Present Value (PV)

Present Value (PV)

Last Updated on 2025-01-22 by Admin

 

Discreet Compounding

 

The formula for the present value with discrete compounding is:

 

$$PV=\frac{FV}{\left( 1+\frac{r}{n} \right)^{nt}}$$

 

Where:

• PV = Present value
• FV = Future value
• r = Interest rate (as a decimal)
• n = Number of compounding periods per year
• t = Time in years

 

This formula calculates the present value of a future cash flow, discounted at an interest rate r, with n compounding periods per year, over a period of t years.

 

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Continuous Compounding

 

The formula for the present value with continuous compounding is:

 

$$PV=FV⋅e^{−𝑟𝑡}$$

$$PV = P \cdot e^{-rt}$$

 

Where:

• PV = Present Value
• FV = Future Value
• r = interest rate (decimal)
• t = time (years)
• e = Euler’s number (approximately 2.71828)

 

This formula calculates the present value of a future cash flow when interest is compounded continuously at a rate r over time t.

 

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Present Value (Discreet Compounding)

 

 

 

 

 

 

 

 

 

 

The given formula is:

$$F_{0}(T) = S_{0}*(1+r)^T$$

 

To solve for

$T$

T, we can follow these steps:

(1) Divide both sides of the equation by

$S_0$

S0​:

$$\frac{F_{0}(T)}{S_{0}} = (1+r)^T$$

 

(2) Take the natural logarithm (ln) of both sides:

$$ln\left( \frac{F_{0}(T)}{S_{0}} \right) = ln\left( (1+r)^T \right)$$

 

(3) Use the logarithmic identity

$\ln(a^b) = b \ln(a)$

ln(ab)=bln(a):

$$ln\left( \frac{F_{0}(T)}{S_{0}} \right) = T*ln(1+r)$$

 

Finally, solve for

$T$

T by dividing both sides by

$\ln(1 + r)$

ln(1+r):

$$T=\frac{ln\left( \frac{F_{0}(T)}{S_{0}} \right)}{ln(1+r)}$$

 

So the formula to find T is:

$$T=\frac{ln\left( \frac{F_{0}(T)}{S_{0}} \right)}{ln(1+r)}$$

 

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