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Time Value of Money: Future value, Present value, Annuities Formulas and Important Numerical

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Introduction to Time Value of Money (TVM)

The Time Value of Money (TVM) is one of the most important concepts in finance.
It states that money available today is worth more than the same amount in the future because it has the potential to earn interest.

Why is money today more valuable?

Because you can invest money now and earn a return and everything else in future is uncertain.
Interest is the reward for waiting or the cost of borrowing money.

Important Terms

  • Principal (P): The original amount of money invested or lent.

  • Interest Rate (i): The percentage charged or earned per time period (e.g., 6% per year).

  • Time (n): Number of periods the money is invested.

  • Future Value (FV): Amount an investment grows to in the future.

  • Present Value (PV): Current value of a future amount.

Future Value (Compounding)

Future Value tells us how much an investment made today will be worth in the future.

1. Simple Interest

Interest is earned only on the original principal.

Simple Interest = Principal × Rate × Time
 

2. Compound Interest

Interest is earned on principal + previous interest (the power of compounding).

Formula: Future Value of a Single Amount

FV = PV (1 + i)^n
 

Where:

  • FV = Future Value

  • PV = Present Value

  • i = Interest Rate per period

  • n = Number of periods

Note: More frequent compounding (monthly/quarterly) results in a higher future value.

Present Value (Discounting)

Present Value is the current worth of a future sum of money.

It answers:

How much should I invest today to receive ₹X in the future?

Formula

PV = FV / (1 + i)^n
 
 

Example on Present value of Money

Alpha Company can invest at 16% interest compounded annually, while Beta Company can invest at 16% interest compounded semi-annually.
Both companies need ₹2,00,000 after 4 years. We will calculate how much each must invest today (Present Value).

Solution

1. Alpha Company – Annual Compounding

  • Time period (n) = 4 years

  • Interest rate (i) = 16% per year

Formula:

PV = Future Value × 1 / (1 + i)ⁿ

 

Calculation

PV = 2,00,000 × 1 / (1.16)⁴
PV = 2,00,000 × 0.55229
PV = ₹1,10,458
 

2. Beta Company – Semi-Annual Compounding

  • Compounding frequency = 2 times per year

  • Time period (n) = 4 × 2 = 8

  • Interest rate per period (i) = 16% ÷ 2 = 8%

Formula:

PV = Future Value × 1 / (1 + i)⁸

Calculation

PV = 2,00,000 × 1 / (1.08)⁸
PV = 2,00,000 × 0.54027
PV = ₹1,08,054
 

Conclusion

Beta Company needs to invest less today because its investment grows faster due to more frequent compounding.

Annuities (Series of Equal Payments)

An annuity is a series of equal payments made at regular intervals of time (e.g., rent, insurance, pension).

1. Future Value of an Ordinary Annuity

The value of all future payments plus interest earned.

FV_annuity = R × [((1 + i)^n – 1) / i]
 

Where R = periodic payment (rent)

2. Present Value of an Ordinary Annuity

The lump sum needed today to provide equal future payments.

PV_annuity = R × [(1 – (1 + i)^(-n)) / i]
 
 

Example: Future Value of an Ordinary Annuity

In the beginning of 2006, the directors of Molloy Corporation decided that the plant facilities must be expanded in a few years.
The company plans to invest ₹50,000 every year, starting on June 30, 2006, into a trust fund that earns 11% interest compounded annually.

Question

How much money will be in the fund on June 30, 2010, after the last deposit has been made?

Solution

  • The first deposit is made at the end of the first year, so it is an ordinary annuity

  • Number of periods (n) = 5 years

  • Interest rate (i) = 11%

  • The last deposit (2010) earns no interest because it is deposited on the final day

From the Future Value of Ordinary Annuity Table,
Future Value Factor (at n = 5, i = 11%) = 6.22780

Formula

FV = Rent × Future Value Annuity Factor

Calculation

FV = 50,000 × 6.22780
FV = ₹3,11,390
 

Conclusion

Molloy Corporation will have ₹3,11,390 in the fund on June 30, 2010.

