Time Value of Money & Interest Rates: Present Value, Future Value, Annuities, Bonds

TAKEAWAYS:

• Money’s value today ≠ its value in the future due to time value of money (TVM)—investors demand compensation for waiting.
• Present value (PV) discounts future cash flows to today’s dollars using the discount rate (opportunity cost of capital).
• Future value (FV) grows today’s cash flows using the interest rate (time preference for money).
• Annuities (equal payments over time) and perpetuities (infinite payments) require special formulas to compute PV/FV.
• Bond pricing links coupon payments, face value, and market interest rates via TVM principles.
• Compounding (reinvesting interest) vs. simple interest affects growth rates—critical for loans, savings, and investments.

1. Why Money Has Time Value

Money today is worth more than the same amount in the future because:

• Opportunity cost: You could invest it and earn returns.
• Risk: Future cash flows are uncertain.
• Inflation: Purchasing power erodes over time.

(A bar chart showing Rs 100 in 2024 buys goods worth Rs 100, Rs 90 in 2026, Rs 80 in 2028, etc., with a downward-sloping trend line.)

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

Key Definitions

• Present Value (PV): Today’s value of a future cash flow, adjusted for discount rate. $$PV=\frac{FV}{(1+r{)}^n}$$ where:

  • $FV$ = Future value
  • $r$ = Discount rate (e.g., 10% = 0.10)
  • $n$ = Number of periods

• Future Value (FV): Value of a current cash flow after earning interest. $$FV=PV\times (1+r{)}^n$$

How It Works

• Discounting: Converting future cash flows to present terms (e.g., valuing a bond’s payments).
• Compounding: Growing present cash flows to future terms (e.g., calculating retirement savings).

FIGURE: Simple vs. Compound Interest Growth

timeline
    title Rs 100 Growing Over 5 Years
    section Simple Interest (10% p.a.)
    2024: Rs 100
    2025: Rs 110
    2026: Rs 120
    2027: Rs 130
    2028: Rs 140
    2029: Rs 150
    section Compound Interest (10% p.a.)
    2024: Rs 100
    2025: Rs 110
    2026: Rs 121
    2027: Rs 133.10
    2028: Rs 146.41
    2029: Rs 161.05

3. Calculating Present and Future Values

Single Sum (Lump Sum)

Example: You win Rs 500,000 in a lottery today. If you invest it at 8% annually, how much will it be worth in 10 years?

FV = 500000 * (1 + 0.08)^10 = Rs 971,837.44

(A step-by-step algebraic expansion showing how annual compounding builds to the formula.)

Annuities (Equal Payments Over Time)

Annuities have two types:

1. Ordinary Annuity: Payments at end of each period (e.g., loan EMI).
2. Annuity Due: Payments at start of each period (e.g., rent).

PV of Annuity Formula: $$P{V}_{annuity}=PMT\times \left[\frac{1-(1+r{)}^{-n}}{r}\right]$$

Example: Calculate PV of Rs 20,000 annual payments for 5 years at 12%.

PV = 20000 * [1 - (1.12)^(-5)] / 0.12 = Rs 79,994.28

FIGURE: Annuity Cash Flows

timeline
    title Ordinary Annuity (Payments at End of Period)
    2024: 0
    2025: Rs 20,000
    2026: Rs 20,000
    2027: Rs 20,000
    2028: Rs 20,000
    2029: Rs 20,000

4. Bonds and Their Pricing

Bonds pay coupon interest (fixed payments) + face value at maturity. Key Terms:

• Par Value: Face amount (e.g., Rs 1,000).
• Coupon Rate: Annual interest (e.g., 8% of Rs 1,000 = Rs 80).
• Yield to Maturity (YTM): Total return if held to maturity.

Bond Pricing Formula: $$BondPrice=\frac{Coupon}{(1+YTM{)}^1}+\frac{Coupon}{(1+YTM{)}^2}+\cdots +\frac{FaceValue+Coupon}{(1+YTM{)}^n}$$

Example: A bond with:

• Par = Rs 1,000
• Coupon = 10% (Rs 100/year)
• Maturity = 5 years
• Market YTM = 12% Calculate Price:

Price = 100/(1.12)^1 + 100/(1.12)^2 + ... + (1000 + 100)/(1.12)^5 = Rs 828.64

(A table showing each Rs 100 coupon discounted back to present value at 12%.)

