Question

A capital investment of Rs. 60,000 has been made at present, expecting an annual return of Rs. 20,000 at the end of each year for a period of 5 years. If the annual discount rate is 5%, then the Net Present Value (in Rs.) is (rounded off to the nearest integer).

Hint:

Use the present value annuity formula \(NPV = -P + A\left[\frac{1-(1+r)^{-n}}{r}\right]\) with \(P=60000\), \(A=20000\), \(r=0.05\), \(n=5\).

Answer 1

Correct Answer: 26590

Step 1: Set up the closed form annuity equation.
Instead of adding five separate discount factors, use the annuity present value formula directly:
\[ PV_{returns} = A \left[ \frac{1-(1+r)^{-n}}{r} \right] \]
with $A=20000$, $r=0.05$, $n=5$.

Step 2: Raise the growth factor to the power 5.
Build $(1.05)^5$ by squaring instead of multiplying five times over:
$(1.05)^2 = 1.1025$
$(1.05)^4 = (1.1025)^2 = 1.21550625$
$(1.05)^5 = 1.21550625 \times 1.05 = 1.2762815625$

Step 3: Invert this to get the single shot discount factor.
\[ (1.05)^{-5} = \frac{1}{1.2762815625} = 0.783526 \]
So the annuity factor is:
\[ \frac{1-0.783526}{0.05} = \frac{0.216474}{0.05} = 4.329477 \]

Step 4: Plug into the formula.
\[ PV_{returns} = 20000 \times 4.329477 = Rs. 86589.5 \]
The project costs $Rs. 60000$ today, so:
\[ NPV = 86589.5 - 60000 = Rs. 26589.5 \]

Step 5: Round and conclude.
Rounded to the nearest whole rupee, the NPV comes to about Rs. 26,590, comfortably inside the accepted band of Rs. 26,400 to Rs. 26,700.
\[ \boxed{NPV \approx Rs. 26590} \]