Compound interest and non-annual periods

Starting investment

\(\displaystyle \$860\)

Annual rate

\(\displaystyle 5\%\)

First year's interest

\(\displaystyle \$43\)

Interest is added at the end of each year. There are no other payments or fees. How much interest is earned in the second year?

Both accounts start with $1,280 at an annual rate of 2.5%. There are no other payments or fees. Move through all three years. Compare the interest added in the second and third years. The compound account adds interest annually

From the second year to the third year, simple interest stays the same. How much does the compound account's interest for the year increase?

Repeat the annual multiplier

For example, start with $1,280 at an annual rate of 2.5%. The principal is the starting amount. Annual compounding adds each year's interest to the balance before the next year's interest is calculated. With a constant annual rate and no other payments or fees, apply the same multiplier once for each complete year. Keep calculator precision until the final answer

1. 1Define the quantities P is the principal, A is the final balance and I is the total interest, all in dollars. The annual percentage number is r, and n counts complete years. The annual multiplier is q
2. 2Keep the balance and add the interest The multiplier includes the whole opening balance as well as the percentage increase \(\displaystyle \begin{aligned}q&=1+\frac{r}{100}\\&=1+\frac{2.5}{100}\\&=1.025\end{aligned}\)
3. 3Use the new balance each year Each closing balance becomes the next year's opening balance \(\displaystyle \begin{aligned}A_1&=1280\times1.025\\&=1312\\A_2&=1312\times1.025\\&=1344.80\\A_3&=1344.80\times1.025\\&=1378.42\end{aligned}\)
4. 4Write repeated multiplication as a power The original principal has been multiplied by the annual multiplier three times. The exponent records those three yearly updates \(\displaystyle A_3=1280(1.025)^3\)
5. 5Separate the balance and total interest The final balance includes the principal. Subtract the principal when the question asks for total interest \(\displaystyle \begin{gathered}A=Pq^n\\I=A-P\end{gathered}\)

Principal

\(\displaystyle \$1{,}460\)

Annual rate

\(\displaystyle 3.5\%\)

Term

\(\displaystyle 3\text{ years}\)

Interest is compounded annually. There are no other payments or fees. Calculate the final balance to the nearest cent

Apply the annual multiplier once for each complete year. Round only the final balance

Principal

\(\displaystyle \$975\)

Annual rate

\(\displaystyle 4.5\%\)

Term

\(\displaystyle 4\text{ years}\)

Interest is compounded annually. There are no other payments or fees. Calculate the total interest to the nearest cent

Find the unrounded final balance first, then subtract the principal

Match the rate to the period

The model keeps a fixed nominal annual rate, with no extra deposits, withdrawals, repayments or fees. Interest remains in the account. A nominal annual rate is allocated equally across the stated compounding periods; it is not an effective annual growth rate

1. 1Define the symbols P is the principal and A the final amount, both in dollars. R is the nominal annual percentage number, k the periods per year, t the years, r the percentage number per period and n the total period count
2. 2Convert both quantities Annual means k=1, half-yearly k=2, quarterly k=4 and monthly k=12. Divide the nominal annual percentage number by k and multiply the years by k. \(\displaystyle \begin{gathered}r=R/k\\n=kt\end{gathered}\)
3. 3Keep the original whole Use the percentage number r without a percent sign inside r/100. The multiplier keeps the balance and adds the period interest \(\displaystyle q=1+r/100\)
4. 4Apply each update Use the unrounded balance from the previous period. The exponent counts all the updates. Keep calculator precision and round only the requested answer \(\displaystyle \begin{gathered}B_0=P\\B_{j+1}=qB_j\\A=Pq^n\\I=A-P\end{gathered}\)

Nominal annual rate

\(\displaystyle 7.2\%\)

Periods per year

\(\displaystyle 4\)

Find the percentage rate for one quarter. Use the nominal annual rate shown

Divide the nominal annual percentage number among the quarters

Term

\(\displaystyle 2.5\text{ years}\)

Periods per year

\(\displaystyle 12\)

How many monthly compounding periods occur over this term?

