PV = 1000
r = 0.08
n = 2
FVIF = (1 + r) ** n
FV = PV * FVIF
PVIF = 1 / ((1 + r) ** n)
PV_again = FV * PVIF
print("FVIF:", round(FVIF, 4))
print("Future Value:", round(FV, 2))
print("PVIF:", round(PVIF, 4))
print("Present Value:", round(PV_again, 2))Lecture 09: Time Value of Money I
NREC4230 Agricultural Finance lecture note on future value, present value, discounting, compounding, FVIF, PVIF, and agricultural finance applications.
Learning objectives
By the end of this lecture, students should be able to:
- Explain why money today is worth more than money in the future.
- Calculate future value using simple and compound interest.
- Calculate present value by discounting future cash flows.
- Use FVIF and PVIF to simplify time value of money calculations.
- Apply time value of money to agricultural finance decisions.
1. Why time value of money matters
Agricultural decisions often involve money paid or received at different points in time. A farmer may pay for seeds today, receive crop revenue after harvest, repay a loan next year, or invest in machinery that generates benefits for several years.
Because these cash flows occur at different times, we cannot compare them directly without adjusting for time.
NoteKey idea
One OMR today is worth more than one OMR next year because money today can be invested, used to reduce debt, or kept as liquidity against risk.
The time value of money is important in agricultural finance because it helps answer questions such as:
- Should a farmer buy machinery now or rent it?
- Is a greenhouse investment profitable?
- How much is a future crop payment worth today?
- Which loan repayment option is cheaper?
- How much should be saved today for a future investment?
2. Interest rate as the price of time
The interest rate measures the reward for waiting or the cost of borrowing.
For a lender, interest is compensation for:
- delaying consumption
- inflation
- default risk
- loss of liquidity
- opportunity cost
For a borrower, interest is the cost of using money before it has been earned.
In agricultural finance, the interest rate may represent:
- bank lending rate
- cost of capital
- opportunity cost of funds
- discount rate used in investment appraisal
- required rate of return
3. Simple interest
Simple interest is calculated only on the original principal.
\[ I = PV \times r \times n \]
where:
| Symbol | Meaning |
|---|---|
| \(I\) | Interest earned or paid |
| \(PV\) | Present value or initial principal |
| \(r\) | Interest rate per period |
| \(n\) | Number of periods |
The future value under simple interest is:
\[ FV = PV \times (1 + rn) \]
Example 1: Simple interest loan
A farmer borrows OMR 1,000 for 2 years at 6% simple annual interest.
\[ I = 1000 \times 0.06 \times 2 = 120 \]
\[ FV = 1000 + 120 = 1120 \]
The farmer repays OMR 1,120 at the end of 2 years.
WarningCommon mistake
Do not apply compound interest when the question explicitly says simple interest.
4. Compound interest
Compound interest means that interest is added to the principal, and future interest is calculated on the new balance.
The compound future value formula is:
\[ FV = PV(1+r)^n \]
where:
| Symbol | Meaning |
|---|---|
| \(FV\) | Future value |
| \(PV\) | Present value |
| \(r\) | Interest rate per period |
| \(n\) | Number of periods |
Example 2: Compound interest saving
A farmer deposits OMR 1,000 in a savings account earning 8% per year for 2 years.
\[ FV = 1000(1+0.08)^2 \]
\[ FV = 1000(1.1664)=1166.40 \]
The investment grows to OMR 1,166.40.
5. Simple vs compound interest
| Feature | Simple interest | Compound interest |
|---|---|---|
| Interest base | Original principal only | Principal plus accumulated interest |
| Growth pattern | Linear | Exponential |
| Formula | \(FV = PV(1+rn)\) | \(FV = PV(1+r)^n\) |
| Common use | Short-term loans, simple calculations | Savings, investments, long-term finance |
Example 3: Comparing simple and compound interest
A farmer invests OMR 2,000 for 3 years at 10%.
Simple interest:
\[ FV = 2000(1+0.10 \times 3)=2600 \]
Compound interest:
\[ FV = 2000(1.10)^3=2662 \]
Compound interest gives a higher future value because interest earns interest.
6. Future value interest factor
The future value formula can be written as:
\[ FV = PV \times FVIF_{r,n} \]
where:
\[ FVIF_{r,n} = (1+r)^n \]
\(FVIF\) means future value interest factor.
Example 4: Using FVIF
A farmer invests OMR 5,000 for 4 years at 7%.
\[ FVIF_{7\%,4} = (1.07)^4 = 1.3108 \]
\[ FV = 5000 \times 1.3108 = 6554 \]
The future value is approximately OMR 6,554.
7. Present value and discounting
Present value answers the reverse question: how much is a future amount worth today?
The present value formula is:
\[ PV = \frac{FV}{(1+r)^n} \]
Discounting converts future money into today’s value.
NoteKey idea
Compounding moves money forward in time. Discounting moves money backward in time.
Example 5: Present value of future crop revenue
A farmer expects to receive OMR 2,000 after 3 years. The discount rate is 6%.
\[ PV = \frac{2000}{(1.06)^3} \]
\[ PV = \frac{2000}{1.1910}=1679.24 \]
The present value is approximately OMR 1,679.24.