Finding the Required Annual Deposit

If the company needs exactly ₹3,00,000 on June 30, 2010, how much must it deposit every year?

Formula

Rent = FV / Future Value Annuity Factor

Calculation

Rent = 3,00,000 / 6.22780
Rent = ₹48,171.10
 

Final Answer

The company must deposit ₹48,171 every year to accumulate ₹3,00,000 by June 30, 2010.

Perpetuities

A Perpetuity is an annuity that pays forever (infinite life).
Examples: perpetual bonds, preferred stock dividends.

Formula

PV = C / i

Where C = constant annual payment

 

Growth Calculations & Doubling Money

1. Compound Annual Growth Rate (CAGR)

gr = (Vn / V0)^(1/n) – 1
 

2. Rule of 72

A shortcut to estimate how long it takes to double money.

Years = 72 / Interest Rate
 

3. Rule of 69 (continuous compounding)

Doubling Period = 0.35 + (69 / Interest Rate)
 

Summary of Important Formulas

Concept Formula Future Value (Single Amount) FV = PV (1 + i)^n Present Value (Single Amount) PV = FV / (1 + i)^n PV of Perpetuity PV = C / i Doubling Period (Rule of 72) Years = 72 / i

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What is Time Value of Money? Future value, Present value, Annuity explained for BBA/MBA Students

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Ever heard the saying, “a rupee today is worth more than a rupee tomorrow”? That’s the single most important idea in all of finance. It’s called the Time Value of Money (TVM), and mastering it is essential for your BBA or MBA exams and your future career.

But what does it really mean?

Simply put, money you have right now is more valuable than the exact same amount in the future. Why? Because you can invest it. That ₹100 you have today could be put in a bank to earn interest, turning it into ₹105 by next year.

This article will break down all the core TVM concepts, formulas, and examples you need to know.

1. Simple vs. Compound Interest: The Two Ways Money Grows

First, let’s understand interest. It’s the fee you pay for borrowing money or the reward you get for saving it. The starting amount is called the Principal (P). (the amount which you give to the bank and gets interest onto it)

What is Simple Interest?

Simple interest is the most basic form. It’s calculated only on the original principal amount.

  • Simple Interest Formula: Interest = P * i * n

    • P = Principal

    • i = Interest rate

    • n = Number of time periods

  • Example: You lend ₹10,000 at 3% simple interest per quarter.

    • Interest per quarter: ₹10,000 * 0.03 * 1 = ₹300

    • Total interest over 5 years (20 quarters): ₹300 * 20 = ₹6,000

What is Compound Interest?

This is where the magic happens. Compound interest is calculated on the principal plus all the interest you’ve already earned. You start earning “interest on your interest.”

  • Example: You invest ₹1,000 at 6% interest compounded annually.

    • End of Year 1: ₹1,000 * 1.06 = ₹1,060

    • End of Year 2: ₹1,060 * 1.06 = ₹1,123.60

      • Notice you earned ₹63.60 in year 2, not just ₹60. That extra ₹3.60 is 6% interest on the ₹60 you earned in year 1.

Key Takeaway: The more frequently interest is compounded (e.g., monthly vs. annually), the faster your money grows.

2. Future Value (FV): What Will Your Money Be Worth?

Future Value (FV) tells you what a single amount of money today will be worth at a specific point in the future. This process is called compounding.

This answers the question: “If I invest ₹X today, how much will I have in 5 years?”

Future Value Formula (for a Single Amount)

FV = P * (1 + i)^n

  • FV = Future Value

  • P = Principal (or Present Value)

  • i = Interest rate per period

  • n = Number of periods

FV Example

  • Problem: A company invests ₹40,00,000 for 5 years at 16% per year, compounded semi-annually. What is the future value?

  • Solution:

    • P: ₹40,00,000

    • i: 16% / 2 = 8% per period (0.08)

    • n: 5 years * 2 periods per year = 10 periods

    • Calculation: FV = 40,00,000 * (1 + 0.08)^10

    • FV = ₹86,35,680

3. Present Value (PV): What is Future Money Worth Today?

Present Value (PV) is the exact opposite of FV. It tells you what a future amount of money is worth in today’s terms. This process is called discounting.

