Time Value of Money: Concepts, Calculations & Applications

Key Concepts and Definitions

1. Time Value of Money (TVM) Basics

The time value of money is the principle that money available today is worth more than the same amount in the future due to its potential earning capacity. This concept is foundational in corporate finance for evaluating investments, loans, and financial planning.

2. Present Value (PV) and Future Value (FV)

• Future Value (FV): The value of a current asset at a future date based on an assumed rate of growth (interest rate).
• Present Value (PV): The current worth of a future sum of money or stream of cash flows given a specified rate of return.

3. Interest Rates and Compounding

• Simple Interest: Interest calculated only on the original principal amount.
• Compound Interest: Interest calculated on the initial principal and also on the accumulated interest of previous periods.

4. Annuities and Perpetuities

• Annuity: A series of equal payments made at regular intervals (e.g., loan payments, lease payments).
• Perpetuity: An annuity with an infinite series of payments (e.g., some types of preferred stocks).

5. Discounting and Net Present Value (NPV)

• Discounting: The process of determining the present value of a payment or series of payments made in the future.
• NPV: A method used to determine the profitability of an investment or project by comparing the present value of cash inflows to the present value of cash outflows.

---

Visualizing TVM Concepts

1. Simple vs. Compound Interest

2. Present Value vs. Future Value

---

Detailed Explanation with Examples

1. Future Value Calculation

The formula for future value with compound interest is: $$FV=PV\times (1+r{)}^t$$ Where:

• $FV$ = Future Value
• $PV$ = Present Value
• $r$ = Interest rate per period
• $t$ = Number of periods

Example: Suppose you invest NPR 100,000 in a bank that offers an annual interest rate of 8%. Calculate the future value after 5 years.

$$FV=100,000\times (1+0.08{)}^5$$ $$FV=100,000\times 1.4693$$ $$FV=146,930$$

So, after 5 years, your investment will grow to NPR 146,930.

---

2. Present Value Calculation

The formula for present value is: $$PV=\frac{FV}{(1+r{)}^t}$$

Example: You need NPR 200,000 in 4 years for a new business venture. If the bank offers a 7% annual interest rate, how much should you invest today?

$$PV=\frac{200,000}{(1+0.07{)}^4}$$ $$PV=\frac{200,000}{1.3108}$$ $$PV=152,590$$

You need to invest NPR 152,590 today to have NPR 200,000 in 4 years.

---

3. Annuities: Ordinary Annuity vs. Annuity Due

An ordinary annuity has payments made at the end of each period, while an annuity due has payments made at the beginning of each period.

Formulas:

• Future Value of Ordinary Annuity: $$F{V}_{OA}=PMT\times \frac{(1+r{)}^t-1}{r}$$
• Future Value of Annuity Due: $$F{V}_{AD}=PMT\times \frac{(1+r{)}^t-1}{r}\times (1+r)$$

Example: Suppose you plan to save NPR 5,000 at the end of each year for the next 10 years in a bank offering 6% annual interest. Calculate the future value.

$$F{V}_{OA}=5,000\times \frac{(1+0.06{)}^{10}-1}{0.06}$$ $$F{V}_{OA}=5,000\times \frac{1.7908-1}{0.06}$$ $$F{V}_{OA}=5,000\times 13.18$$ $$F{V}_{OA}=65,900$$

The future value of your savings will be NPR 65,900.

---

4. Perpetuity Calculation

The formula for the present value of a perpetuity is: $$PV=\frac{PMT}{r}$$

Example: A company issues a preferred stock that pays a NPR 5,000 dividend annually forever. If the required rate of return is 10%, what is the present value of this stock?

$$PV=\frac{5,000}{0.10}$$ $$PV=50,000$$

The present value of the preferred stock is NPR 50,000.

---

Comparison Table: Simple vs. Compound Interest

---

Real-World Applications

1. eSewa and Khalti: Loan Repayments

Both eSewa and Khalti offer digital loan services where borrowers repay loans in installments. The time value of money is used to calculate the Equated Monthly Installments (EMIs). For example, if you take a loan of NPR 500,000 at 12% annual interest for 3 years, the EMI is calculated using the present value of an annuity formula. This ensures that the lender earns interest over time while the borrower repays in manageable chunks.

2. Ncell and NTC: Investment in Infrastructure

Telecom companies like Ncell and NTC use TVM to evaluate long-term investments in infrastructure such as 5G networks. For instance, if Ncell plans to invest NPR 10 billion in a new network that will generate cash flows over 10 years, they use NPV to determine if the investment is profitable. If the NPV is positive, the investment is viable.

3. Daraz and NEPSE: Stock Valuation

Daraz (an e-commerce platform) and NEPSE (Nepal Stock Exchange) rely on TVM to value stocks. For example, if a company like Daraz Nepal expects to pay dividends of NPR 2,000 annually forever, investors use the perpetuity formula to determine the stock's fair value. If the required return is 8%, the stock's value is NPR 25,000.

4. Banks: Loan Approval

Banks use TVM to assess loan applications. For example, if a customer applies for a NPR 2 million loan to be repaid in 5 years at 9%, the bank calculates the EMI to ensure the loan is sustainable. The bank also uses NPV to determine if the loan's interest revenue justifies the risk.

---

Worked Example: Kathmandu Retail Shop

Scenario: Mr. Sharma owns a retail shop in Kathmandu. He wants to expand his business by purchasing new inventory worth NPR 500,000. He has two options:

1. Option 1: Borrow NPR 500,000 from a bank at 10% annual interest, repayable in 3 years.
2. Option 2: Save money from his current profits and invest it in a fixed deposit that offers 8% annual interest.

Assumptions:

• Mr. Sharma can save NPR 150,000 annually from profits.
• The bank loan requires equal annual repayments (annuity).

Step 1: Calculate Loan Repayments (Option 1)

Using the Present Value of Annuity formula: $$PV=PMT\times \frac{1-(1+r{)}^{-t}}{r}$$ $$500,000=PMT\times \frac{1-(1+0.10{)}^{-3}}{0.10}$$ $$500,000=PMT\times 2.4869$$ $$PMT=\frac{500,000}{2.4869}$$ $$PMT=201,000$$

Mr. Sharma must repay NPR 201,000 annually for 3 years.

Step 2: Calculate Savings Growth (Option 2)

Using the Future Value of Annuity formula: $$FV=PMT\times \frac{(1+r{)}^t-1}{r}$$ $$FV=150,000\times \frac{(1+0.08{)}^3-1}{0.08}$$ $$FV=150,000\times 3.246$$ $$FV=486,900$$

After 3 years, Mr. Sharma will have NPR 486,900, which is NPR 13,100 short of his goal.

Step 3: Decision Making

• Option 1 (Loan): Requires repaying NPR 201,000 annually but provides immediate funds.
• Option 2 (Savings): Falls short by NPR 13,100 but avoids debt.

Mr. Sharma may choose Option 1 if he can manage the annual repayments, or explore additional financing to cover the shortfall in Option 2.

---

Exam Tip

1. Master the Formulas: Ensure you can derive and apply the formulas for PV, FV, annuities, and perpetuities under exam conditions.
2. Practice Numerical Problems: The exam often includes calculations, so practice with different scenarios (e.g., loans, investments, leases).
3. Understand the Concepts: Know when to use simple vs. compound interest, ordinary annuity vs. annuity due, and NPV vs. IRR.
4. Real-World Application: Relate TVM concepts to real-life examples like loans, investments, and business decisions.
5. Watch Units: Always ensure your final answers are in the correct units (e.g., NPR, years) and double-check calculations for arithmetic errors.

---

Summary Table of Key Formulas