Time Value of Money
Time Value of Money (TVM)
The fundamental principle of the time value of money is that a sum of money is worth more now than the same sum will be worth at a future date due to its earnings potential in the interim. This potential earning capability is often referred to as the opportunity cost of not having the money now.
1. Future Value (FV)
Future Value is the value of a current asset at a specified date in the future based on an assumed rate of growth.
Formula:
$\qquad FV = PV \times (1 + r)^n$
Where:
- $FV$ = Future Value
- $PV$ = Present Value (the initial amount)
- $r$ = Interest rate per period (expressed as a decimal)
- $n$ = Number of periods (e.g., years)
Example: If you invest ₹1,000 today at an annual interest rate of 5% for 10 years, the future value would be:
$\qquad FV = 1000 \times (1 + 0.05)^{10} \approx ₹1,628.89$
2. Present Value (PV)
Present Value is the current value of a future sum of money or stream of cash flows given a specified rate of return. It’s essentially the reverse of future value.
Formula:
$\qquad PV = \frac{FV}{(1 + r)^n}$
Where:
- $PV$ = Present Value
- $FV$ = Future Value (the amount to be received in the future)
- $r$ = Discount rate per period (expressed as a decimal)
- $n$ = Number of periods
Example: If you are promised to receive ₹1,000 in 5 years, and the discount rate is 8% per year, the present value of that ₹1,000 is:
$\qquad PV = \frac{1000}{(1 + 0.08)^5} \approx ₹680.58$
3. Interest Factor
An interest factor is a multiplier used to calculate the future or present value of a single sum or an annuity. It simplifies TVM calculations, especially when using tables (though less common now with calculators and software).
There are various types of interest factors, including:
- Future Value Interest Factor (FVIF): $(1 + r)^n$ (used in the future value formula)
- Present Value Interest Factor (PVIF): $\frac{1}{(1 + r)^n}$ (used in the present value formula)
You can find these factors in financial tables or calculate them directly using the formulas above.
4. Doubling Period
The doubling period is the time it takes for an investment to double in value at a given interest rate, assuming compounding. A common approximation for this is the Rule of 72.
Approximate Formula (Rule of 72):
$\qquad \text{Doubling Period} \approx \frac{72}{r}$
Where:
- Doubling Period is in years
- $r$ is the annual interest rate (as a percentage, not a decimal)
More Precise Formula:
$\qquad \text{Doubling Period} = \frac{\log(2)}{\log(1 + r)}$
Where:
- $r$ is the interest rate per period (as a decimal)
- $\log$ is the logarithm function (can be natural logarithm or base-10, as long as it’s consistent)
Example: Using the Rule of 72, an investment growing at 8% per year will approximately double in $72 / 8 = 9$ years.
Using the more precise formula:
$\qquad \text{Doubling Period} = \frac{\log(2)}{\log(1 + 0.08)} \approx \frac{0.693}{0.077} \approx 9.01 \text{ years}$
5. Annuity
An annuity is a series of equal payments or receipts made at regular intervals over a specified period.
There are two main types of annuities we often consider in TVM:
- Ordinary Annuity: Payments occur at the end of each period.
- Annuity Due: Payments occur at the beginning of each period.
Future Value of an Ordinary Annuity (FVOA):
$\qquad FVOA = PMT \times \frac{(1 + r)^n - 1}{r}$
Where:
- $FVOA$ = Future Value of the Ordinary Annuity
- $PMT$ = Payment amount per period
- $r$ = Interest rate per period
- $n$ = Number of periods
Present Value of an Ordinary Annuity (PVOA):
$\qquad PVOA = PMT \times \frac{1 - (1 + r)^{-n}}{r}$
Where:
- $PVOA$ = Present Value of the Ordinary Annuity
- $PMT$ = Payment amount per period
- $r$ = Discount rate per period
- $n$ = Number of periods
For an Annuity Due, you simply multiply the formulas for an ordinary annuity by $(1 + r)$:
$\qquad FVAD = FVOA \times (1 + r) = PMT \times \frac{(1 + r)^n - 1}{r} \times (1 + r)$
$\qquad PVAD = PVOA \times (1 + r) = PMT \times \frac{1 - (1 + r)^{-n}}{r} \times (1 + r)$