What Is an Ordinary Annuity?
An ordinary annuity is a series of equal payments made at the end of regularly spaced periods (monthly, quarterly, semi‑annual, or annual) over a fixed time horizon. Common real‑world examples include bond coupon payments (usually semiannual), many dividend payouts, mortgage payments, and many consumer loan payments. The defining feature is timing: payments occur at the end of each period. When payments occur at the beginning of each period, the stream is an annuity due.
Key takeaways
– Ordinary annuity = equal payments at the end of each period.
– Present and future values depend on the periodic interest rate, the payment amount, and the number of periods.
– An annuity due is worth more than an identical ordinary annuity because payments arrive sooner.
– Ordinary annuities describe timing of cash flows and are distinct from insurance annuity products (although insurance annuities produce similar cash flows).
How an ordinary annuity works
– Parties agree on the payment amount, frequency, and number of payments (term).
– Each payment is made at the end of a period.
– The value of the annuity today (present value, PV) equals the sum of the discounted payments at the prevailing interest rate — i.e., what those future payments are worth in today’s dollars.
– The future value (FV) is the accumulated value of the payments at some future date, if each payment is compounded at the given rate.
Key formulas (periodic rate r, number of periods n, payment per period PMT)
– Present value of an ordinary annuity:
PV = PMT * [1 − (1 + r)^(-n)] / r
– Future value of an ordinary annuity:
FV = PMT * [ (1 + r)^n − 1 ] / r
– Present value of an annuity due (payments at beginning of period):
PV_due = PV_ordinary * (1 + r)
(For annuity due, each payment is effectively shifted one period earlier, increasing PV by one period’s interest.)
Present value example (ordinary annuity)
Suppose an ordinary annuity pays $50,000 annually for 5 years, and the annual discount rate is 7% (r = 0.07, n = 5, PMT = 50,000).
1. Compute the discount factor:
[1 − (1 + 0.07)^(-5)] / 0.07 ≈ 4.10014
2. Multiply by the payment:
PV ≈ 50,000 × 4.10014 ≈ $205,000 (rounded)
If this were an annuity due with the same payments and rate, PV_due ≈ 205,000 × 1.07 ≈ $219,357. Because payments arrive earlier, an annuity due is worth more.
Practical steps — how to evaluate or calculate an ordinary annuity
1. Gather inputs
– PMT: the fixed cash flow per period.
– Term: total number of payments (n).
– Rate: the nominal interest rate and payment frequency (convert to periodic rate r if needed).
2. Convert rates and periods if frequency differs from quoted rate
– If payments are monthly and you have an annual rate APR, r = APR/12 and n = years × 12.
3. Decide which value you need
– Present value (PV): how much the future payments are worth today.
– Future value (FV): how much the stream will accumulate to at a future date.
4. Use the appropriate formula or a financial tool
– Manual formulas (see above).
– Financial calculator: enter n, i (periodic rate), PMT; compute PV or FV.
– Excel/Google Sheets:
– PV: =PV(rate, nper, pmt, [fv], [type]) with type = 0 for ordinary annuity.
– FV: =FV(rate, nper, pmt, [pv], [type]) with type = 0.
5. Interpret results and compare alternatives
– Compare PVs at your required discount rate.
– If offered an annuity due vs ordinary annuity with identical payments, the due is more valuable to the recipient.
– Factor in inflation, taxes, credit risk, liquidity, and fees.
Worked example — future value of an ordinary annuity
If you deposit $1,000 at the end of each year for 10 years into an account that earns 5% annually:
FV = 1,000 * [ (1.05)^10 − 1 ] / 0.05 ≈ 1,000 * 12.578 = $12,578.
When comparing annuity due vs ordinary annuity — which is better?
– For the recipient of the cash flows (the payee), an annuity due is generally preferable because payments arrive earlier and thus have higher PV.
– For the payer (the person making payments), an ordinary annuity is usually preferable since payments are made at the end of each period.
– Which is “better” therefore depends on whether you are receiving or paying and on other factors like liquidity needs and interest rates.
Common types of ordinary annuities
– Mortgage or auto loan payments (most are ordinary annuities).
– Bond coupon payments (often semiannual, paid in arrears).
– Regular dividend distributions paid at period end.
– Other fixed contractual payouts made at period end.
Practical considerations and cautions
– Payment timing matters: be explicit whether payments are at period end (ordinary) or at beginning (due).
– Match the rate and period: use periodic rates consistent with payment frequency.
– Account for inflation and taxes: nominal payments can lose real value; taxes can reduce the effective return.
– Credit/default risk: corporate bond coupons are subject to issuer risk; annuity contracts have counterparty risk.
