In dynamic environments like DeFi, rates fluctuate based on market conditions (e.g., lending demand/supply). Accurate APY representation requires predictive modeling or real-time updates.
Fees and Deductions: Protocol fees, gas costs, and withdrawal charges directly reduce net returns, necessitating a "net APY" calculation distinct from gross APY.
Continuous Compounding: The theoretical upper bound of compounding is continuous compounding, represented by the limit of (1 + 1/n)^n as n approaches infinity. This yields an APY of e - 1 (approx. 71.828%), where 'e' is Euler's number.
Underlying Mechanisms: DeFi APY calculations are often derived programmatically within smart contracts. For lending protocols, this is based on the ratio of interest generated to total assets. For staking, it's block rewards distributed per staked token. [Liquidity pools](/ja/terms/liquidity-pools) factor in trading fees and emission schedules.

### Improved ELI5 Analogy

Original Analogy: APR is the interest you get if you withdraw your profit immediately. APY is the interest you get if you leave your profit in the account so it can earn even more money (snowball effect).

Improved Analogy: Imagine you have a tree. APR is like the fixed number of apples the tree produces each year. APY is like planting the seeds from those apples. The next year, your tree is bigger and produces even more apples because the apples you harvested (your profit) helped grow the tree itself. It's a "profit-generating-profit" effect.

### Key Expert Concepts

Mathematics of Compound Interest: Detailed derivation of APY = (1 + Periodic Rate)^n - 1 and its relation to APR. Discussion of the limit as n approaches infinity.
Time Value of Money (TVM): Explicitly connecting APY to TVM, explaining how reinvesting earnings leverages opportunity cost and future growth potential.
Financial Instrument Mechanics: Analysis of APR/APY calculation in DeFi: lending/borrowing (supply/demand, utilization), staking (block rewards, total stake), and liquidity pools (trading fees, emissions). On-chain vs. off-chain calculations.
Risk-Adjusted Returns & Impermanent Loss: Understanding that higher APY can correlate with higher risk. Comparing APY against potential impermanent loss for liquidity providers.
Economic Incentives: How APR/APY are used to attract liquidity, encourage participation, and influence user actions through yield generation.
* Variable Rate Dynamics: Technical challenges in representing APY for variable rates, including the use of historical data, predictive models, or real-time display.

🧒 5歳でもわかるように説明

🤓 Expert Deep Dive

📚 出典