01:640:103 Chap.12 Compounding Frequency, APY, Annuities and Loans
Compounding Frequency, APY, Annuities and Loans
Match the rate to the period
Banks quote an annual percentage rate, the APR, but often compound monthly, quarterly or weekly. The rate per period is the APR divided by the number of periods per year, and the number of periods is that count times the years.
A 4.8% APR compounded monthly adds 0.4% each month, not 4.8%, and $2,500 grows to about $2,622.68 in a year rather than $2,620.00 with yearly compounding.
The effective rate
The APY, or annual percentage yield, is the rate of an account compounding once a year that grows money exactly as fast.
For an APR r compounded m times a year, APY = (1 + r/m)^m minus 1. At a fixed APR, more frequent compounding means a higher APY: an 18% APR becomes about 19.56% compounded monthly. Savers want the highest APY, borrowers the lowest, and offers should be compared by APY.
Annuities and loans
An annuity is a series of equal payments at regular intervals.
A savings plan with deposits at the end of each period grows to FV = PMT((1 + i)^n minus 1)/i, where n counts deposits, not the gaps between them. An installment loan is the same idea seen from the start: the amount you can borrow is the sum of the present values of your payments, and solving for PMT gives the payment.
In the guide's example an $18,000 car loan at 6% APR over 60 months costs $347.99 a month.
Savings plans versus single deposits
A single deposit grows by one growth factor per period. An annuity is many deposits, each growing for a different length of time: the first deposit earns interest for n minus 1 periods and the last earns none. The annuity formula adds all of those compound-interest amounts at once.
In the guide's example $200 a month for four years at 6% APR compounded monthly reaches about $10,819.57, of which $9,600 is deposits and the rest interest. Solving the same formula for the payment tells you how much to save each period to reach a target.
Loans work the same way in reverse.
A shorter term raises each payment but lowers the total interest: the same $18,000 at 6% APR costs about $547.59 a month over 36 months, with roughly $1,713 of interest, against $2,879 of interest over 60 months.
What this chapter covers
- 01
Compounding more often than yearly
- 02
APR versus APY
- 03
Fixed deferred annuities
- 04
Counting payments correctly
- 05
Installment loans and monthly payments
Worked example · free
Which savings account pays more?
- 1Account 1: monthly factor 1 + 0.06/12 = 1.005.
- 2APY = 1.005^12 − 1 ≈ 6.168%, just below Account 2's 6.2%.
- 2After a year: $5,308.39 in Account 1 against $5,310.00 in Account 2.
Key terms
- APR
- The APR is the annual percentage rate a lender or bank quotes before compounding is taken into account.
- APY
- The APY is the annual percentage yield, the equivalent rate if interest were compounded once a year.
- Annuity
- An annuity is a run of identical payments spaced evenly in time.
- Installment loan
- An installment loan is borrowed at once and repaid in equal payments at regular intervals.
- Effective rate
- The effective rate is another name for the APY, the true yearly growth after compounding.
- Compounding period
- A compounding period is the interval after which interest is added to the balance, such as a month or a quarter.
- Fixed deferred annuity
- A fixed deferred annuity is a savings plan with equal deposits at the end of each period, each earning compound interest.
Compounding Frequency, APY, Annuities and Loans FAQ
Why is the APY higher than the APR when interest compounds monthly?
Each month's interest is added to the balance and earns interest in later months. Over a year that interest on interest lifts the total growth above the quoted annual rate.
How many deposits are there from age 30 through age 62 with one deposit a year?
Thirty-three. There are thirty-two one-year gaps between the first and last deposit, but a deposit sits at both ends, so the count of payments is one more than the count of gaps.
What changes if a loan is repaid monthly instead of yearly?
The rate per period becomes the APR divided by 12 and the number of payments becomes 12 times the years. The loan formula itself stays the same.
How do I find the monthly payment on a loan?
Use the present value form of the annuity formula with the amount borrowed as the present value, the monthly rate as i and the number of monthly payments as n, then solve for the payment.
Exam move
For every finance question write i and n per period on the first line, decide whether money arrives at the end or is received now, and pick the formula that matches.
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