📖 年金：普通年金 vs 预付年金

📌 Topic: The Two Faces of Annuities — Beginning vs End, One (1+r) Apart

I. From Single to Multiple Cash Flows: Why Annuities?

L089 and L090 covered PV/FV calculations for single cash flows — one payment, one starting point, one ending point.

In the real world, most scenarios involve a stream of equal payments: - Monthly mortgage payments - Annual insurance premiums - Monthly pension payouts - Semi-annual bond coupon payments

These series of equal, equally-spaced cash flows are called annuities.

🔑 The essence of an annuity: treat multiple cash flows as a "bundle" and compute PV or FV in one shot — no need to discount each payment individually.

II. The Core Distinction Between the Two Types

2.1 Ordinary Annuity

Payments occur at the END of each period

Timeline (5-year ordinary annuity, PMT = 1,000 each):

t=0      t=1      t=2      t=3      t=4      t=5
|--------|--------|--------|--------|--------|
        PMT      PMT      PMT      PMT      PMT
       (end)    (end)    (end)    (end)    (end)

Typical applications: - Loan repayments (you borrow, repayments start at the end of the first period) - Bond coupons (paid at the end of each period) - Salaries (paid at month-end)

2.2 Annuity Due

Payments occur at the BEGINNING of each period

Timeline (same 5-year span, PMT = 1,000 each):

t=0      t=1      t=2      t=3      t=4      t=5
|--------|--------|--------|--------|--------|
PMT      PMT      PMT      PMT      PMT
(beg)    (beg)    (beg)    (beg)    (beg)

Typical applications: - Rent (paid at the start of the month) - Insurance premiums (paid at the start of the period) - Leases (pay first, use later)

III. The Key Relationship: Only One (1+r) Apart

This is one of the most frequently tested concepts on CFA Level I:

3.1 Why (1+r)?

Intuition (using PV as an example):

In an ordinary annuity, each PMT occurs at period-end. In an annuity due, each PMT occurs at period-beginning.

Each payment in an annuity due is one period earlier than its counterpart in an ordinary annuity → discounted one fewer period → larger present value.

Consider a 5-payment annuity: - Ordinary: first payment at end of period 1 → discounted 1 period; fifth payment at end of period 5 → discounted 5 periods - Annuity Due: first payment at t=0 (immediate) → no discounting; fifth payment at end of period 4 → discounted 4 periods

Every payment is discounted one period less → the entire PV is multiplied by (1+r).

🧠 Visual trick: Shift the ordinary annuity timeline one space to the left, and you get an annuity due. After the shift, each PMT is one period closer to t=0 → PV increases by a factor of (1+r).

IV. Ordinary Annuity PV Formula (The Foundation)

$$\boxed{PV_{\text{ordinary}} = PMT \times \frac{1 - (1+r)^{-n}}{r}}$$

$$= PMT \times PVIFA(r, n)$$

• PVIFA = PV Interest Factor of Annuity
• Meaning: the total present value (at t=0) of n payments of $1 each, discounted at rate r

Calculation Example

Receive $10,000 at the end of each year for 5 years, discount rate 6%. What is the present value of these cash flows?

$$PV = 10{,}000 \times \frac{1 - (1.06)^{-5}}{0.06}$$

$$= 10{,}000 \times \frac{1 - 0.7473}{0.06}$$

$$= 10{,}000 \times \frac{0.2527}{0.06}$$

$$= 10{,}000 \times 4.2124 = 42{,}124$$

Answer: $42,124

V. Deriving Annuity Due from Ordinary Annuity

5.1 Annuity Due PV

Using the same example, but with payments at the beginning of each year:

$$PV_{\text{due}} = PV_{\text{ordinary}} \times (1 + r)$$

$$= 42{,}124 \times 1.06 = 44{,}651$$

💡 Receiving money at the beginning of each year is worth $2,527 more (≈6%) — because each of the 5 payments is discounted one year less.

5.2 Annuity Due FV

First compute ordinary annuity FV, then multiply by (1+r):

$$\boxed{FV_{\text{ordinary}} = PMT \times \frac{(1+r)^n - 1}{r}}$$

$$\boxed{FV_{\text{due}} = FV_{\text{ordinary}} \times (1+r)}$$

Example: Deposit $10,000 at the beginning of each year, interest rate 6%, for 5 years. Future value?

First, ordinary annuity FV (assuming end-of-year deposits):

$$FV_{\text{ord}} = 10{,}000 \times \frac{(1.06)^5 - 1}{0.06}$$

$$= 10{,}000 \times \frac{1.3382 - 1}{0.06}$$

$$= 10{,}000 \times 5.6371 = 56{,}371$$

Then adjust for annuity due:

$$FV_{\text{due}} = 56{,}371 \times 1.06 = 59{,}753$$

Answer: $59,753 — because deposits at the start of each year earn one extra year of compounding.

VI. Quick Comparison Table

VII. TI BA II Plus Key Operations

Switching Annuity Mode (HIGH-RISK EXAM TRAP!)

⚠️ After finishing an annuity due problem, immediately switch back to END mode! The next question is likely an ordinary annuity. Forgetting to turn BGN off = every subsequent answer wrong.

Compute Ordinary Annuity PV (Example: PMT=10,000, n=5, I/Y=6, find PV)

Compute Annuity Due PV (same parameters, only mode differs)

VIII. Common Exam Pitfalls

Trap 1: BGN Mode Left On 🔴🔴🔴

The most common fatal mistake on exam day. After solving an annuity due problem, forget to switch back to END — all subsequent ordinary annuity answers are wrong. ✅ After every annuity due problem, immediately press 2nd BGN → 2nd SET to return to END.

