Time Value of Money: Understanding Present & Future Value, Annuities & More #Finance #Investment #TimeValueOfMoney

 

Lecture on Time Value of Money (TVM) – Comprehensive with Detailed Numerical Examples

The Time Value of Money (TVM) is a fundamental financial concept that states that a dollar today is worth more than a dollar in the future due to its earning potential. This principle plays a crucial role in investment analysis, loan amortization, retirement planning, and financial decision-making.

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1. Concepts of Time Value of Money (TVM)

1.1 Present Value (PV) – Discounting Future Cash Flows

The Present Value (PV) refers to the current worth of a future sum of money, given a certain discount rate. This process is called discounting.

Formula for Present Value

$$PV = \frac{FV}{(1 + r)^t}$$

Where:

• $PV$ = Present Value
• $FV$ = Future Value
• $r$ = Discount rate (interest rate) per period
• $t$ = Number of periods

Numerical Example 1: Present Value of a Single Cash Flow

A company expects to receive $5,000 in 4 years, and the discount rate is 10%. What is the present value?

$$PV = \frac{5000}{(1.10)^4}$$ $$PV = \frac{5000}{1.4641} = 3,416.49$$

So, $5,000 received in 4 years is worth $3,416.49 today at a 10% discount rate.

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Numerical Example 2: Comparing Present Value at Different Interest Rates

You will receive $1,000 in 5 years. Compare the present value when the discount rate is 5% vs. 10%.

For 5% rate:

$$PV = \frac{1000}{(1.05)^5} = \frac{1000}{1.2763} = 783.53$$

For 10% rate:

$$PV = \frac{1000}{(1.10)^5} = \frac{1000}{1.6105} = 620.92$$

As the discount rate increases, the present value decreases.

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1.2 Future Value (FV) – Compounding Money Over Time

The Future Value (FV) tells us how much an investment today will grow over time.

Formula for Future Value

$$FV = PV \times (1 + r)^t$$

Numerical Example 3: Future Value of an Investment

If you invest $2,000 today at 8% annual interest for 6 years, how much will you have?

$$FV = 2000 \times (1.08)^6$$ $$FV = 2000 \times 1.5869 = 3,173.80$$

So, after 6 years, your investment will grow to $3,173.80.

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Numerical Example 4: Future Value with Different Interest Rates

Compare the future value of $2,000 after 5 years at 6% vs. 12% interest rates.

For 6% rate:

$$FV = 2000 \times (1.06)^5 = 2000 \times 1.3382 = 2,676.40$$

For 12% rate:

$$FV = 2000 \times (1.12)^5 = 2000 \times 1.7623 = 3,524.60$$

A higher interest rate results in a higher future value.

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1.3 Annuities – Series of Equal Cash Flows

1.3.1 Future Value of an Ordinary Annuity

This is used when equal payments are made at the end of each period.

Formula for Future Value of an Annuity

$$FV_A = P \times \frac{(1 + r)^t - 1}{r}$$

Numerical Example 5: Future Value of Monthly Savings

You deposit $500 at the end of each year into a savings account earning 7% interest for 5 years.

$$FV_A = 500 \times \frac{(1.07)^5 - 1}{0.07}$$ $$FV_A = 500 \times \frac{1.4026 - 1}{0.07} = 500 \times 5.751$$ $$FV_A = 2,875.50$$

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1.3.2 Present Value of an Annuity (PV of Annuity)

The Present Value of an Annuity determines today’s worth of a series of future cash flows.

Formula for Present Value of an Annuity

$$PV_A = P \times \frac{1 - (1 + r)^{-t}}{r}$$

Numerical Example 6: Present Value of a Pension

You will receive $10,000 per year for 8 years, and the discount rate is 5%. What is its present value?

$$PV_A = 10000 \times \frac{1 - (1.05)^{-8}}{0.05}$$ $$PV_A = 10000 \times \frac{1 - 0.6768}{0.05} = 10000 \times 6.464$$ $$PV_A = 64,640$$

So, the present value of the annuity is $64,640.

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1.4 Perpetuities – Annuities That Last Forever

A perpetuity is a type of annuity with infinite payments.

Formula for Perpetuity

$$PV = \frac{P}{r}$$

Numerical Example 7: Present Value of a Perpetual Income

A university receives $2,000 annually from an endowment fund, and the discount rate is 8%. What is the present value of this fund?

$$PV = \frac{2000}{0.08} = 25000$$

So, the perpetuity is worth $25,000 today.

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1.5 Effective Interest Rate (EIR) – True Interest Rate Considering Compounding

The Effective Interest Rate (EIR) accounts for compounding within a year.

Formula for EIR

$$EIR = \left(1 + \frac{r}{n} \right)^n - 1$$

Where:

• $r$ = Stated annual interest rate
• $n$ = Number of compounding periods per year

Numerical Example 8: Comparing Compounded Interest Rates

A bank offers 10% annual interest, compounded monthly. What is the EIR?

$$EIR = \left(1 + \frac{0.10}{12} \right)^{12} - 1$$ $$EIR = \left(1.0083 \right)^{12} - 1$$ $$EIR = 1.1047 - 1 = 10.47\%$$

So, the effective interest rate is 10.47%.

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Conclusion

The Time Value of Money (TVM) is a core concept in finance, affecting investment, retirement, and financial planning.

✅ Present Value (PV): Determines today's worth of future money.
✅ Future Value (FV): Measures how money grows over time.
✅ Annuities: Series of equal cash flows over time (Ordinary, Due, Perpetuity).
✅ Effective Interest Rate (EIR): True cost of borrowing/lending money considering compounding.

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