Question

How do you find the present value of the given future payment at the specified interest rate: Rs. 5000 due in two years at $9\dfrac{1}{2}\%$ compounded daily?

Answer 1

Hint: First convert the rate given in mixed fraction into the improper function. Use compound interest formula for the calculation of principal amount P given by: \[A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}\]. Here, A is the amount or future payment, P is the principal/present value, r is the rate per annum, t is the time in years and n is number of times the interest is given in one year. Use the formula: ${{\left( 1+x \right)}^{y}}=1+xy$, if x << 1, to calculate the approx value.

Complete step by step solution:
Here we have been provided with the future amount that will be paid after two years, compounded daily at the rate of $9\dfrac{1}{2}\%$. We are asked to find its present value that means the principal value.
We know that compound interest is given by the formula: \[A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}\], where A is the amount to be paid, t is the number of years, r is the rate, P is the principal/present value and n is the number of times the interest is compounded in a year. It is given that the interest is compounded daily so the number of times it will be compounded in a year will be 365.
We have been given: A = Rs 5000, t = 2 years, n = 365, r = $9\dfrac{1}{2}\%$. Converting r into improper fraction we get r = $\dfrac{19}{2}\%=\dfrac{19}{2\times 100}$. So substituting all the values in the given formula we get,
\[\begin{align}
  & \Rightarrow 5000=P{{\left( 1+\dfrac{19}{2\times 100\times 365} \right)}^{2\times 365}} \\
 & \Rightarrow 5000=P{{\left( 1+\dfrac{19}{73000} \right)}^{730}} \\
 & \Rightarrow P=\dfrac{5000}{{{\left( 1+\dfrac{19}{73000} \right)}^{730}}} \\
\end{align}\]
Here, we can write the denominator in the R.H.S in the form of negative exponent so the expression becomes:
\[\Rightarrow P=5000{{\left( 1+\dfrac{19}{73000} \right)}^{-730}}\]
We can see that $\dfrac{19}{73000}$ << 1 so we can use the binomial relation ${{\left( 1+x \right)}^{y}}=1+xy$ for approximation.
\[\begin{align}
  & \Rightarrow P=5000\left( 1+\left( -730\times \dfrac{19}{73000} \right) \right) \\
 & \Rightarrow P=5000\left( 1-\dfrac{19}{100} \right) \\
 & \Rightarrow P=5000\times \left( \dfrac{81}{100} \right) \\
 & \therefore P=4050 \\
\end{align}\]
Hence, the present value is Rs. 4050.

Note: Note that the value we have obtained in not exact but is approximated. In actuality the value of P will be nearly Rs. 4135. This value will come if you will use the calculator for the calculations because it is almost impossible for us to find the value of ${{730}^{th}}$ power of \[\left( 1+\dfrac{19}{73000} \right)\] in the denominator. If this type of question will be asked then you might be provided with the calculator or the value will be given in the question. In other cases we don’t have any other option but to use the binomial formula.