This answers the question: “I need to have ₹5,00,000 in 10 years. How much do I need to invest today to get there?”

Present Value Formula (for a Single Amount)

PV = FV / (1 + i)^n

  • PV = Present Value

  • FV = Future Value (the amount you want in the future)

  • i = Discount rate (interest rate) per period

  • n = Number of periods

PV Example

  • Problem: You need ₹2,00,000 in 4 years. How much must you invest today if the interest rate is 16% compounded semi-annually?

  • Solution:

    • FV: ₹2,00,000

    • i: 16% / 2 = 8% per period (0.08)

    • n: 4 years * 2 periods per year = 8 periods

    • Calculation: PV = 2,00,000 / (1 + 0.08)^8

    • PV = ₹108,054

      • This means ₹108,054 invested today at 8% per period will grow to ₹2,00,000 in 4 years.

4. Annuities: The Power of Equal Payments

An Annuity is simply a series of equal payments made at equal time intervals. Think of a car loan, a mortgage payment, or a regular monthly saving plan.

  • Ordinary Annuity: Payments are made at the end of each period (this is the most common type).

Future Value of an Annuity (FVA)

This tells you the total value of a stream of regular payments at a future date.

  • Formula: FVA = R * [((1 + i)^n - 1) / i]

  • R = Rent (the amount of each equal payment)

  • Example: You invest ₹50,000 every year for 5 years in a fund that earns 11% annually. How much will you have at the end?

    • R: ₹50,000

    • i: 0.11

    • n: 5

    • Calculation: FVA = 50,000 * [((1 + 0.11)^5 - 1) / 0.11]

    • FVA = ₹311,390

Present Value of an Annuity (PVA)

This tells you the value today of a series of future payments. This is how loan amounts are calculated and how lottery jackpots are valued.

  • Formula: PVA = R * [(1 - (1 + i)^-n) / i]

  • R = Rent (the amount of each payment)

  • Example: A retirement plan offers you ₹5,00,000 per year for 32 years. The interest rate is 9%. How much would you need to pay for this plan today?

    • R: ₹5,00,000

    • i: 0.09

    • n: 32

    • Calculation: PVA = 5,00,000 * [(1 - (1 + 0.09)^-32) / 0.09]

    • PVA = ₹52,04,000 (approx.)

5. Perpetuities: An Annuity That Lasts Forever

A Perpetuity is a special type of annuity where the equal payments continue forever.

Present Value of a Perpetuity Formula

There is no future value (it’s infinite!), but the present value is surprisingly simple.

PV = C / i

  • C = Cash flow (payment) per period

  • i = Interest rate (or discount rate)

  • Example: What is the present value of an investment that pays ₹500 every year, forever, if the discount rate is 5%?

    • C: ₹500

    • i: 0.05

    • Calculation: PV = 500 / 0.05 = ₹10,000

      • This means ₹10,000 invested today at 5% would pay you ₹500 every year without ever touching the principal.

6. Quick Tools: Growth Rate & The Doubling Rules

How to Calculate Compound Growth Rate

This formula finds the average annual growth rate (r) of something (like sales or dividends) over time.

  • Formula: Vn = Vo * (1 + r)^n

    • Vn = Ending Value

    • Vo = Starting Value

    • n = Number of periods

    • (You must solve for r)

  • Example: A dividend grows from ₹21 to ₹31 over 5 years. The growth rate (r) is 8.09%.

Quick Doubling Period Rules

Need a fast way to estimate how long it takes for an investment to double? Use these rules.

  1. Rule of 72: A simple and popular estimate.

    • Formula: Years to Double ≈ 72 / i (use the whole number interest rate)

    • Example: At 10% interest, your money will double in 72 / 10 = 7.2 years.

  2. Rule of 69: A slightly more accurate formula.

    • Formula: Years to Double ≈ 0.35 + (69 / i)

    • Example: At 10% interest: 0.35 + (69 / 10) = 7.25 years.

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