Unit 4 Introduction to the Time Value of Money

Time Value of Money (TVM)

The idea that money today is worth more than the same amount of money in the future because today's money can be invested and earn interest.

Present Value (PV)

The value of money today, at Time 0.

Future Value (FV)

The amount an investment will be worth at a future point in time.

Interest Rate (r)

The rate used to relate money at different points in time. It represents the return earned per period.

Time Period (t or N)

The number of periods over which money earns interest.

Simple Interest

Interest calculated only on the original principal.

Compound Interest

Interest earned on the original principal AND previously earned interest.

⭐ Future Value Interest Factor (FVIF)

The factor used to grow money into the future.

⭐ FVIF = (1 + r)^t

⭐ Future Value Formula

Calculates what money today will grow to in the future.

⭐ FV = PV(1 + r)^t

What does (1 + r)^t represent?

The future value interest factor (FVIF) — how much $1 today grows to after t periods.

What happens to FV when the interest rate increases?

FV increases, all else equal.

What happens to FV when time increases?

FV increases, all else equal.

Compounding

Moving money FORWARD in time.

⭐ PV → FV

⭐ Compounding Rule

When moving money from the present to the future:

⭐ PV → FV = Compounding

Example of Compounding

If you invest $1,000 today and want to know its value 5 years from now, you are compounding.

Present Value Calculation

Determines what a future amount of money is worth today.

Discounting

Moving money BACKWARD in time.

⭐ FV → PV

⭐ Discounting Rule

When moving money from the future back to today:

⭐ FV → PV = Discounting

Example of Discounting

If you'll receive $10,000 five years from now and want to know what it's worth today, you are discounting.

⭐ Present Value Formula

Calculates what a future amount is worth today.

⭐ PV = FV / (1 + r)^t

⭐ Present Value Interest Factor (PVIF)

The factor used to discount future money back to today.

⭐ PVIF = 1 / (1 + r)^t

Relationship Between FVIF and PVIF

They are reciprocals of each other.

⭐ PVIF = 1 / FVIF

What happens when you discount money?

You move money backward through time to determine its present value.

What happens when you compound money?

You move money forward through time to determine its future value.

⭐ Easy TVM Direction Rule

⭐ Forward = Compound = Find FV
⭐ Backward = Discount = Find PV

Why is PV normally lower than FV?

With a positive interest rate, money today can grow over time, so the present value is lower than its corresponding future value.

Relationship Between PV and FV

PV and FV are the same amount of money expressed at different points in time.

Effect of Interest Rate on PV

Holding everything else constant:

⭐ r ↑ → PV ↓

A higher discount rate makes future money worth less today.

Effect of Time on PV

Holding everything else constant and assuming a positive interest rate:

⭐ t ↑ → PV ↓

Money farther in the future is worth less today.

Discount Rate

The interest rate used to convert a future value into a present value.

⭐ Interest Rate Formula

Used when PV, FV, and time are known but the interest rate is unknown.

⭐ r = (FV / PV)^(1/t) − 1

What does the interest rate formula tell you?

The rate of return needed for PV to grow into FV over a certain number of periods.

Example: Doubling Your Money

If $10,000 becomes $20,000 in 6 years:

⭐ r = (20,000 / 10,000)^(1/6) − 1

= approximately 12.25%

Implied Interest Rate

The rate of return implied by the relationship between PV, FV, and time.

⭐ Rule of 72

An approximation for how long it takes money to double.

⭐ Years to double ≈ 72 / Interest Rate (%)

Rule of 72 Example

At an 8% return:

72 ÷ 8 ≈ 9 years

So money takes approximately 9 years to double.

BA II Plus TVM Keys

N, I/Y, PV, PMT, FV

N on the BA II Plus

The number of periods.

PV on the BA II Plus

The present value or value at Time 0.

I/Y on the BA II Plus

The interest rate per period

For your calculator setup, enter a percentage as a whole number:

⭐ 6% → enter 6, NOT .06

FV on the BA II Plus

The future value.

⭐ BA II Plus Sign Convention

Cash inflows and cash outflows must have opposite signs.

For these lump-sum TVM problems:

⭐ PV and FV should have opposite signs.

