#### Time Value of Money

The concept of the time value of money is critical in financial planning as it helps in determining the worth of money received or invested at different points in time. The time value of money considers the impact of interest or opportunity cost on the present value of money. The following are some important points related to the time value of money:

• Money available in the present time is worth more than the same amount in the future.

• The preference for receiving money in the present is due to the ability to earn returns on it.

• The time value of money received earlier is higher than that received later.

• Financial decisions often involve cash flows spread over different periods, and the time value of money is a critical principle in such decisions.

The time value of money is calculated using different formulas such as present value, future value, annuity, and perpetuity. These formulas take into account the factors such as interest rate, the period of investment, and the amount invested or received.

Financial planners use the concept of the time value of money to help clients make informed decisions related to investment, retirement planning, loan repayment, and other financial decisions. By understanding the time value of money, investors can calculate the future value of their investments or determine the amount of money they need to save to meet their financial goals.

#### Calculate the following

**Present Value (PV):**

The present value is the current value of future cash flows, discounted at a certain rate. It is the amount that would be required today to generate the same cash flows as the future amount. The present value is calculated to assess the value of an investment or an annuity at a particular point in time.

Formula: PV = FV / (1 + r)^n

Where, FV = Future Value r = Rate of Return n = Number of Years

*Example:* Suppose you are considering investing in a bond that will pay Rs. 10,000 after 5 years. If the interest rate is 8%, what is the present value of the bond?

Using the formula, PV = 10,000 / (1 + 0.08)^5 = Rs. 6,710.81

*Calculation using Excel: *You can use the PV function in Excel to calculate the present value. The PV function syntax is as follows: PV(rate, nper, pmt, [fv], [type])

where, rate = Interest rate per period nper = Total number of payment periods pmt = Payment made each period fv = Future value of the investment (optional) type = Timing of payments (optional, 0 or 1)

To calculate the present value of the bond in Excel, the formula would be: =PV(8%, 5, 0, 10000, 0)

The output of this formula will be Rs. 6,710.81.

**Future Value (FV):**

The future value is the value of an investment or an annuity at a specific point in the future, based on a certain rate of interest or return. It is calculated by estimating the value of the current investment or cash flows in the future, with a certain rate of interest or return.

Formula: FV = PV x (1 + r)^n

Where, PV = Present Value r = Rate of Return n = Number of Years

*Example: *Suppose you invest Rs. 5,000 in a fixed deposit account that pays an annual interest rate of 7% for a period of 5 years. What is the future value of the investment?

Using the formula, FV = 5,000 x (1 + 0.07)^5 = Rs. 7,711.85

*Calculation using Excel*: To calculate the future value of the investment in Excel, you can use the FV function. The syntax for the FV function is as follows: FV(rate, nper, pmt, [pv], [type])

where, rate = Interest rate per period nper = Total number of payment periods pmt = Payment made each period pv = Present value of the investment (optional) type = Timing of payments (optional, 0 or 1)

To calculate the future value of the investment in Excel, the formula would be: =FV(7%, 5, -5000, 0, 0)

The output of this formula will be Rs. 7,711.85.

**Rate of Return:**

Rate of return is the percentage change in the value of an investment over a given period of time. It is the measure of the profitability of an investment and is used to compare different investment opportunities. The rate of return can be calculated using the formula:

Rate of return = (Ending value – Beginning value + Income) / Beginning value

Where, Ending value = the value of the investment at the end of the investment period Beginning value = the value of the investment at the beginning of the investment period Income = the income generated by the investment during the investment period

*Example:*

Suppose you invest Rs. 10,000 in a stock at the beginning of the year and at the end of the year, the value of your investment is Rs. 12,000. Additionally, you received a dividend of Rs. 500 during the year. Then the rate of return on your investment for the year would be:

Rate of return = (12,000 – 10,000 + 500) / 10,000 Rate of return = 25%

*Detailed Steps using Excel:*

- Open a new Excel sheet and enter the beginning value of the investment in cell A1. For example, enter “10000” (without quotes).
- Enter the ending value of the investment in cell A2. For example, enter “12000” (without quotes).
- Enter the income generated by the investment during the investment period in cell A3. For example, enter “500” (without quotes).
- In cell A4, enter the formula for calculating the rate of return: =(A2-A1+A3)/A1
- Press Enter to calculate the rate of return.
- The calculated rate of return will appear in cell A4.
- Periodic Investments or Pay-outs:

4. **Periodic investments or pay-outs **

refer to the regular investments or pay-outs made at a fixed interval of time. These could be monthly, quarterly, semi-annually or annually. The calculation of periodic investments or pay-outs involves the use of the future value of an annuity formula.