Trap 2: Missing "Beginning/End" Cues in the Question

• "at the beginning of each year" → Annuity Due
• "at the end of each year" → Ordinary Annuity
• "first payment today" / "immediate" → Annuity Due
• "first payment in one year" / "starting next year" → Ordinary Annuity

Trap 3: Using PV_due as Ordinary PV in Subsequent Calculations

If the question asks "monthly payments at the beginning, what is the total present value?" — using END mode will understate the result. You must either turn BGN on, or compute in END mode and multiply by (1+r).

Trap 4: Calculator Output ≠ Manual Formula

PV computed in BGN mode should match PV computed in END mode multiplied by (1+r). If they differ → check whether you're in the correct mode.

IX. Practice Examples

📝 Example 1: Identifying Annuity Type

Classify each scenario as ordinary annuity or annuity due:

📝 Example 2: Ordinary Annuity PV

You win a prize: $50,000 paid at the end of each year for 20 years. Discount rate is 5%. What is the present value?

Solution:

$$PV = 50{,}000 \times \frac{1 - (1.05)^{-20}}{0.05}$$

$$= 50{,}000 \times \frac{1 - 0.3769}{0.05}$$

$$= 50{,}000 \times 12.4622 = 623{,}110$$

Answer: ≈ $623,110

💡 Although total payments over 20 years equal $1,000,000, the present value today is only about $623K due to the time value of money.

📝 Example 3: Annuity Due PV (Same Problem, Beginning-of-Year Payments)

Same scenario, but payments at the beginning of each year. What is the PV now?

Solution:

$$PV_{\text{due}} = 623{,}110 \times 1.05 = 654{,}266$$

Answer: ≈ $654,266 — $31K more than end-of-year payments, exactly 5% of PV_ord.

📝 Example 4: Annuity Due FV

You deposit $12,000 at the beginning of each year into an account earning 8% annually. How much will you have after 30 years?

Solution: First compute ordinary annuity FV (assuming end-of-year deposits)

$$FV_{\text{ord}} = 12{,}000 \times \frac{(1.08)^{30} - 1}{0.08}$$

(1.08)³⁰ ≈ 10.0627

$$= 12{,}000 \times \frac{10.0627 - 1}{0.08} = 12{,}000 \times 113.2832 = 1{,}359{,}399$$

Then adjust for annuity due:

$$FV_{\text{due}} = 1{,}359{,}399 \times 1.08 = 1{,}468{,}151$$

Answer: ≈ $1,468,151

📈 Comparison: end-of-year deposits = $1,359,399; beginning-of-year deposits = $1,468,151. Over 30 years, the difference is nearly $109K — the power of one extra year of compounding on every deposit.

X. Key Formula Summary

🧠 Memory aid: Due = Ordinary × (1+r) — two faces, one factor.

XI. Practice Questions

Q1 (Concept Identification)

Which of the following is an ordinary annuity?

A. Rent paid on the 1st of each month B. Annual insurance premium paid at the start of the year C. Monthly mortgage payments made at the end of each month D. First installment paid immediately upon purchase

Q2 (Annuity Due PV)

You purchase an annuity product that pays $8,000 at the beginning of each year for 10 years. The discount rate is 6%. The present value is closest to:

A. $58,881 B. $62,413 C. $66,000 D. $72,000

Q3 (Due vs Ordinary Comparison)

For identical PMT, n, and r, which statement is correct?

A. PV_due < PV_ordinary, and FV_due < FV_ordinary B. PV_due > PV_ordinary, and FV_due > FV_ordinary C. PV_due > PV_ordinary, but FV_due < FV_ordinary D. PV_due = PV_ordinary, and FV_due = FV_ordinary

Q4 (Annuity Due FV)

You invest $5,000 at the beginning of each year at 7% annual interest. The future value after 20 years is:

A. $204,977 B. $219,325 C. $225,000 D. $234,678

Q5 (Trap: BGN Mode)

You just solved an annuity due problem and the BGN indicator is still on screen. You now need to compute the PV of an ordinary annuity (end-of-period payments). If you enter the parameters and press CPT PV without changing modes:

A. The result is correct — mode does not affect PV calculations B. The result will be overstated — the calculator treats each PMT as a beginning-of-period payment C. The result will be understated D. The calculator will display an error

XII. After-Class Reflection

L089–L090 built the PV/FV framework for single cash flows. L091 enters the world of annuities — treating a stream of equal cash flows as a single unit of valuation.

Ordinary annuity (end-of-period) and annuity due (beginning-of-period) differ by just one (1+r) factor — seemingly simple, yet it is the foundation of the entire annuity framework.

L092 will introduce perpetuities: if payments never stop, how do you compute PV? This leads to one of the most elegant formulas in CFA Level I...

📎 Answers

Explanations:

• Q1: Monthly mortgage payments at month-end → end-of-period → Ordinary Annuity. A/B/D are all beginning-of-period payments → Annuity Due.

• Q2: First compute ordinary PV: $8,000 × [1−(1.06)⁻¹⁰]/0.06 = $8,000 × 7.3601 = $58,881. Then multiply by 1.06: $58,881 × 1.06 = $62,413.

• Q3: Annuity due payments are always one period earlier → discounted less, compounded more → both PV and FV are larger.

• Q4: First compute ordinary FV: $5,000 × [(1.07)²⁰−1]/0.07 = $5,000 × 40.9955 = $204,977. Then multiply by 1.07 → $219,325.

• Q5: BGN mode treats the first PMT as occurring at t=0 (immediate, no discounting). The result will be higher than the correct ordinary annuity PV. You must switch back to END mode.