Count every monthly update across the whole term

Predict: two accounts start with $2,000 at the same 6% nominal annual rate for two years. No payments or fees occur. One compounds annually and the other quarterly. Which should finish higher?

Keep $2,000, the 6% nominal annual rate and two-year term fixed. Try 1, 2, 4 and 12 periods per year for annual, half-yearly, quarterly and monthly compounding. Other settings divide the year into k equal periods. Rates and balances are rounded only for display

Which observation fits this fixed-rate, no-fee model?

Principal

\(\displaystyle \$2{,}400\)

Nominal annual rate

\(\displaystyle 6\%\)

Compounding

\(\displaystyle \text{quarterly}\)

Term

\(\displaystyle 2\text{ years}\)

Calculate the final balance, rounding only the answer to cents. The nominal rate is fixed and there are no payments or fees

Convert both the rate and period count before raising the multiplier

Principal

\(\displaystyle \$1{,}600\)

Nominal annual rate

\(\displaystyle 7.2\%\)

Compounding

\(\displaystyle \text{monthly}\)

Term

\(\displaystyle 1\text{ year}\)

Find total interest to cents. The nominal rate is fixed, interest remains invested and there are no other payments or fees

Calculate the full-precision final balance before removing the principal

A balance table is repeated multiplication

A fictional $750 account compounds half-yearly at 6% nominal p.a. with no payments or fees. Each half-year uses 3%, so the multiplier is 1.03. Keep the full balance for the next step; a displayed cent value is only a summary

1. 1Start with the principal The opening row is period zero \(\displaystyle B_0=750\)
2. 2First update Apply the half-yearly multiplier once \(\displaystyle B_1=750(1.03)=772.50\)
3. 3Second update The next opening balance includes the earlier interest. The unrounded result is retained even though it is displayed to cents \(\displaystyle \begin{gathered}B_2=772.50(1.03)\\=795.675\\\approx\$795.68\end{gathered}\)
4. 4Third update Use 795.675, not the displayed 795.68. A power of the multiplier reproduces the same sequence \(\displaystyle \begin{gathered}B_3=750(1.03)^3\\=819.54525\\\approx\$819.55\end{gathered}\)

Principal

\(\displaystyle \$900\)

Nominal annual rate

\(\displaystyle 8\%\)

Compounding

\(\displaystyle \text{quarterly}\)

Find the balance after the third quarter to cents. Keep full precision between updates; no other payments or fees occur

Apply the quarter multiplier three times from the principal

Principal

\(\displaystyle \$1{,}250\)

Nominal annual rate

\(\displaystyle 8\%\)

Compounding

\(\displaystyle \text{half-yearly}\)

Term

\(\displaystyle 2\text{ years}\)

Find the final amount after the complete term to cents. The rate is nominal, with no payments or fees

Count two half-yearly periods in each year and match the period rate

A student models $1,500 at 8% nominal p.a., compounded quarterly for three years. There are no other payments or fees. Which line first breaks the model?

An account advertises 6% effective annual growth. Can we obtain its monthly percentage rate just by dividing six by twelve?

Test a model and report its limits

Investigate a fictional $2,000 deposit at 6% nominal p.a. over two years. Compare annual, quarterly and monthly compounding. Predict the order, build a table of full-precision updates or use the multiplier formula, and report the final amounts and interest to cents. Explain why the nominal rate can stay fixed while the effective growth changes

1. 1Choose and state assumptions Keep the rate fixed, retain interest and exclude payments and fees. State the compounding frequency and the final-only rounding rule
2. 2Evaluate and modify Next suppose a $20 fee is deducted at the end of each year. Modify the year-end update before carrying that balance into year two. Compare results and explain why subtracting $40 only at the end is not the same model
3. 3Check plausibility Interest is positive in the fee-free model. Under the same nominal rate, more frequent compounding gives a slightly higher balance. This does not establish which real product is best when fees, rates or access conditions differ