TVM Calculations and Applications

Building on the fundamental TVM concepts, this section covers advanced applications that are heavily tested on the CFP exam. Understanding NPV, IRR, and annuity calculations is essential for retirement planning, investment analysis, and capital budgeting decisions.

Net Present Value (NPV)

Net Present Value (NPV) measures the difference between the present value of cash inflows and the present value of cash outflows. It answers: "How much value does this investment create or destroy in today's dollars?"

NPV Formula: NPV = Present Value of Inflows - Initial Investment

Decision Rule:

NPV Calculation Example:

A client is considering an investment that costs $100,000 today and will generate the following cash flows over 5 years. The required rate of return is 8%.

Calculator Steps (TI BA II Plus):

1. Press CF, then 2nd CLR WORK to clear
2. CF0 = -100,000 (initial investment as outflow)
3. C01 = 25,000, F01 = 1
4. C02 = 30,000, F02 = 1
5. C03 = 30,000, F03 = 1
6. C04 = 35,000, F04 = 1
7. C05 = 40,000, F05 = 1
8. Press NPV, I = 8, then CPT

Result: NPV = $24,237.65

Since NPV is positive, the investment exceeds the 8% required return and creates value.

Exam Tip: Always remember that NPV is calculated using a specified discount rate (the required return). If the discount rate changes, the NPV changes.

Internal Rate of Return (IRR)

Internal Rate of Return (IRR) is the discount rate at which the NPV equals zero. It represents the actual return generated by the investment.

Decision Rule:

Using the same example above:

Calculator Steps (TI BA II Plus):

1. After entering cash flows (as shown above)
2. Press IRR, then CPT

Result: IRR = 15.24%

Since IRR (15.24%) exceeds the required return (8%), the investment should be accepted.

NPV vs. IRR: Key Differences

Exam Tip: When NPV and IRR give conflicting rankings for mutually exclusive projects, NPV is generally preferred because it measures actual value creation.

Annuities: Ordinary vs. Annuity Due

An annuity is a series of equal periodic payments. Understanding the distinction between ordinary annuities and annuities due is critical for the CFP exam.

Ordinary Annuity (END mode)

• Payments occur at the END of each period
• Most common type in financial planning
• Examples: Mortgage payments, car loan payments, investment account contributions at month-end

Annuity Due (BEGIN mode)

• Payments occur at the BEGINNING of each period
• Examples: Rent payments, insurance premiums, lease payments, retirement withdrawals

Converting Between Annuity Types

Key Relationship:

• Annuity Due Value = Ordinary Annuity Value x (1 + i)

This works for both present value and future value calculations.

Example: If an ordinary annuity has a PV of $50,000 at 6% interest:

• Annuity Due PV = $50,000 x 1.06 = $53,000

Practical Example: Retirement Savings (Ordinary Annuity)

Problem: A client, age 35, wants to save for retirement at age 65. She can contribute $500 per month at the end of each month to an investment account earning 7% annually. How much will she have at retirement?

Calculator Inputs:

• N = 360 (30 years x 12 months)
• I/Y = 7/12 = 0.5833 (monthly rate)
• PV = 0 (starting from nothing)
• PMT = -500 (monthly contribution, outflow)
• Solve for FV = ?

Solution: FV = $610,987.61

Important: Set your calculator to END mode for ordinary annuity calculations.

Practical Example: Retirement Income (Annuity Due)

Problem: A retiree has $800,000 in savings and wants to receive payments at the beginning of each month for 25 years. If the account earns 5% annually, how much can she withdraw monthly?

Calculator Inputs:

• Set calculator to BGN (beginning) mode
• N = 300 (25 years x 12 months)
• I/Y = 5/12 = 0.4167 (monthly rate)
• PV = -800,000 (current savings, outflow from account)
• FV = 0 (deplete account completely)
• Solve for PMT = ?

Solution: PMT = $4,651.83 per month

Important: Retirement withdrawals typically use annuity due because retirees need money at the beginning of each period to pay living expenses.

Serial Payments (Inflation-Adjusted Annuities)

Serial payments are annuity payments that increase each period to maintain purchasing power. This is crucial for long-term retirement planning.

Calculation Method: Use the "inflation-adjusted" or "real" interest rate:

Real Rate = [(1 + Nominal Rate) / (1 + Inflation Rate)] - 1

Example: If investments earn 7% and inflation is 3%: Real Rate = (1.07 / 1.03) - 1 = 3.88%

Use this real rate in your TVM calculation to determine payments that will maintain purchasing power.

Uneven Cash Flows

Many financial planning scenarios involve uneven cash flows that do not fit the standard annuity pattern. Use the Cash Flow (CF) function:

Common Scenarios:

• Investment projects with varying returns
• Retirement with different income phases
• Education funding across multiple years

Calculator Approach:

1. Enter CF0 (initial cash flow, typically negative)
2. Enter each subsequent cash flow (C01, C02, etc.)
3. Enter frequency (F) if cash flows repeat
4. Calculate NPV or IRR as needed

Deferred Annuities

A deferred annuity is an annuity that does not begin payments until some point in the future. This requires a two-step calculation:

Step 1: Calculate the present value of the annuity at the time payments begin Step 2: Discount that value back to today

Example: A client will receive $2,000 per month for 20 years, but payments do not start for 10 years. At 6% annual interest, what is this worth today?

Step 1: PV of annuity at Year 10 (when payments begin)

• N = 240 (20 years x 12)
• I/Y = 0.5 (6%/12)
• PMT = 2,000
• FV = 0
• Solve: PV at Year 10 = -$279,161.54

Step 2: Discount back to today

• N = 120 (10 years x 12)
• I/Y = 0.5
• FV = 279,161.54
• PMT = 0
• Solve: PV today = -$153,257.83

Key Calculator Settings Checklist

Before every TVM calculation, verify:

Application in Financial Planning