BAFS Study Notes: Time Value of Money
Hello everyone! Welcome to one of the most important concepts in finance: the Time Value of Money (TVM). Sounds complicated? Don't worry! The main idea is super simple: money you have today is worth more than the same amount of money in the future.
Why? Because you can use the money you have today to earn more money! You could put it in a bank to earn interest, or invest it. This chapter will teach you how to compare money across different points in time, which is a key skill for making smart personal and business financial decisions.
Let's unlock the secrets of how money grows over time!
The Core Idea: A Dollar Today is Worth More Than a Dollar Tomorrow
Imagine your friend offers you a choice: get $100 today or get $100 one year from now. Which one would you choose? Most people would take the $100 today! This is the core of the Time Value of Money.
Here’s why:
- Opportunity to Earn Interest: You can deposit the $100 in a bank today and it will grow. If the bank offers 2% interest, you'll have $102 in a year. Your $100 today has the potential to become more than $100 in the future.
- Inflation: Prices of goods and services tend to rise over time (this is called inflation). The $100 you have today can buy more things than the same $100 will be able to buy in a year.
Think of it like this: Money today is like a seed. You can plant it (invest it), and over time, it will grow into a bigger tree (more money!).
Key Takeaway
Money has a 'time value' because of its potential to earn interest. This is why we can't simply compare money from different time periods directly.
Future Value (FV) and The Magic of Compounding
Future Value (FV) is the value of a sum of money at a specific date in the future, assuming it earns interest. In simple terms, it's about figuring out how much your money will grow to become.
What is Compounding?
Compounding is the process of earning interest not only on your original money (the principal) but also on the interest you've already earned. It’s like a snowball rolling downhill – it gets bigger and bigger, faster and faster! This is why Albert Einstein reportedly called compounding the "eighth wonder of the world".
Example: You deposit $1,000 in a bank account that pays 5% interest per year.
- After Year 1: You earn $1,000 x 5% = $50 interest. Your new balance is $1,050.
- After Year 2: You now earn interest on $1,050! So, you get $1,050 x 5% = $52.50 interest. Your new balance is $1,102.50. Notice you earned more interest in the second year!
- After Year 3: You earn interest on $1,102.50. So, you get $1,102.50 x 5% = $55.13 interest. Your new balance is $1,157.63.
That extra interest you earn on previous interest is the power of compounding!
The Future Value Formula
Instead of calculating year by year, we can use a simple formula. Don't worry if it looks scary at first, we'll break it down!
$$FV = PV \times (1 + r)^n$$Where:
- FV = Future Value (the amount you'll have in the future)
- PV = Present Value (the amount you start with today)
- r = Interest rate per period (usually per year)
- n = Number of periods (usually the number of years)
Step-by-Step Calculation Example
Question: You invest $5,000 today in an account that pays 4% interest per year. What will be the future value of your investment after 3 years?
- Identify the variables:
PV = $5,000
r = 4% or 0.04
n = 3 years - Write down the formula:
$$FV = PV \times (1 + r)^n$$ - Substitute the values into the formula:
$$FV = \$5,000 \times (1 + 0.04)^3$$ $$FV = \$5,000 \times (1.04)^3$$ $$FV = \$5,000 \times 1.124864$$ - Calculate the final answer:
$$FV = \$5,624.32$$
So, your $5,000 will grow to become $5,624.32 in 3 years!
Key Takeaway
Compounding is the process of growing money forward in time to find its Future Value (FV). It’s all about earning interest on your interest.
Present Value (PV) and The Art of Discounting
Present Value (PV) is the current value of a future sum of money. It’s the exact opposite of Future Value. Instead of asking "How much will my money grow to?", we ask "How much money do I need today to reach a certain amount in the future?".
What is Discounting?
Discounting is the process of finding the Present Value. It's like "rewinding" the growth of money. If compounding is a snowball rolling forwards downhill, discounting is figuring out how small that snowball was at the very top of the hill.
This is useful for knowing what a future promise of money is actually worth to you right now.
The Present Value Formula
We can get the PV formula just by rearranging the FV formula. It's the same idea, just looking at it from a different angle.
$$PV = \frac{FV}{(1 + r)^n}$$Where the variables mean the same thing:
- PV = Present Value (the amount it's worth today)
- FV = Future Value (the amount you will receive in the future)
- r = Interest rate (also called the 'discount rate' here)
- n = Number of periods
Step-by-Step Calculation Example
Question: You want to have $10,000 in your savings account in 3 years for a trip. If the bank offers an interest rate of 5% per year, how much money do you need to deposit today (Present Value)?
- Identify the variables:
FV = $10,000
r = 5% or 0.05
n = 3 years - Write down the formula:
$$PV = \frac{FV}{(1 + r)^n}$$ - Substitute the values into the formula:
$$PV = \frac{\$10,000}{(1 + 0.05)^3}$$ $$PV = \frac{\$10,000}{(1.05)^3}$$ $$PV = \frac{\$10,000}{1.157625}$$ - Calculate the final answer:
$$PV = \$8,638.38$$
This means you need to invest $8,638.38 today to have $10,000 in 3 years. See? The $10,000 in the future is only worth $8,638.38 today!