Future value of an annuity formula: FV = PMT x [((1+r)^n – 1)/r]

Where, FV = Future Value PMT = Periodic payment r = Rate of return n = Number of periods

*Example*:

Suppose you want to save Rs. 10,000 every year for the next 5 years, and you expect to earn a rate of return of 8% per year on your investments. Then the future value of your investment at the end of 5 years would be:

PMT = Rs. 10,000 r = 8% n = 5

FV = 10,000 x [((1+8%)^5 – 1)/8%] FV = Rs. 63,359

*Detailed Steps using Excel:*

- Open a new Excel sheet and enter the periodic payment in cell A1. For example, enter “10000” (without quotes).
- Enter the rate of return in cell A2. For example, enter “8%” (without quotes).
- Enter the number of periods in cell A3. For example, enter “5” (without quotes).
- In cell A4, enter the formula for calculating the future value of the investment: =FV(A2/12,A3*12,-A1)
- Press Enter to calculate the future value of the investment.
- The calculated future value of the investment will appear in cell A4.

**Period of the loan (NPER):**

NPER or Number of Periods is used to calculate the time required to repay a loan. It is calculated based on the periodic payment, the present value of the loan, the interest rate, and the future value of the loan.

Formula:

=NPER(rate, pmt, pv, [fv], [type])

where, rate: interest rate per period pmt: periodic payment pv: present value fv: future value (optional) type: payment timing (optional, 0 or 1)

*Example:*

Let’s say you want to take a loan of Rs. 5,00,000 for a tenure of 5 years. The interest rate offered by the bank is 10% p.a. compounded annually, and you will be making monthly payments. Calculate the number of payments you need to make to repay the loan.

Solution:

- The interest rate per period = 10%/12 = 0.0083
- The total number of periods = 5 x 12 = 60
- The monthly payment = PMT(rate, nper, pv, [fv], [type]) = PMT(0.0083, 60, 500000) = Rs. 10,718.17 (rounded off)
- The present value of the loan = -500000
- The future value of the loan = 0

Using the formula =NPER(0.0083, -10718.17, 500000, 0, 0) in Excel, we get NPER as 60.

Therefore, the number of payments you need to make to repay the loan is 60 months or 5 years.

**6a. Ordinary Annuity:**

An Ordinary Annuity is a series of fixed payments made at the end of each period for a fixed number of periods. The payments can be made monthly, quarterly, yearly, or any other time interval.

Formula:

=PMT(rate, nper, pv, [fv], [type])

where, rate: interest rate per period nper: number of periods pv: present value or principal amount fv: future value (optional) type: payment timing (optional, 0 or 1)

*Example:*

Let’s say you want to invest in a scheme where you will receive Rs. 10,000 every year for the next 5 years. If the interest rate offered by the scheme is 8% p.a., what is the present value of this investment?

Solution:

- The interest rate per period = 8%
- The number of periods = 5
- The future value of the investment = 0
- The payment amount = Rs. 10,000

Using the formula =PMT(8%, 5, 0, 10000, 0) in Excel, we get a payment of Rs. 2,580.68 (rounded off)

Now, to calculate the present value of the investment, we can use the PV function in Excel.

Formula:

=PV(rate, nper, pmt, [fv], [type])

where, rate: interest rate per period nper: number of periods pmt: periodic payment fv: future value (optional) type: payment timing (optional, 0 or 1)

Using the formula =PV(8%, 5, -2580.68, 0, 0) in Excel, we get the present value of the investment as Rs. 10,000.

Therefore, the present value of the investment is Rs. 10,000.

**6b. Annuity Due:**

An annuity due is an investment that requires periodic payments at the beginning of each period, rather than at the end. Annuities due are often used for leases, insurance, and other financial products. The formula for calculating the present value of an annuity due is similar to that of an ordinary annuity, except that the payments are discounted to the present value at the beginning of each period instead of at the end.

The formula for calculating the present value of an annuity due:

PV = (PMT x (1 – (1 + r)^-n) / r) x (1 + r)

where:

PV = present value of the annuity due

PMT = periodic payment

r = interest rate per period

n = number of periods

*Example:*

Suppose you are considering investing in an annuity due that pays Rs. 5,000 at the beginning of each month for the next 5 years. The annual interest rate for this investment is 8%. What is the present value of this annuity due?

Step 1: Determine the number of periods.

Since the payments are made monthly and the investment is for 5 years, the total number of periods is 5 x 12 = 60.

Step 2: Determine the interest rate per period.

The annual interest rate is 8%, so the monthly interest rate is 8% / 12 = 0.6667%.

Step 3: Calculate the present value of the annuity due.

Using the formula above, we get:

PV = (5000 x (1 – (1 + 0.6667%)^-60) / 0.6667%) x (1 + 0.6667%)

PV = Rs. 245,173.51

Therefore, the present value of the annuity due is Rs. 245,173.51.

**Perpetuity:**

A perpetuity is a type of investment that pays a fixed amount of money to the investor at regular intervals, indefinitely. The formula for calculating the present value of a perpetuity is similar to that of an ordinary annuity, except that there is no defined end date for the payments.

Formula for calculating the present value of a perpetuity:

PV = PMT / r

where:

PV = present value of the perpetuity

PMT = periodic payment

r = interest rate per period

*Example:*

Suppose you are considering investing in a perpetuity that pays Rs. 10,000 every year, indefinitely. The annual interest rate for this investment is 6%. What is the present value of this perpetuity?

Using the formula above, we get:

PV = 10,000 / 6%

PV = Rs. 166,666.67

Therefore, the present value of the perpetuity is Rs. 166,666.67.