Microeconomics — Chapter 8: Basic Concepts in Value and Time

💰 Time value of money:
A dollar today is worth more than a dollar in the future because:
• It can be invested and earn interest
• It can be consumed now instead of waiting
• There is a risk that we will not receive the money in the future

🔄 Discounting:
Translating a sum of money from the future to the present.
Answers the question: "What is the value today of a sum we will receive in the future?"
The formula: PV = FV / (1+r)ⁿ

📈 Compounding:
Translating a sum of money from the present to the future.
Answers the question: "How much will a sum I have today be worth in n years?"
The formula: FV = PV × (1+r)ⁿ

🔢 Compounding calculation:
FV = PV × (1+r)¹
FV = 1,000 × 1.10 = $1,100

🔢 Discounting calculation:
PV = FV / (1+r)¹
PV = 1,100 / 1.10 = $1,000

📊 PV = Present Value
Present value - the value in todays terms of any sum of money (whether in the present, the past or the future).

📊 FV = Future Value
Future value - the value of a sum of money at a specific future point in time.

⬇️ High interest = low present value
When the interest rises, the denominator (1+r)ⁿ grows, and therefore the present value PV = FV/(1+r)ⁿ shrinks.
In other words: future money is worth less when there is a higher return on investments.

⬇️ More time = lower present value
The greater the number of years (n), the larger the denominator (1+r)ⁿ, and therefore the smaller the present value.
Money we will receive in 10 years is worth less than money we will receive in one year.

🔢 Compounding calculation for two years:
FV = PV × (1+r)²
FV = 1,000 × (1.10)² = 1,000 × 1.21 = $1,210
Note: it is not 1,200 because there is compound interest!

💡 Compound interest:
In the first year: 1,000 × 10% = 100
In the second year: 1,100 × 10% = 110 (not 100!)
The interest is calculated on the principal + the interest already accrued.

🎯 Discount factor:
The expression 1/(1+r)ⁿ is called the "discount factor".
PV = FV × [1/(1+r)ⁿ]
For example: at 10% interest for one year, the discount factor is 1/1.10 = 0.909

🔢 Discount factor calculation:
Discount factor = 1/(1+r)ⁿ
= 1/(1+0.20)¹ = 1/1.20 = 0.833
That is: $1 in one year is worth $0.833 today.

🤔 Opportunity cost:
If I have $1,000 and I can deposit it in a bank at 10%, then in any other use of the money I "lose" that 10%.
Therefore the interest rate = the opportunity cost of money.

📋 Discounting formula:
PV = FV / (1+r)ⁿ
• PV - present value
• FV - future value
• r - interest rate
• n - number of periods

📋 Compounding formula:
FV = PV × (1+r)ⁿ
• FV - future value
• PV - present value
• r - interest rate
• n - number of periods

🔢 Calculation:
PV = FV / (1+r)²
PV = 2,420 / (1.10)²
PV = 2,420 / 1.21 = $2,000

⚖️ Comparison:
1,000 today = 1,000
1,050 in one year = 1,050/1.10 = 954.55 today
1,000 today is worth more!
Because 1,000 > 954.55

🏦 Discount rate = opportunity cost:
The entrepreneurs best alternative is to deposit in the bank and receive 8%.
Therefore, in order for the project to be worthwhile, it needs to yield at least 8%.
Discount rate = 8%.

✅ The decision rule:
• If PV(receipts) > cost → worthwhile
• If PV(receipts) < cost → not worthwhile
Important: it is not enough that the sum of receipts is large - they must be discounted first!