Microeconomics — Chapter 8: Value and Time — The Discounting Operation
Microeconomics — Chapter 8: Value and Time — The Discounting Operation. Practice questions to deepen understanding of the discounting operation. Online economics practice with full solutions and step-by-step explanations.
Discounting practice — the discounting operation, discount factor, present value, comparing amounts at different times.
Questions on discounting computations, discount factor, and comparison of amounts at different times.
🔢 What is the present value of $5,500 we will receive in one year, at 10% interest?
🔢 Calculation:
PV = FV / (1+r)¹
PV = 5,500 / 1.10 = $5,000
🔢 What is the present value of $12,100 we will receive in two years, at 10% interest?
🔢 Calculation:
PV = FV / (1+r)²
PV = 12,100 / (1.10)² = 12,100 / 1.21 = $10,000
🔢 What is the present value of $6,000 we will receive in one year, at 20% interest?
🔢 Calculation:
PV = FV / (1+r)¹
PV = 6,000 / 1.20 = $5,000
📊 The discount factor for two years at 10% interest is:
📊 Discount factor calculation:
Factor = 1/(1+r)²
= 1/(1.10)² = 1/1.21 ≈ 0.826
📊 The discount factor for one year at 5% interest is:
📊 Calculation:
Factor = 1/(1+0.05)¹ = 1/1.05 ≈ 0.952
⚖️ Compare: $10,000 today versus $11,000 in one year (15% interest):
⚖️ Comparison:
PV of 11,000 in one year = 11,000/1.15 = 9,565
10,000 > 9,565
10,000 today is worth more!
⚖️ Compare: $10,000 today versus $10,500 in one year (5% interest):
⚖️ Comparison:
PV of 10,500 in one year = 10,500/1.05 = 10,000
10,000 = 10,000
Exactly equal! (Still 10,500 is worth slightly more because we have more money in the future)
🔢 An entrepreneur receives $3,000 in one year and $3,000 in two years. What is the total present value at 10% interest?
🔢 Calculation:
PV₁ = 3,000/1.10 = 2,727
PV₂ = 3,000/1.21 = 2,479
Total = 2,727 + 2,479 = $5,207
📈 Which discount factor is higher?
📈 Comparison:
• 1 year, 5%: 1/1.05 = 0.952
• 2 years, 5%: 1/1.1025 = 0.907
• 1 year, 15%: 1/1.15 = 0.870
• 2 years, 10%: 1/1.21 = 0.826
One year at 5% = the highest
🔢 What is the present value of $1,000 we will receive in 3 years, at 10% interest?
🔢 Calculation:
PV = 1,000 / (1.10)³
= 1,000 / 1.331 ≈ $751
💰 An entrepreneur buys a machine for $8,000 and receives $4,500 in one year and $4,500 in two years. What is the present value of the receipts at 10% interest?
💰 Calculation:
PV₁ = 4,500/1.10 = 4,091
PV₂ = 4,500/1.21 = 3,719
Total = 4,091 + 3,719 = $7,810
✅ Continuing the previous question - is the investment worthwhile?
✅ The decision rule:
PV of receipts = 7,810
Investment cost = 8,000
7,810 < 8,000
The investment is not worthwhile!
🔄 If the interest rate were 5% instead of 10%, what would happen to the present value of the receipts?
🔄 Inverse relation:
Lower interest → higher discount factor → higher present value
At 5%: PV₁=4,286, PV₂=4,082 → total 8,367 > 8,000
At 5% interest the investment IS worthwhile!
🔢 What is the present value of $10,000 we will receive in two years at 5% interest?
🔢 Calculation:
PV = 10,000 / (1.05)²
= 10,000 / 1.1025 ≈ $9,070
🔢 What is the present value of $10,000 we will receive in two years at 15% interest?
🔢 Calculation:
PV = 10,000 / (1.15)²
= 10,000 / 1.3225 ≈ $7,561
📊 What is the difference in the present value of $10,000 (in 2 years) between 5% interest and 15% interest?
📊 Comparison:
At 5% interest: PV = 9,070
At 15% interest: PV = 7,561
Difference = 9,070 - 7,561 = $1,509
The interest rate has a major effect!
🎯 Which receipt is worth more today (at 10% interest)?
🎯 Comparison:
PV of 5,000 in one year = 5,000/1.10 = 4,545
PV of 5,500 in two years = 5,500/1.21 = 4,545
Exactly equal!
💡 Why is a receipt of 5,000 in one year equal to 5,500 in two years (at 10% interest)?
💡 The explanation:
If we receive 5,000 in one year and deposit it in the bank:
5,000 × 1.10 = 5,500 in two years
Therefore we are indifferent between the two options!
🔢 An investment costs $15,000 and yields $18,000 in two years. At 10% interest, is it worthwhile?
🔢 Calculation:
PV = 18,000 / (1.10)² = 18,000 / 1.21 = 14,876
14,876 < 15,000
The investment is not worthwhile!
🔢 Continuing the previous question - at what interest rate will the investment become worthwhile?
🔢 Calculation:
We need: PV ≥ 15,000
18,000/(1+r)² = 15,000
(1+r)² = 1.2
1+r = 1.0954
r = 9.54%
At an interest rate lower than 9.54% it is worthwhile!