We’ve already seen a key weakness of the payback method: it ignores the time value of money. The investment methods from this point forward do take the time value of money into account.

Time matters because the value of money can shrink or be put at risk over time. Common reasons include:

The risk that a business may never get its money back if it is owed for a long time
The opportunity cost of forgone interest when money is tied up
The risk of inflation, which reduces the purchasing power of money over time

The time value of money is the idea that money received today is worth more than the same amount received in the future.

For example, suppose you receive $$10$ today and use it to generate a profit of $$5$ , giving you $$15$ . In that case:

$$10 today = $15 in one week$

and equivalently:

$$15 in one week = $10 today$

Simple interest

Definitions

Simple interest: Interest is calculated only on the original principal amount. It doesn’t “build on itself” over time.

So, if an organization borrowed $$10, 000$ on 1 January 2024 at an interest rate of $10%$ , the interest charged each year would stay the same because the principal used in the calculation never changes.

Formula (simple interest)

$V = P (1 + r \times n)$

Where:

$V$ = future value receivable or payable
$P$ = present value (principal)
$r$ = interest rate (decimal)
$n$ = number of periods

Example 1
KTA borrowed $$10, 000$ to be paid back after $5$ years at an interest rate of $10%$ per year. How much will KTA need to repay at the end of year $5$ ?

Solution

(spoiler)

$V V = 10, 000 (1 + 0.1 \times 5) = 15, 000$

The amount payable at the end of year $5$ is $$15, 000$ .

Compounding

Definitions

Compound interest: Interest is calculated on the principal plus any interest already earned. In other words, interest earns interest.

If the principal is $$100$ and the interest earned in year $1$ is $$10$ , then the new balance is:

$100 + 10 = 110$

Interest in year $2$ is then calculated on $$110$ .

Formula (compound interest)

$V = P (1 + r)^{n}$

Example 2
P. Martins invested $$100$ for $3$ years at a compound interest rate of $10%$ . What is the value of the investment after $3$ years?

Solution

(spoiler)

$V V V = 100 (1 + 0.1)^{3} = 100 \times 1.331 = 133.10$

This means $$100$ today is equivalent to $$133.10$ after $3$ years.

Equivalent rate of interest

Definitions

Equivalent interest rates: Used when interest is compounded more frequently than once per year. The goal is to convert a nominal annual rate (with a compounding frequency) into an effective annual rate.

Example 3
KTA has an annual nominal interest rate of $10%$ , compounded quarterly. What is the effective annual interest rate?

Solution

(spoiler)

Quarterly rate:

$\frac{10%}{4} = 2.5%$

Convert to decimal:

$2.5% = 0.025$

Apply the compounding formula:

$(1 + 0.025)^{4} = 1.1038$

Remove $1$ :

$1.1038 - 1 = 0.1038$

Convert back to a percentage:

$0.1038 \times 100% = 10.38%$

The effective annual interest rate is $10.38%$ .

Terminal values

Definitions

Terminal values: When there are multiple cash flows, each cash flow compounds for a different length of time. Earlier deposits earn interest for more periods than later deposits.

Example 4
KTA deposits $$1, 000$ at the beginning of each year for $5$ years at an interest rate of $10%$ and withdraws the total at the beginning of year $6$ . How much will KTA be able to withdraw?

Solution

(spoiler)

$(1.10)^{5} + (1.10)^{4} + (1.10)^{3} + (1.10)^{2} + (1.10)^{1} 6.7156 \times 1, 000 = 6.7156 = 6, 715.60$

KTA will withdraw $$6, 715.60$ .

Sinking funds

Definitions

Sinking fund: Calculation works in the opposite direction of a terminal value. Instead of finding the accumulated total from regular deposits, you start with a target future amount and solve for the regular deposit needed.

Example 5
If KTA wants to accumulate $$6, 715.60$ over $5$ years at $10%$ , what annual deposit is required?

Solution

(spoiler)

$\frac{6 , 715.60}{6.7156} = 1, 000$

The required annual deposit is $$1, 000$ .

Always pay attention to timing, especially whether deposits are made at the beginning or end of a period.

