# Present Value and Discounting

##### Basic Concepts

Interest is often termed as the time value of money. The basic idea of interest is quite familiar.  If you invest ₹1000 in a bank account that pays you 8% interest per year, then at the end of one year you will have in your account the principal or original amount plus interest at 8% for the total of ₹1080. If interest rate is r, expressed as a decimal, then your initial investment would be multiplied by (1+r) after one year.

Under a simple interest rule, money invested for a period different from one year accumulates interest proportional to the total time of the investment. So after 2 years, the total interest due is 2r times the original investment and so forth. In other words, the investment produces interest equal to $$r$$ times the original investment every year. Usually, partial years are treated in a proportional manner; that is, a fraction $$f$$ of 1 year, interest of $$r.f$$ times the original investment is earned. The general rule for simple interest is that of an amount ‘x’ is left in an account at simple interest r, the value after n years is

V = (1+rn) x

If the proportional rule holds for fractional years, then after any time t (measured in years), the account value is

V = (1+rt) x

The account grows linearly with time as shown above; the account value at any time is just the sum of principal amount and the accumulated interest, which is proportional to time.

If we talk about Compound Interest most banks and loans employ some form of compounding- producing compounding interest. Again consider an account that pays interest at a rate of r per year. If interest is compounded yearly, then after one year, the first year’s interest is added to the original principal to define a larger principal base for the second year. Thus during the second year, the account earns interest on interest. This is the compounding effect, which is continued year after year.

Under yearly compounding, money left in an account is multiplied by (1+r) after one year. After the second year, it grows by another factor of $$\left(1 +r \right)$$ to $$\left(1 +r \right)^{2}$$. After n years, such an account will grow to $$\left( 1 + r \right)^{n}$$ times its original value, and this is the analytic expression for the account growth under compound interest. This expression is said to exhibit geometric growth because of the $$n^{th}$$ -power form.

Figure 1: Simple Interest leads to linear growth over time whereas compound interest leads to an accelerated increase defined by geometric growth. The figure shows both cases for an interest rate of 10%.

Figure 1 shows a graph of a $100 investment over time when it earns 10% interest under simple and compound interest rules. The figure shows the characteristic shapes of linear growth tor simple interest and of accelerated upward growth for compound interest. Note that under compounding, the value doubles in about 7 years. There is a cute little rule that can be used to estimate the effect of interest Compounding (More exactly, at 7% and 10 years, account increases by a factor of 1.97, whereas at 10% and 7 years it increases by a factor of 1.95) • The Seven Ten rule- Money invested at 7% per year doubles in approximately 10 years. Also, money invested at 10% per year doubles in approximately 7 years. #### PRESENT VALUE As discussed above that money invested today leads to increased value in the future as a result of interest. The formulas above show how to determine this future value. That whole set of concepts and formulas can be reversed in time to calculate the value that should be assigned now, in the present, to money that is to be received at a later time. This reversal is the essence of the extremely important concept of present value. To introduce this concept, consider two situations: (1) You will receive$110 in 1 year; (2) you receive $100 now and deposit it in a bank account for 1 year at 10% interest. Clearly, these situations are identical after 1 year you will receive$110. We can restate this equivalence by saying that $110 received in 1 year is equivalent to the receipt of$100 now when the interest rate is 10% or we say that the $110 to be received in 1 year has a present value of$100 in general, $1 to be received a year in the future has a present value of$1/(1+ r), where r is the interest rate.

A similar transformation applies to future obligations such as the repayment of debt. Suppose that, for some reason, you have an obligation to pay someone $100 in exactly 1 year. This obligation can be regarded as a negative cash flow that occurs at the end of the year. To calculate the present value of this obligation, you determine how much money you would need now in order to cover the obligation. This is easy to determine. If the current yearly interest rate is r, you need$100/(1+r). If that amount of money is deposited in the bank now, it will grow to $100 at the end of the year. You can then fully meet the obligation. The present value of the obligation is therefore$100/(1+r).

The process of evaluating future obligations as an equivalent present value is alternatively referred to as discounting. The present value of a future monetary amount is less than the face value of that amount, so the future value must be discounted to obtain the present value. The factor by which the future value must be discounted is called the discount factor. The 1 year discount factor is $$dt\; =\; \frac{1}{\left( 1\; +\; r \right)}$$ where r is the 1-year interest rate. So if an amount A is to be received in 1 year, the present value is the discounted amount $$d_{k}A$$.

The formula for present value depends on the interest rate that is available from a bank or other source. If that source quotes rates with compounding, then such a compound interest rate should be used in the calculation of present value. As an example, suppose that the annual interest rate, is compounded at the end of each of m equal periods each year; and suppose that a cash payment of amount A will be received at the end of the $$k^{th}$$ period, Then the appropriate discount factor is

$$d_{k}\; =\; \frac{1}{\left[ 1\; +\; \left( \frac{r}{m} \right) \right]^{k}}$$

The present value of a payment of A to be received k periods in the future is $$d_{k}A$$.

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