– Liquidity and surrender features: many insurance annuity products restrict access or charge fees for early withdrawal.
– If using an insurance annuity product (not just the timing concept), read fees, guarantees, mortality credits, and payout options carefully.
Decision checklist for choosing or valuing an annuity stream
1. Identify exact payment schedule (amount, first payment timing, frequency, and total number).
2. Determine discount rate that reflects opportunity cost and risk (after taxes if you’re comparing after‑tax cash flows).
3. Use the correct formula or tool (ordinary vs due) and compute PV/FV.
4. Adjust for taxes, inflation, and fees.
5. Compare alternatives (lump sum now vs annuity stream) using PV; consider liquidity needs and risk tolerance.
6. If considering an insurance annuity product, evaluate guarantees, insurer strength, fees and surrender terms.
Bottom line
An ordinary annuity describes equal payments made at the end of each period and is widely used to model loans, bond coupons, dividends, and other recurring payments. Valuing an ordinary annuity requires three main inputs — payment amount, discount rate (periodic), and number of periods — and can be done with closed‑form formulas, calculators, or spreadsheet functions. Timing matters: an identical annuity due will always be worth more to the recipient because payments are received earlier.
Source
Adapted and summarized from Investopedia — “Ordinary Annuity” (Zoe Hansen): https://www.investopedia.com/terms/o/ordinaryannuity.asp
What Is an Ordinary Annuity?
An ordinary annuity is a sequence of equal payments made at the end of each period for a fixed number of periods. Payments can be monthly, quarterly, semiannually, or annually. Common real‑world examples include bond coupon payments (typically semiannual), many dividend schedules, and most mortgage payments.
The opposite timing is an annuity due, where payments occur at the beginning of each period (rent is a common annuity‑due example). Timing matters because receiving a cash flow earlier increases its present value.
Key features
– Equal payments (level payment amount each period).
– Fixed number of periods (finite term) for ordinary annuities; infinite sequences are treated as perpetuities.
– Payments occur at period end (ordinary annuity) unless specified as an annuity due.
– Valuation depends on the discount rate (the rate of return you could earn elsewhere).
Core formulas and what they mean
Notation
– PMT = payment amount per period
– r = interest rate per period (if annual rate is given and payments are monthly, divide by 12)
– n = total number of payments (periods)
– PV = present value (value today) of the annuity
– FV = future value (value at the end of n periods)
Present value of an ordinary annuity
PV = PMT * [1 − (1 + r)^−n] / r
– Use this to find how much a future stream of end‑of‑period payments is worth today.
Future value of an ordinary annuity
FV = PMT * [(1 + r)^n − 1] / r
– Use this to compute how much a sequence of deposits will accumulate to at a future date.
Annuity due adjustment
– For an annuity due (payments at beginning of period): multiply the ordinary‑annuity PV or FV by (1 + r).
– PV_annuity_due = PV_ordinary * (1 + r)
– FV_annuity_due = FV_ordinary * (1 + r)
Solving for the payment given PV (loan amortization / mortgage)
PMT = PV * [r / (1 − (1 + r)^−n)]
– This is the formula used to calculate level loan payments (ordinary annuity assumption).
Practical steps: valuing or using an ordinary annuity
1. Identify the payment schedule (PMT), timing (end of period?), frequency (monthly/quarterly/annual), and total number of payments (n).
2. Convert the quoted interest rate to the correct per‑period rate (r). If 6% annually and payments are monthly, r = 0.06/12.
3. Choose the appropriate formula depending on whether you need PV, FV, or PMT.
4. Use a spreadsheet (Excel), calculator, or a financial calculator. Excel examples:
– PV: =PV(rate, nper, pmt, [fv], [type]) where type = 0 for ordinary annuity
– FV: =FV(rate, nper, pmt, [pv], [type])
– PMT: =PMT(rate, nper, pv, [fv], [type])
5. Check units and compounding. Match period definitions across rate, n, and PMT.
6. Consider taxes, inflation, credit/default risk, and fees before making decisions based solely on nominal PV/FV.
Worked examples
1) Present value example (ordinary annuity)
Problem: You will receive $50,000 at the end of each year for 5 years. The appropriate discount rate is 7% annually. What is the present value?
Steps:
– PMT = 50,000; r = 0.07; n = 5
– PV = 50,000 * [1 − (1 + 0.07)^−5] / 0.07
– Compute (1.07)^−5 ≈ 0.7130. Then [1 − 0.7130] / 0.07 ≈ 4.1002.
– PV ≈ 50,000 * 4.1002 ≈ $205,010.
Interpretation: Receiving $50,000 at year‑end for five years is worth about $205,010 today at a 7% discount rate.