5. Comparing Interest Types

Real-World Tie:

• Ncell’s EMI Plan: Uses simple interest for monthly payments on phones (e.g., 2% monthly on balance).
• NEPSE’s Fixed Deposits: Use compound interest (e.g., 8% annually compounded).

6. In the Real World

1. eSewa’s Loan Repayments:

   • When you take a loan via eSewa, the app calculates your monthly EMI using the PV of an annuity formula to ensure you repay the principal + interest over time.
   • Example: A Rs 50,000 loan at 15% APR for 2 years → EMI = Rs 2,380/month (calculated via annuity formula).

2. Khalti’s Investment Plans:

   • Khalti’s savings plans (e.g., 12% annual compound interest) use future value calculations to project how Rs 1,000/month grows over 5 years.
   • Worked Example:

     FV = 1000 * [(1.12)^60 - 1] / 0.12 = Rs 197,880 (for 5 years)
     

3. Daraz’s Inventory Financing:

   • Sellers use present value to decide if financing inventory (e.g., Rs 200,000 for 6 months at 18% APR) is cheaper than paying upfront.
   • Calculation:

     PV = 200000 / (1 + 0.18/2)^12 = Rs 165,000 (present value of future payments)
     

7. Worked Example: Lumbini Hotel’s Bond

Scenario: Lumbini Hotel issues a 7-year bond with:

• Par = Rs 1,000
• Coupon = 8% (Rs 80/year)
• Market Price = Rs 850 Find YTM: Use trial-and-error or financial calculator:
• Test YTM = 9%:

  PV = 80/(1.09)^1 + ... + (1000 + 80)/(1.09)^7 ≈ Rs 850.30 (close to Rs 850)
  

• Answer: YTM ≈ 9.0%.

Why It Matters:

• If YTM > Coupon Rate (8%), bond trades at a discount (Rs 850 < Rs 1,000).
• Investors earn higher returns by holding the bond until maturity.

8. Exam Tip: How This Unit Is Tested

1. Direct Calculations:

   • Expect 1–2 questions on PV/FV of single sums or annuities (e.g., "Calculate FV of Rs 10,000 at 12% for 4 years").
   • Tip: Memorize formulas and use the TVM solver on your calculator (e.g., BA II+).

2. Bond Pricing:

   • Questions often ask for YTM, current yield, or bond price given coupon, par, and market rate.
   • Tip: Plug in numbers systematically. If YTM is missing, solve iteratively.

3. Annuity vs. Perpetuity:

   • Distinguish between finite annuities (e.g., loan payments) and perpetuities (e.g., preferred stock dividends).
   • Tip: Perpetuity PV = Dividend / Required Return.

4. Real-World Scenarios:

   • Expect 1 question linking TVM to business decisions (e.g., "Should a business lease or buy equipment?").
   • Tip: Always justify answers with NPV or IRR logic.

5. Common Pitfalls:

   • Miscounting periods: Is it monthly (n=12), quarterly (n=4), or annual?
   • Ignoring compounding frequency: Annual vs. semi-annual affects FV/PV.
   • Tip: Clarify the time horizon (e.g., "compounded annually" vs. "monthly").

6. Time Management:

   • Spend 10–15 minutes on each numerical question. Use scrap paper to organize steps.
   • For bonds, list cash flows clearly before discounting.

9. Practice Question (From Past Exams)

Question: City Bank’s 10% coupon bond (par Rs 1,000, 15 years to maturity) sells for Rs 900. What is the current yield? Solution:

1. Current Yield = Annual Coupon / Market Price = (10% × 1,000) / 900 = 11.11%.
2. Note: Current yield ≠ YTM. YTM accounts for capital gains/losses; current yield ignores them.

Answer: 11.11% (rounded to 2 decimal places).