Time value of money

Money today > same amount in future
Reasons: risk of nonpayment, opportunity cost, inflation

Simple interest

Interest only on original principal
Formula: $V = P (1 + r \times n)$

Compound interest

Interest on principal + accumulated interest
Formula: $V = P (1 + r)^{n}$

Equivalent interest rates

Converts nominal rate (compounded more than annually) to effective annual rate
Effective rate formula: $(1 + \frac{r}{m})^{m} - 1$ , where $m$ = compounding periods per year

Terminal values

Multiple cash flows, each compounding for different periods
Earlier deposits earn more interest than later ones

Sinking funds

Find regular deposit to reach a future target amount
Reverse of terminal value calculation
Timing of deposits (beginning vs. end of period) affects results

Next | 8.2 Present value and net present value

The time value of money

We’ve already seen a key weakness of the payback method: it ignores the time value of money. The investment methods from this point forward do take the time value of money into account.

Time matters because the value of money can shrink or be put at risk over time. Common reasons include:

The risk that a business may never get its money back if it is owed for a long time
The opportunity cost of forgone interest when money is tied up
The risk of inflation, which reduces the purchasing power of money over time

The time value of money is the idea that money received today is worth more than the same amount received in the future.

For example, suppose you receive $$10$ today and use it to generate a profit of $$5$ , giving you $$15$ . In that case:

$$10 today = $15 in one week$

and equivalently:

$$15 in one week = $10 today$

Simple interest

Definitions

Simple interest: Interest is calculated only on the original principal amount. It doesn’t “build on itself” over time.

Formula (simple interest)

$V = P (1 + r \times n)$

Where:

$V$ = future value receivable or payable
$P$ = present value (principal)
$r$ = interest rate (decimal)
$n$ = number of periods

Example 1
KTA borrowed $$10, 000$ to be paid back after $5$ years at an interest rate of $10%$ per year. How much will KTA need to repay at the end of year $5$ ?

Solution

(spoiler)

$V V = 10, 000 (1 + 0.1 \times 5) = 15, 000$

The amount payable at the end of year $5$ is $$15, 000$ .

Compounding

Definitions

Compound interest: Interest is calculated on the principal plus any interest already earned. In other words, interest earns interest.

If the principal is $$100$ and the interest earned in year $1$ is $$10$ , then the new balance is:

$100 + 10 = 110$

Interest in year $2$ is then calculated on $$110$ .

Formula (compound interest)

$V = P (1 + r)^{n}$

Example 2
P. Martins invested $$100$ for $3$ years at a compound interest rate of $10%$ . What is the value of the investment after $3$ years?

Solution

(spoiler)

$V V V = 100 (1 + 0.1)^{3} = 100 \times 1.331 = 133.10$

This means $$100$ today is equivalent to $$133.10$ after $3$ years.

Equivalent rate of interest

Definitions

Equivalent interest rates: Used when interest is compounded more frequently than once per year. The goal is to convert a nominal annual rate (with a compounding frequency) into an effective annual rate.

Example 3
KTA has an annual nominal interest rate of $10%$ , compounded quarterly. What is the effective annual interest rate?

Solution

(spoiler)

Quarterly rate:

$\frac{10%}{4} = 2.5%$

Convert to decimal:

$2.5% = 0.025$

Apply the compounding formula:

$(1 + 0.025)^{4} = 1.1038$

Remove $1$ :

$1.1038 - 1 = 0.1038$

Convert back to a percentage:

$0.1038 \times 100% = 10.38%$

The effective annual interest rate is $10.38%$ .

Terminal values

Definitions

Terminal values: When there are multiple cash flows, each cash flow compounds for a different length of time. Earlier deposits earn interest for more periods than later deposits.

Example 4
KTA deposits $$1, 000$ at the beginning of each year for $5$ years at an interest rate of $10%$ and withdraws the total at the beginning of year $6$ . How much will KTA be able to withdraw?

Solution

(spoiler)

$(1.10)^{5} + (1.10)^{4} + (1.10)^{3} + (1.10)^{2} + (1.10)^{1} 6.7156 \times 1, 000 = 6.7156 = 6, 715.60$

KTA will withdraw $$6, 715.60$ .

Sinking funds

Definitions

Sinking fund: Calculation works in the opposite direction of a terminal value. Instead of finding the accumulated total from regular deposits, you start with a target future amount and solve for the regular deposit needed.

Example 5
If KTA wants to accumulate $$6, 715.60$ over $5$ years at $10%$ , what annual deposit is required?

Solution

(spoiler)

$\frac{6 , 715.60}{6.7156} = 1, 000$

The required annual deposit is $$1, 000$ .

Always pay attention to timing, especially whether deposits are made at the beginning or end of a period.

CGMA BA1

8. Investment appraisal

Our CGMA course is currently in development and is a work-in-progress.