2) Present value if payments are an annuity due
Same cash flows, but payments arrive at the beginning of each year (annuity due). Multiply the ordinary PV by (1 + r):
– PV_annuity_due = 205,010 * 1.07 ≈ $219,360.
Because payments arrive sooner, the annuity due is worth more.
3) Future value example (regular savings)
Problem: You deposit $200 at the end of each month for 10 years into an account earning 6% annually, compounded monthly. What is the future value?
Steps:
– PMT = 200; monthly r = 0.06/12 = 0.005; n = 10*12 = 120
– FV = 200 * [ (1 + 0.005)^120 − 1 ] / 0.005
– Compute FV using a calculator or spreadsheet; result ≈ $34,974. (Exact value will depend on rounding.)
If you changed to an annuity due (deposit at beginning of each month), multiply FV by (1 + r) → slightly larger.
4) Mortgage payment example (ordinary annuity)
Problem: You borrow $200,000 at 4% annual interest, monthly payments for 30 years. What is the monthly payment?
Steps:
– PV = 200,000; r_monthly = 0.04/12 ≈ 0.0033333; n = 360
– PMT = 200,000 * [r / (1 − (1 + r)^−n)]
– PMT ≈ 200,000 * [0.0033333 / (1 − (1.0033333)^−360)] ≈ $954.83.
This payment schedule is an ordinary annuity because payments are typically made at the end of each month.
Is an ordinary annuity better than an annuity due?
– For the payee (recipient): annuity due is better because payments arrive earlier.
– For the payer: ordinary annuity is better because payments are delayed.
– From a valuation standpoint, an annuity due always has a higher PV than an otherwise identical ordinary annuity (PV difference = PV_ordinary * r).
– Which is “better” depends on your role and preferences (liquidity needs, inflation expectations, reinvestment rates). If you can invest earlier payments at attractive returns, annuity due is superior for the recipient.
Practical decision steps when comparing annuity types
1. Determine your role: Are you the payer or receiver?
2. Compute PV or FV for both timing conventions using the relevant discount/interest rate.
3. Adjust for taxes and inflation: earlier payments reduce inflation risk and provide earlier utility.
4. Consider reinvestment possibilities: if you can reinvest earlier payments at a rate higher than the discount rate, annuity due yields extra benefit.
5. Evaluate legal/contractual features, default risk, and fees.
Common examples and use cases
– Ordinary annuities: mortgage payments, car loans, many bond coupon schedules (coupon paid at period end), certain dividend distributions.
– Annuity due examples: rent (paid at period start), insurance premiums paid at the beginning of coverage, some lease payments.
– Financial planning: fixed pension payouts and structured settlements are often modeled as annuities (timing must be verified).
Limitations, risks, and practical caveats
– Constant payments and constant interest rates are simplifying assumptions; real life often includes variable rates or payment amounts.
– Inflation erodes purchasing power; nominal PVs should be interpreted in nominal terms.
– Taxes and fees reduce net cash flows—always assess after‑tax impacts.
– Credit/default risk matters for private annuity payors—government bonds vs. corporate obligations differ in risk.
– Liquidity: annuity streams can be illiquid; selling them often involves discounts.
Tools that make annuity work easier
– Excel/Google Sheets: PV, FV, PMT, RATE, NPER functions. Remember type=0 (end‑of‑period) for ordinary annuity, type=1 for annuity due.
– Financial calculators (TI BA II Plus, HP 12C): standard TVM keys use end‑of‑period conventions unless you set the calculator to begin mode.
– Online annuity calculators for quick scenarios.
Additional example: calculating PMT for an annuity due
If you need an equal payment at the beginning of each period that equates to a given PV, use:
PMT_annuity_due = PV / {[(1 − (1 + r)^−n) / r] * (1 + r)}
(derived by solving PV_annuity_due = PMT * [(1 − (1 + r)^−n) / r] * (1 + r))
Summary and practical takeaways
– An ordinary annuity pays at the end of each period; it is widely used in loans and many investment cash flows.
– Valuations hinge on the discount rate and payment timing. Earlier payments are worth more due to the time value of money.
– Use PV and FV formulas (or spreadsheet/financial calculator functions) to compare alternatives and make informed decisions.
– Choose the right timing convention when modeling: ordinary annuity (end) vs. annuity due (beginning) has a predictable effect (annuity due PV or FV = ordinary PV or FV * (1 + r)).
– Always adjust for taxes, inflation, and risk before drawing final conclusions.
Sources
– Investopedia: “Ordinary Annuity” (https://www.investopedia.com/terms/o/ordinaryannuity.asp)
– Standard finance formulas and TVM conventions (e.g., Excel PV/FM/PMT functions)
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