Time Value of Money: Concepts, Calculations & Applications

TAKEAWAYS

• Money’s value changes over time due to inflation, interest, and opportunity cost, so financial decisions must account for time value.
• Present Value (PV) and Future Value (FV) are calculated using the formula $FV=PV(1+r{)}^n$ or $PV=\frac{FV}{(1+r{)}^n}$, where $r$ is the discount rate and $n$ is the time period.
• Annuities (regular payments like loan EMIs or rent) have distinct formulas for ordinary annuities (payments at period-end) vs. annuities due (payments at period-start).
• Loan amortization splits payments into interest and principal, reducing debt systematically—critical for mortgages, car loans, and business financing.
• Effective Annual Rate (EAR) and Annual Percentage Rate (APR) differ: EAR accounts for compounding, while APR is the nominal rate.
• Real-world applications include Khalti’s interest calculations, Nepal Rastra Bank’s loan structuring, and Nepse’s dividend discount models.

1. Why Does Money Have Time Value?

Money’s value changes over time due to three key factors:

• Inflation: Purchasing power erodes (e.g., ₹100 today buys less in 5 years).
• Interest: Lenders charge for delayed payment (e.g., bank loans, deposits).
• Opportunity Cost: Investing today could yield returns (e.g., ₹100 invested at 10% grows to ₹110 in a year).

2. Core Concepts: Present Value (PV) and Future Value (FV)

Future Value (FV)

The value of money at a future date, calculated as: $$FV=PV\times (1+r{)}^n$$ where:

• $PV$ = Present Value
• $r$ = Interest rate per period (as decimal)
• $n$ = Number of periods

Example: If you invest ₹50,000 at 8% annual interest for 3 years: $$FV=50,000\times (1+0.08{)}^3=₹62,985.60$$

Present Value (PV)

The current worth of future money, calculated as: $$PV=\frac{FV}{(1+r{)}^n}$$

Example: What is the PV of ₹100,000 to be received in 5 years at 6%? $$PV=\frac{100,000}{(1+0.06{)}^5}=₹74,725.82$$

TABLE: PV vs. FV Comparison

3. Annuities: Regular Payments Over Time

An annuity is a series of equal payments (e.g., loan EMIs, rent, pensions). Two types:

1. Ordinary Annuity: Payments at end of each period (e.g., monthly loan payments).
2. Annuity Due: Payments at start of each period (e.g., rent paid in advance).

Future Value of an Ordinary Annuity (FVA)

$$FVA=PMT\times \frac{(1+r{)}^n-1}{r}$$ where $PMT$ = Payment per period.

Example: You deposit ₹2,000 monthly in a bank at 12% annual interest (1% monthly) for 5 years. What’s the FV? $$FVA=2,000\times \frac{(1+0.01{)}^{60}-1}{0.01}=₹1,56,210.20$$

Present Value of an Annuity (PVA)

$$PVA=PMT\times \frac{1-(1+r{)}^{-n}}{r}$$

Example: What’s the PV of a ₹5,000 annual rent for 3 years at 10%? $$PVA=5,000\times \frac{1-(1+0.10{)}^{-3}}{0.10}=₹12,262.45$$

MERMAID DIAGRAM: Annuity Types

flowchart TD
    A["Present Value of an Annuity (PVA)"] --> B["₹5,000 × \(\frac{1 - (1 + 0.10)^{-3}}{0.10}\) = ₹12,262.45"]
    A --> C["Formula: PVA = PMT × [1 - (1 + r)^(-n)] / r"]
    B --> D["Example: ₹5,000 annual payments for 3 years at 10% discount rate"]

4. Loan Amortization: How EMIs Work

When you take a loan (e.g., from NMB Bank or Global IME), each EMI consists of:

1. Interest (calculated on remaining principal).
2. Principal repayment (reduces the loan balance).

Example: ₹500,000 loan at 9% annual (0.75% monthly) for 5 years (60 months).

• Monthly EMI: $$EMI=P\times \frac{r(1+r{)}^n}{(1+r{)}^n-1}=500,000\times \frac{0.0075(1.0075{)}^{60}}{(1.0075{)}^{60}-1}=₹10,630.50$$

TABLE: Amortization Schedule (First 3 Months)

5. Effective Annual Rate (EAR) vs. Annual Percentage Rate (APR)

Example: A bank offers 12% APR compounded monthly. What’s the EAR? $$EAR={\left(1+\frac{0.12}{12}\right)}^{12}-1=12.68\%$$

In the Real World

1. Khalti’s Interest Calculations

   • When you use Khalti’s "Khalti Loan" (e.g., ₹50,000 for 12 months at 1.5% monthly), the time value of money determines your EMI. The loan’s PV is discounted to reflect future repayments, and amortization tables show how much of each EMI goes to interest vs. principal.

2. Nepal Rastra Bank’s Loan Structuring

   • NRB uses PV calculations to assess project viability. For example, a ₹100 million infrastructure loan at 8% for 10 years must show that future cash flows (discounted at 8%) cover the loan’s PV.

3. Nepse’s Dividend Discount Model (DDM)

   • Companies like Nepal Bank Limited use PV of future dividends to determine stock prices. If a stock pays ₹5 annual dividends growing at 5%, its intrinsic value is: $$PV=\frac{5}{(0.12-0.05)}=₹83.33\text{ (using Gordon Growth Model)}$$

6. Perpetuities: Infinite Annuities

A perpetuity is an annuity with infinite payments (e.g., some preference shares or consol bonds). $$PV=\frac{PMT}{r}$$ Example: A bond pays ₹1,000 annually forever. At 8% discount rate: $$PV=\frac{1,000}{0.08}=₹12,500$$

7. Comparing Investments: Net Present Value (NPV)

NPV determines if an investment is profitable by comparing PV of cash inflows to initial cost. $$NPV=\sum \frac{C{F}_t}{(1+r{)}^t}-InitialInvestment$$ Example: Kathmandu Retail Shop invests ₹200,000 in a new product line with expected cash flows:

At 10% discount rate: $$NPV=\frac{60,000}{1.10}+\frac{70,000}{1.10^2}+\frac{80,000}{1.10^3}+\frac{50,000}{1.10^4}-200,000=₹12,850.50$$ Since NPV > 0, the investment is profitable.

Exam Tip

1. Memorize the 4 key formulas:

   • $FV=PV(1+r{)}^n$
   • $PV=\frac{FV}{(1+r{)}^n}$
   • $FVA=PMT\times \frac{(1+r{)}^n-1}{r}$
   • $PVA=PMT\times \frac{1-(1+r{)}^{-n}}{r}$

2. Watch for annuity types:

   • Ordinary vs. Annuity Due: The latter’s PV/FV is higher by a factor of $(1+r)$.

3. Loan amortization questions:

   • Always calculate interest first, then principal, and verify the ending balance.

4. NPV vs. IRR:

   • NPV tells you if a project is profitable (use at 10% discount rate in Nepal).
   • IRR finds the rate where NPV = 0 (often asked in PU/TU exams).

5. Real-world applications:

   • Banks (NMB, Standard Chartered): Use PV to price loans.
   • Nepse: Uses DDM for stock valuation.
   • E-sewa/Khalti: Apply time value to digital payment interest.

6. Common pitfalls:

   • Mismatched periods: Ensure $r$ and $n$ units match (e.g., monthly $r$ for monthly $n$).
   • Ignoring compounding: Always use EAR for accurate comparisons.
   • Sign errors in NPV: Cash inflows are +, outflows are -.

MERMAID DIAGRAM: Accounting Cycle (Time Value Applications)