The time value of money (TVM) is a fundamental concept in finance that recognizes the idea that a sum of money today has a different value than the same sum in the future. The time value of money is influenced by two key processes: compounding and discounting.

1. Compounding:
   • Definition: Compounding refers to the process by which an investment’s earnings, as well as the interest on those earnings, earn interest over time.
   • Formula: The future value (FV) of an investment compounded annually can be calculated using the formula:
     
     $\mathrm{FV}=\mathrm{PV}\times (1+R\mathrm{<br>}{)}^{\mathrm{n<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;\; font-size:\; 16px;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal"></span></span></span></span></span>}}$

   • 
     $FV=PV×(1+r)n where:$

     
     $\mathrm{<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal"></span></span></span></span></span>}$

   • 
     $\mathrm{<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">F</span><span\; class="mord\; mathnormal">V</span></span></span></span></span><span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;">\; is\; the\; future\; value\; of\; the\; investment,</span>}$

     • PV is the present value (initial investment),

     • r is the interest rate per compounding period, and

       n is the number of compounding periods.

   • Example: If you invest $1,000 at an annual interest rate of 5% compounded annually for 3 years, the future value would be calculated as:
     
     $\mathrm{FV}\mathrm{<br>}=1000\times (1+0.05{)}^3$

     FV=1000×(1+0.05)3

2. Discounting:
   • Definition: Discounting is the process of determining the present value of a future sum of money, considering the time value of money.
   • Formula: The present value (PV) of a future sum discounted annually can be calculated using the formula:
     
     $\mathrm{PV}\mathrm{<br>}=\frac{\mathrm{FV}}{(1+{<span\; style="text-transform:\; math-auto;">r<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;\; font-size:\; 16px;">)</span></span>}^n}$

     PV=(1+r)nFV​ where:

     PV is the present value,

     • FV is the future value,
       

       r is the discount rate per period, and

       n is the number of periods.

   • Example: If you want to determine the present value of $1,000 to be received in 3 years with a discount rate of 5%, the present value would be calculated as:
     
     $\mathrm{PV}=\frac{1000}{(1+0.05{)}^3}$

     PV=(1+0.05)31000​

Both compounding and discounting are crucial in various financial calculations, such as determining the future value of investments, assessing the present value of future cash flows, and making informed decisions about borrowing, lending, or investing. These processes are fundamental to time-sensitive financial decisions, allowing individuals and businesses to evaluate the true worth of money over different time periods.