The Present Value is a core dimension of the Time Value of Money. It is defined as the current value (the value at time “O”) of a future amount of money or a series of future cash flows, evaluated at a given interest rate. The fundamental principle is that money received in the future is not worth as much as an equal amount received today. This difference in value across time periods arises because money received today can be invested to earn a return, or because of factors like risk and preference for current consumption.
The process of determining the present value of a future payment or a series of future payments is called Discounting. Discounting is essentially the reverse of compounding. It involves comparing an initial outflow with the sum of the present values (PV) of future inflows at a given rate of interest, specifically at time “O”. Discounting explicitly takes into account the time value of money.
Present Value calculations can be applied to several circumstances:
- A single flow.
- Un-even Multiple Flows.
- An Annuity.
- A Perpetuity.
Present Value of a Single Amount
The present value of a future cash inflow or outflow is the amount of current cash flow that is equivalent in desirability, to the decision maker, to a specified amount of cash to be received or paid at the future date. It is the current value of a future amount of money calculated at a given interest rate.
The formula for calculating the present value (PV) of a single future amount (FVn) received ‘n’ periods from now, at an interest rate ‘i’ (or ‘k’ or ‘r’), is given as: PV = FVn / (1+i)n or PV = FVn * (1/1+i)n or PV = FVn * (1+r)-n or PV = FV * PVIF(r,n)
Here:
- PV = Present Value
- FV = Future Value (FVn represents the future value in year ‘n’)
- i, k, or r = interest rate, discount rate, or required rate of return
- n = number of periods or duration
The term (1/(1+i)n) is called the discounting factor or the present value interest factor (PVIFi,n). Tables exist that provide the value of PVIF(i,n) for various combinations of interest rates (‘i’) and periods (‘n’). Examples of such tables are provided in the sources. For instance, the PVIF for Re 1 at 10% for 1 period is 0.909.
Examples from the sources illustrate the calculation of present value for a single amount:
- Suppose you are promised ₹1000 six years hence, and the interest rate is 10%. The present value is calculated as 1000 * (1/1.10)6 = 1000 * (1/1.771561) = 1000 * 0.56449 = 564.5.
- Similarly, the present value of ₹1000 receivable 6 years hence at a 10% discount rate is 1000 * PVIF10%, 6 = 1000 * 0.5645 = 564.5.
- The present value of ₹1000 receivable 3 years hence at a 10% interest rate is calculated by discounting year by year: Year 3 value is ₹1000, Year 2 value is ₹1000 * (1/1.10), Year 1 value is ₹1000 * (1/1.10)2, and the Present Value (Value now) is 1000 * (1/1.10)3 = 1000 * 0.751 = 751.
- The present value of ₹1000 receivable 20 years hence if the discount rate is 8 percent is 1000 * (1/1.08)20. This can also be calculated using PVIF values, for example, 1000 * PVIF8%,10 * PVIF8%,10 = (1000 * 0.463 * 0.463) = 214.
For calculations involving discounting multiple times within a year (e.g., semi-annually, quarterly), the formula is modified to include ‘m’ (number of times discounting is done per year).
Present Value of an Annuity
An annuity is defined as a series of equal payments or receipts occurring over a specified number of periods. The time period between two successive payments is called the payment period or rent period. While the word ‘annuity’ often implies annual payments, it can also refer to semi-annual, quarterly, or other frequencies.
The present value of an annuity (PVAn) is the sum of the present values of each individual cash flow in the annuity stream. For an ordinary annuity where cash flows occur at the end of each period, the formula for its present value is given as: PVAn = A * [{(1+k)n – 1} / {k(1+k)n}] or PVAn = A * [{1 – (1/(1+r)n)}/r] or S = A * [{1 – (1+i)-n}/i] or pn = R * [{(1+i)n – 1} / {i(1+i)n}] or P = C * PVIFAr,n (when C is the constant periodic flow)
Here:
- PVAn, S, or pn = Present value of the annuity
- A, R, or C = Constant periodic flow or amount of each instalment
- k, r, or i = Discount Rate or interest rate per period
- n = Number of periods or duration of the annuity
The term [{1- (1/1+r)n}/r] or similar variations is called the present value interest factor for an annuity. Tables for the Present Value Interest Factor of an (ordinary) Annuity of Re 1 per period at i% for n periods, PVIFA(i,n), are provided.
An example from the sources calculates the present value of a 4-year annuity of ₹10000 discounted at 10%. Using the formula: PVAn = 10000 * [{(1.10)4 – 1} / {0.10(1.10)4}] = 10000 * [{(1.4641) – 1} / {0.10(1.4641)}] = 10000 * [0.4641 / 0.14641] = 10000 * 3.17008 = 31700. Another example shows the calculation for receiving ₹1000 annually for 3 years at a 10% discount rate by summing the present values of each individual payment: [1,000 * (1/1.10) + 1,000 * (1/1.10)2 + 1,000 * (1/1.10)3] = [1,000 * 0.9091 + 1,000 * 0.8264 + 1,000 * 0.7513] = 2,486.70.
Present Value of a Perpetuity
A perpetuity is a special type of annuity where the stream of cash flows continues indefinitely. Despite the infinite duration, the value of a perpetuity is finite. This is because cash flows expected far into the future have extremely low present values when discounted. Examples of perpetuities include fixed coupon payments on permanently invested (irredeemable) sums of money or perpetual scholarships paid from an endowment fund.
The present value of a perpetuity is calculated using the formula: P∞ = A / r Or Present value of perpetuity = Perpetuity / Interest rate
Here:
- P∞ = Present value of perpetuity
- A = Constant periodic flow (Perpetuity)
- r = Discount rate or Interest rate
Applications of Present Value
The concept of present value is essential for applying discounted cash flow techniques. These techniques, such as Net Present Value (NPV), Internal Rate of Return (IRR), Profitability Index (PI), and Discounted Payback Period, all rely on discounting future cash flows to their present value to make them comparable to current investments or other cash flows occurring at different points in time.
Net Present Value (NPV) is a key application. NPV is defined as the difference between the present value of cash inflows and the present value of cash outflows over a period of time. It is used in capital budgeting to analyze the profitability of a projected investment. The NPV calculation involves determining the net cash inflow in each year, selecting a discount rate (often the cost of capital), finding the discount factor for each year, determining the present values of the net cash flows, and summing them up. A positive or zero NPV suggests accepting the project, while a negative NPV suggests rejecting it. The NPV method correctly postulates that cash flows arising at different time periods differ in value and are comparable only with their present values.
Other applications mentioned in the sources include:
- Valuation of preference shares. The value of redeemable preference shares is the present value of future expected dividend payments and maturity value, discounted at the required return.
- Valuation of bonds. The price of a bond is the present value of its future coupon payments (often treated as an annuity) and the final face value payment, discounted at the appropriate yield.
- Leasing decisions. Comparing lease vs. buy options involves calculating the present value of the cash flows associated with each option, such as loan repayments, interest, tax shields, and residual values.
- Evaluating credit policies in working capital management, which involves considering the opportunity cost of investment in receivables, calculated using present value principles.
- Certainty Equivalent Approach in risk analysis, which adjusts cash flows using certainty equivalent coefficients and then discounts them at the risk-free rate to find the NPV.
In summary, Present Value is a fundamental concept in finance, representing the current worth of future money or cash flows. It is calculated through the process of discounting, using formulas that incorporate the future value, the discount rate, and the number of periods. This concept extends to single sums, annuities (series of equal payments), and perpetuities (annuities that last forever). Present Value calculations are vital for evaluating investments, valuing assets, and making sound financial decisions across various areas of financial management.