Quick Review Box
- Future Value (FV): Finding the future worth of money you have today. (Moving money FORWARD in time)
- Present Value (PV): Finding the current worth of money you will receive in the future. (Moving money BACKWARD in time)
Key Takeaway
Discounting is the process of bringing future money back to the present to find its Present Value (PV). It helps you understand the true worth of future cash flows today.
Net Present Value (NPV) - Making Smart Investment Decisions
Now that you understand PV and FV, you can learn a powerful tool for making decisions: Net Present Value (NPV). Businesses use this all the time to decide if a project or investment is worth doing.
The idea is simple: You compare the cost of the investment today with the value of all the money it will bring you in the future. But remember, we can't just add up future money! We first have to discount all future cash inflows to find their present value.
The NPV Formula and Decision Rule
NPV = (Sum of the Present Values of all future cash inflows) - Initial Investment
After you calculate the NPV, you use this simple rule:
- If NPV is positive (> 0): Accept the project. It is expected to be profitable and will add value.
- If NPV is negative (< 0): Reject the project. It is expected to be a loss-maker.
- If NPV is zero (= 0): You are indifferent. The project is expected to earn exactly the required rate of return, but no more.
Step-by-Step NPV Calculation
Question: A company is considering a project that costs $2,000 today. The project is expected to generate cash inflows of $900 in Year 1, $800 in Year 2, and $700 in Year 3. The company's required rate of return (discount rate) is 6%. Should the company accept the project?
This looks like a lot, but it's just repeating the PV calculation you already know. Let's do it step-by-step.
- Find the PV of EACH future cash inflow:
PV of Year 1 cash flow: $$PV = \frac{\$900}{(1.06)^1} = \$849.06$$ PV of Year 2 cash flow: $$PV = \frac{\$800}{(1.06)^2} = \$711.99$$ PV of Year 3 cash flow: $$PV = \frac{\$700}{(1.06)^3} = \$587.72$$ - Sum (add up) all the Present Values:
Total PV of inflows = $849.06 + $711.99 + $587.72 = $2,148.77 - Subtract the Initial Investment:
NPV = Total PV of inflows - Initial Investment
NPV = $2,148.77 - $2,000
NPV = $148.77 - Make a decision based on the NPV rule:
The NPV is $148.77, which is positive. Therefore, the company should accept the project.
Common Mistake Alert!
A common mistake is forgetting to discount the future cash flows. Remember, you can't just add $900 + $800 + $700 and compare it to the cost. You MUST find the present value of each inflow first!
Key Takeaway
NPV is a decision-making tool. It tells you the value that an investment will add to you in today's dollars, helping you choose profitable projects.
Nominal Rate vs. Effective Rate of Return
When you see an interest rate advertised, is it the full story? Not always! This is where we need to distinguish between two types of rates.
Nominal Rate of Return
The Nominal Rate is the stated or advertised annual interest rate. It's the simple rate that doesn't include the powerful effect of compounding within the year. It's the "headline" number.
Example: A bank advertises "12% interest per year". This 12% is the nominal rate.
Effective Rate of Return
The Effective Rate (also called the Effective Annual Rate or EAR) is the true or actual annual interest rate that you earn. It takes into account the effect of compounding that happens more than once a year (e.g., semi-annually, quarterly, or monthly).
Why is there a difference?
If interest is compounded more frequently than once a year, you start earning interest on your interest sooner. This makes your money grow faster, so the *effective* rate you earn is higher than the *nominal* rate.
Let's illustrate with a simple example. You have $1,000 and the nominal rate is 12% per year.
- Case 1: Compounded Annually (once a year)
Interest = $1,000 x 12% = $120.
Your total is $1,120. The effective rate is 12%. - Case 2: Compounded Semi-Annually (twice a year)
The rate for each 6-month period is 12% / 2 = 6%.
After 6 months: $1,000 x 6% = $60 interest. Your balance is $1,060.
After the next 6 months: $1,060 x 6% = $63.60 interest. Your final balance is $1,123.60.
Total interest earned is $123.60. So the effective rate is 12.36%!
As you can see, even though the nominal rate was 12% in both cases, you earned more money when it was compounded more frequently. This means the effective rate was higher.
Quick Review Box
- Nominal Rate: The advertised, simple annual rate.
- Effective Rate: The true, actual annual rate you earn after accounting for compounding.
- If interest is compounded more than once a year, then the Effective Rate will always be higher than the Nominal Rate.
Key Takeaway
The Effective Rate gives you a more accurate picture of your investment's return than the Nominal Rate because it includes the effect of compounding. It allows for a true comparison between different investment options.