8. Present value interest factor
The present value formula can be written as:
\[ PV = FV \times PVIF_{r,n} \]
where:
\[ PVIF_{r,n} = \frac{1}{(1+r)^n} \]
\(PVIF\) means present value interest factor.
Example 6: Using PVIF
A farmer will receive OMR 10,000 in 5 years. The discount rate is 8%.
\[ PVIF_{8\%,5} = \frac{1}{(1.08)^5}=0.6806 \]
\[ PV = 10000 \times 0.6806 = 6806 \]
The future OMR 10,000 is worth approximately OMR 6,806 today.
9. Relationship between interest rate and present value
Present value decreases when the discount rate increases.
| Future amount | Years | Discount rate | Present value |
|---|---|---|---|
| OMR 1,000 | 3 | 3% | OMR 915.14 |
| OMR 1,000 | 3 | 6% | OMR 839.62 |
| OMR 1,000 | 3 | 10% | OMR 751.31 |
The same future payment becomes less valuable today when the discount rate is higher.
TipExam tip
If the question asks what happens to present value when the discount rate increases, the answer is: present value decreases.
10. Relationship between time and present value
Present value also decreases when the payment is further in the future.
| Future amount | Discount rate | Years | Present value |
|---|---|---|---|
| OMR 1,000 | 6% | 1 | OMR 943.40 |
| OMR 1,000 | 6% | 3 | OMR 839.62 |
| OMR 1,000 | 6% | 5 | OMR 747.26 |
The longer the delay, the lower the present value.
11. Agricultural example: delayed payment for produce
A buyer offers a vegetable farmer two payment options:
| Option | Payment |
|---|---|
| A | OMR 4,500 today |
| B | OMR 5,000 after one year |
The farmer’s discount rate is 12%.
Present value of Option B:
\[ PV = \frac{5000}{1.12}=4464.29 \]
Option A gives OMR 4,500 today. Option B is worth OMR 4,464.29 today.
Therefore, Option A is slightly better in present value terms.
WarningCommon mistake
Do not compare OMR 4,500 today with OMR 5,000 next year directly. Convert both to the same time period first.
12. Agricultural example: future machinery replacement
A farmer expects to replace irrigation equipment in 4 years. The expected replacement cost is OMR 8,000. The farmer can earn 5% per year on savings.
How much should be saved today?
\[ PV = \frac{8000}{(1.05)^4} \]
\[ PV = \frac{8000}{1.2155}=6581.65 \]
The farmer should save approximately OMR 6,581.65 today.
13. Agricultural example: investment growth
A farmer saves OMR 3,000 today for a greenhouse project. The money earns 6% annually for 5 years.
\[ FV = 3000(1.06)^5 \]
\[ FV = 3000(1.3382)=4014.68 \]
The savings grow to approximately OMR 4,014.68 after 5 years.
14. Python calculation example
The following Python code calculates FV, PV, FVIF, and PVIF.
Expected output:
FVIF: 1.1664
Future Value: 1166.4
PVIF: 0.8573
Present Value: 1000.015. Spreadsheet interpretation
In Excel or Google Sheets, the same calculations can be done with formulas.
| Task | Formula logic | Example |
|---|---|---|
| Future value | =PV*(1+r)^n |
=1000*(1+0.08)^2 |
| Present value | =FV/(1+r)^n |
=1000/(1+0.08)^2 |
| FVIF | =(1+r)^n |
=(1+0.08)^2 |
| PVIF | =1/(1+r)^n |
=1/(1+0.08)^2 |
Students should understand the financial logic before relying on spreadsheet functions.
16. Common mistakes
WarningMistake 1: Mixing percentage and decimal form
Use 8% as 0.08 in formulas, not 8.
WarningMistake 2: Using the wrong direction
Use future value when moving money forward. Use present value when moving money backward.
WarningMistake 3: Ignoring the number of periods
If interest is annual and the investment lasts 5 years, then \(n=5\). If interest is monthly, the number of periods must be monthly.
WarningMistake 4: Comparing cash flows at different dates
Convert all cash flows to the same point in time before comparing them.
17. Practice questions
Short-answer questions
- Why is OMR 1 today worth more than OMR 1 next year?
- What is the difference between simple interest and compound interest?
- What does FVIF measure?
- What does PVIF measure?
- What happens to present value when the discount rate increases?
Applied questions
A farmer invests OMR 2,500 for 3 years at 7% compound interest. Calculate the future value.
A farmer will receive OMR 6,000 after 4 years. The discount rate is 9%. Calculate the present value.
A buyer offers OMR 3,000 today or OMR 3,400 after one year. If the discount rate is 10%, which option is better?
A farmer borrows OMR 4,000 for 2 years at 5% simple interest. Calculate total repayment.
A farmer saves OMR 1,200 today at 6% annual compound interest. How much will it become after 5 years?
18. Key takeaways
- Time value of money is central to agricultural finance.
- Future value moves money forward in time.
- Present value discounts future money into today’s value.
- Compound interest grows faster than simple interest.
- FVIF and PVIF simplify calculations.
- Higher discount rates reduce present value.
- Longer waiting periods reduce present value.
- Agricultural decisions involving loans, investments, delayed payments, and savings require time value of money calculations.
Source note
This lecture note is prepared for NREC4230 Agricultural Finance using course materials on agricultural finance, financial formulas, and time value of money applications.