Daily Compounding Interest and Effective Annual Rate
Jim invested 5800 in an account that pays an annual interest rate of 2.6%, compounded daily. Assume there are 365 days in each year. Answer each part. (a) Find the amount in the account after one year, assuming no withdrawals are made. Do not round any intermediate computations, and round your answer to the nearest cent. (b) Find the effective annual interest rate, expressed as a percentage. Do not round any intermediate computations, and round your answer to the nearest hundredth of a percent.
What This Problem Teaches
- How to apply the compound interest formula with high-frequency compounding
- The distinction between nominal and effective interest rates
- Why more frequent compounding increases returns (and by how much)
- Precision in financial calculations and appropriate rounding techniques
- How banks structure interest-bearing accounts in real practice
Solution: Method 1 — Direct Formula Application
Step 1 — Identify the compound interest formula components
For compound interest, we use A = P(1 + r/n)^(nt) where:
P = $5800(principal)r = 0.026(annual rate as decimal)n = 365(compounding frequency per year)t = 1(time in years)
Step 2 — Calculate the amount after one year
Substitute into the formula:
A = 5800(1 + 0.00007123287671...)^365
A = 5800(1.00007123287671...)^365
Step 3 — Compute the compound factor
Calculate (1.00007123287671...)^365:
Step 4 — Find the final amount
Multiply by the principal:
A = 5952.57653224...
A = $5952.58 (rounded to nearest cent)
Step 5 — Calculate the effective annual interest rate
The effective rate formula is (1 + r/n)^n - 1:
Effective rate = 1.02627543659... - 1
Effective rate = 0.02627543659...
Effective rate = 2.63% (rounded to nearest hundredth)
Solution: Method 2 — Step-by-Step Daily Rate Analysis
Step 1 — Calculate the daily interest rate
Divide the annual rate by the number of days:
Step 2 — Express as a daily multiplier
Each day, the account balance gets multiplied by:
Step 3 — Apply compounding for 365 days
After 365 days, the original principal has been multiplied by this factor 365 times:
Step 4 — Calculate final balance
Apply to Jim's initial investment:
Step 5 — Determine the effective annual rate
The effective rate is the total multiplier minus 1:
Part (b): The effective annual interest rate is 2.63%
Verification
Let's check our work using an independent calculation approach:
Check Part (a)
Using the exact compound interest formula with full precision:
= 5800 × (1.00007123287671232877)^365
= 5800 × 1.02627543659043836
= $5952.57653224...
Rounded to the nearest cent: $5952.58 ✓
Check Part (b)
Verify the effective rate calculation:
Effective rate = $152.58 ÷ $5800 = 0.026306896... = 2.63%
This matches our formula result ✓
Sanity check
The effective rate (2.63%) should be slightly higher than the nominal rate (2.60%) due to compounding. The difference of 0.03 percentage points is reasonable for daily compounding at this rate level.
Common Pitfalls
This treats 2.6% as the daily rate instead of the annual rate, leading to astronomical growth. Always divide the annual rate by the number of compounding periods per year.
This ignores the compounding effect entirely. The effective rate must account for earning interest on previously earned interest.
Then: (1.000071)^365 = 1.02625...
Leading to: A = 5800 × 1.02625 = $5952.25
Premature rounding introduces error that compounds over 365 calculations. Keep full precision until the final step, especially in financial calculations where accuracy matters.
The Underlying Pattern
This problem demonstrates the general compound interest formula:
And the effective annual rate formula:
Key insights:
- As compounding frequency increases, the effective rate approaches
e^r - 1(continuous compounding) - For daily compounding at 2.6%, we get nearly the maximum possible effective rate for this nominal rate
- The "boost" from daily vs. annual compounding is modest: 2.63% vs. 2.60%
- Higher nominal rates see proportionally larger compounding effects
For any nominal rate r with daily compounding, the effective rate is always (1 + r/365)^365 - 1. This formula works for any time period by adjusting the exponent accordingly.
Real Applications
- Banking: Most savings accounts and CDs use daily compounding. Understanding the effective rate helps you compare accounts that advertise different nominal rates and compounding frequencies.
- Credit cards: Many cards compound interest daily on outstanding balances. A 24% APR becomes an effective rate of about 27.1% with daily compounding—significantly higher than the advertised rate.
- Investment planning: When projecting long-term growth, using effective rates gives more accurate forecasts than nominal rates, especially for frequently compounded accounts.
What If?
With monthly compounding: P = $5800, r = 0.026, n = 12, t = 1
A = 5800(1 + 0.026/12)^(12×1) = 5800(1.002166667)^12 = 5800 × 1.026196 = $5951.94
Effective rate = (1 + 0.026/12)^12 - 1 = 1.026196 - 1 = 2.62%
Monthly: $5951.94 and 2.62%
Daily: $5952.58 and 2.63%
Difference: Daily compounding earns $0.64 more
We want A = $6000, so: 6000 = P(1 + 0.026/365)^365
(1 + 0.026/365)^365 = 1.02627543659...
P = $6000 ÷ 1.02627543659 = $5846.65
$5846.65 × 1.02627543659 = $6000.00 ✓
Jim should invest $5846.65 today
A = 5800(1 + 0.026/365)^(365×3) = 5800(1.00007123...)^1095
(1.00007123...)^1095 = 1.08136294...
A = 5800 × 1.08136294 = $6271.85
Total return: ($6271.85 - $5800) ÷ $5800 = 8.14%
Amount: $6271.85
3-year effective rate: 8.14%
For continuous compounding: A = Pe^(rt) where e ≈ 2.71828
e^(0.026×1) = e^0.026 = 1.02635971...
A = $5800 × 1.02635971 = $5952.89
Continuous: $5952.89
Daily: $5952.58
Continuous earns $0.31 more, showing daily compounding is very close to the theoretical maximum.
Frequently Asked Questions
Nominal rate is the stated annual percentage, while effective rate is what you actually earn when compounding is factored in. With daily compounding at 2.6% nominal, you earn more than 2.6% because interest compounds 365 times per year. In this problem, the effective rate is 2.63%, slightly higher than the 2.6% nominal rate.
Use the formula A = P(1 + r/n)^(nt), where P is principal, r is annual rate as a decimal, n is compounding frequency per year, and t is time in years. For daily compounding, n = 365. In this example, A = 5800(1 + 0.026/365)^(365×1) = $5952.58.
Each compounding period adds interest to your principal, and then you earn interest on that larger amount in the next period. Daily compounding means you earn interest on interest 365 times per year instead of just once. The effect is small but measurable—here, daily compounding earns about $0.37 more than simple annual compounding would.
2026-08-13