Example:
PV = −1,000
FV = +2,000

Why does the BA II Plus use opposite signs?

Because one amount represents money going out and the other represents money coming in.

PMT for Lump-Sum TVM Problems

If there are no recurring payments, enter:

⭐ PMT = 0

⭐ P/Y Setting for This Class

⭐ P/Y = 1

This means you manually adjust N and I/Y when compounding occurs more than once per year.

Why set P/Y = 1?

It allows you to manually make N and I/Y match the compounding period rather than having the calculator convert them automatically.

Clearing TVM Before a New Problem

Clear old TVM values before starting a new problem so previous numbers don't affect your answer.

On BA II Plus:

⭐ 2nd → FV (CLR TVM)

Annual Compounding

Interest is compounded once per year.

Semiannual Compounding

Interest is compounded twice per year.

Quarterly Compounding

Interest is compounded 4 times per year.

Monthly Compounding

Interest is compounded 12 times per year.

Daily Compounding

Interest is compounded approximately 365 times per year.

⭐ Monthly Compounding Conversion

Because your professor uses P/Y = 1, manually convert both:

⭐ N = Years × 12
⭐ I/Y = Annual Rate ÷ 12

⭐ Quarterly Compounding Conversion

⭐ N = Years × 4
⭐ I/Y = Annual Rate ÷ 4

⭐ Semiannual Compounding Conversion

⭐ N = Years × 2
⭐ I/Y = Annual Rate ÷ 2

Example: 6 Years at 11% Compounded Monthly
Convert both values:

⭐ N = 6 × 12 = 72

⭐ I/Y = 11 ÷ 12 = 0.91667

⭐ Matching Period Rule

N and I/Y must refer to the SAME length of period.

If N is measured in months, I/Y must be the monthly rate.

⭐ Easy Compounding Conversion Rule
When there are more compounding periods:

⭐ Multiply N
⭐ Divide I/Y

Example for monthly:

N → ×12
I/Y → ÷12

Excel FV Function

Used to calculate future value. ⭐ FV(rate, nper, pmt, pv)

Excel PV Function

Used to calculate present value.

⭐ PV(rate, nper, pmt, fv)

Excel RATE Function

Used to calculate the interest rate. ⭐ RATE(nper, pmt, pv, fv)

Excel NPER Function

Used to calculate the number of periods.

⭐ NPER(rate, pmt, pv, fv)

⭐ Number of Periods Formula

Used when you know PV, FV, and r but need to find time.

⭐ t = ln(FV / PV) / ln(1 + r)

Example 8 — Number of Periods

You have $1,833.69 today, need $4,341, and earn 9% annually.

BA II Plus:

PV = −1,833.69
FV = 4,341
I/Y = 9
PMT = 0
CPT → N

Answer:

⭐ N = 10 years

Example 6 — Finding Rate of Return

Invest $37,548.54 today and receive $136,771 in 15 years.

BA II Plus:

N = 15
PV = −37,548.54
PMT = 0
FV = 136,771
CPT → I/Y

Answer:

⭐ I/Y = 9%

Example 7 — Finding an Implied Rate

$10,000 doubles to $20,000 in 6 years.

BA II Plus:

N = 6
PV = −10,000
PMT = 0
FV = 20,000
CPT → I/Y

Answer:

⭐ 12.25%

⭐ TVM Master Rule
Before solving any TVM problem, ask:

1. What am I solving for? → PV, FV, I/Y, or N
2. Am I moving forward or backward in time?
3. Do N and I/Y use the same period?
4. Is PMT = 0 for a lump-sum problem?
5. Are PV and FV opposite signs on the calculator?

⭐ Formulas/rules I would actually memorize

⭐ FV = PV(1 + r)^t
⭐ PV = FV / (1 + r)^t
⭐ r = (FV/PV)^(1/t) − 1
⭐ t = ln(FV/PV) / ln(1 + r)
⭐ Forward → Compounding → PV → FV
⭐ Backward → Discounting → FV → PV
⭐ Multiply N, divide I/Y for more compounding periods
⭐ N and I/Y must use the same period
⭐ PV and FV = opposite signs on BA II Plus
⭐ PMT = 0 for these lump-sum problems