The Real Math Behind APY vs APR in Crypto Staking

Somewhere between the marketing page and your actual wallet balance, something gets lost. A protocol advertises 18% APY on ETH staking. You deposit 10 ETH and wait a year. At the end, you haven't earned 1.8 ETH. You've earned something close — maybe 1.74 ETH, maybe 1.62 ETH depending on how often rewards are distributed and compounded. That gap isn't fraud. It's math. And most people staking crypto have never seen the formulas that explain it.

This piece goes through the actual mechanics — the conversion between APR and APY, what compounding frequency does to real yields, and why the number on the landing page is almost never the number you'll see in your portfolio.

APR Is a Simple Rate. APY Is What Compounding Does to It.

Annual Percentage Rate (APR) is the base interest rate applied to your principal over a year, with no assumption about reinvestment. If you stake 100 tokens at 12% APR, and rewards are paid out once at year-end and never compounded, you end with 112 tokens. Clean, linear, unsurprising.

Annual Percentage Rate of Yield (APY) layers compounding on top. It answers a different question: if you continuously reinvest your rewards, what is the effective annual growth rate? The formula that converts APR to APY is:

APY = (1 + APR / n)^n − 1

Where n is the number of compounding periods per year. This is not complicated, but the implications compound (pun intended) quickly.

Take 12% APR with monthly compounding (n = 12):

APY = (1 + 0.12 / 12)^12 − 1
    = (1.01)^12 − 1
    = 1.12683 − 1
    = 12.68%

With daily compounding (n = 365):

APY = (1 + 0.12 / 365)^365 − 1
    ≈ 12.747%

With continuous compounding — the theoretical limit — you use Euler's number:

APY = e^(APR) − 1 = e^0.12 − 1 ≈ 12.75%

The difference between daily and continuous compounding at 12% APR is about 0.003%. Negligible. But these formulas are precisely why platforms advertising APY are showing you a better-looking number than the raw APR, and why the conversion direction matters enormously when you're trying to compare products.

The Protocol Is Showing You APY, But Paying You APR (Sometimes)

Here is where things get genuinely tricky. Most DeFi protocols and centralized exchanges advertise APY because it's a higher number — the effect of compounding makes the headline figure larger and more attractive. But the rewards they pay out are calculated on an APR basis, usually distributed at specific intervals.

If a protocol pays staking rewards weekly and shows you 20% APY, what they typically mean is: the underlying APR is such that if you manually compound your weekly rewards every week for 52 weeks, you'd achieve 20% APY. The formula to reverse this — to find the APR underlying an advertised APY with weekly compounding — is:

APR = n × ((1 + APY)^(1/n) − 1)

For 20% APY with n = 52 (weekly payouts):

APR = 52 × ((1.20)^(1/52) − 1)
    = 52 × (1.003513 − 1)
    = 52 × 0.003513
    = 18.27%

So the "20% APY" protocol is actually paying you 18.27% APR in periodic distributions. You only hit 20% if you claim and restake every single week without missing one. Compounding is not automatic in most staking setups — it requires action on your part, gas costs if you're on-chain, or trust in a custodial platform's auto-compound feature.

Gas Costs and the Compounding Frequency Illusion

On Ethereum mainnet, restaking rewards to compound them isn't free. During normal congestion, claiming and restaking might cost $5–15 in gas. During spikes, easily $40+. For small stakers, this creates a real optimization problem.

Say you're staking $2,000 worth of tokens at a protocol advertising 20% APY. Your actual reward per week at 18.27% APR is about $7.03. If gas costs $8 to claim and restake, you're net-negative on every compounding transaction. The "optimal compounding interval" isn't "as often as possible" — it's the frequency where the marginal yield from compounding exceeds the gas cost.

The math for optimal compounding frequency with gas cost G, principal P, and APR r is approximate but useful:

Optimal intervals per year ≈ √(r × P / (2 × G))

For P = $2,000, r = 0.1827, G = $10:

≈ √(0.1827 × 2000 / 20)
≈ √(18.27)
≈ 4.27

Roughly quarterly compounding is optimal given these parameters. Weekly compounding would cost more in gas than the extra yield earned from more frequent reinvestment. This is a real calculation most small stakers never make, and platforms never tell you about it because it reveals the gap between advertised and achievable returns.

Validator Uptime, Penalties, and Effective Yield Reduction

In Proof-of-Stake systems like Ethereum, there's a third factor that further separates advertised yield from actual yield: validator performance. Ethereum's staking rewards aren't fixed — they depend on validator uptime, attestation inclusion rate, and whether any slashing events occur.

The base reward for an Ethereum validator is proportional to the inverse square root of total ETH staked on the network. As more ETH enters staking, yields compress. The formula is complex and varies per epoch, but the directional point is important: the APY Lido or similar protocols advertise is a trailing average, not a guaranteed forward rate.

A validator offline for 2% of the year doesn't just miss 2% of rewards — it also accrues inactivity penalties that further reduce net yield. Empirically, a validator with 98% uptime sees effective yield roughly 2.5–3% below what perfect uptime would produce, not just 2%.

When you stake via a liquid staking protocol, these penalties are socialized across all depositors — you share in their dilution. When you run your own validator, the risk (and occasionally the benefit of above-average performance) is yours alone.

The Liquidity Premium That Isn't Always There

One more layer: lockup periods. Some protocols offer higher APY in exchange for locked staking — your assets are illiquid for 30, 90, or 180 days. The premium for this illiquidity is often framed as a pure yield improvement, but it has an opportunity cost that the APY figure doesn't capture.

If you lock 10 ETH for 90 days at 22% APY versus liquid staking at 18% APY, the gross math favors the lock-up. But if ETH's price drops 15% during those 90 days, you've missed the window to rotate into a different position or simply buy more at lower prices. The locked yield's true cost includes the optionality you surrendered.

Crypto-native investors often use a hurdle rate comparison: the yield premium over liquid staking should exceed the annualized value of being able to react to market events. That's not a formula you'll find on any protocol's documentation, but it's the honest framework for evaluating locked versus liquid staking choices.

Converting Between APR and APY — Quick Reference

To close with something genuinely useful, here are the three core conversions you'll use repeatedly:

APR to APY (with n compounding periods):

APY = (1 + APR / n)^n − 1

APY to APR (to find the underlying rate):

APR = n × ((1 + APY)^(1/n) − 1)

Effective yield after m compounding events at APR r, per-period gas cost G, principal P:

Effective Yield = P × (1 + r/m)^m − P − (m × G)

Divide by P to get effective rate. This third formula is the one to internalize. It converts the advertised number into a realistic personal outcome given your position size and the gas environment you're operating in.

What to Actually Look For When Evaluating Staking Yields

The discipline here isn't complex but it requires intentionality. Before depositing into any staking protocol: identify whether the advertised rate is APR or APY (if it's not explicitly labeled, assume APY because that's the more attractive number and that's why platforms use it); determine the reward distribution frequency and whether compounding is automatic or manual; estimate your position size against gas costs to find your true optimal compounding interval; check whether the rate is fixed or variable and what the trailing 30/60/90-day average looks like; and understand any lock-up terms and price-in the illiquidity cost honestly.

The advertised yield is the ceiling. Gas costs, imperfect compounding, validator performance variance, and lockup opportunity costs all work as floors. Your actual yield sits somewhere in between — and the gap is determined by math that platforms have no incentive to walk you through.

Now you've seen the math. The gap should make more sense.