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## Q. 5.8

Solve the problem given in Example 5.2 by the I.R.R. method.

## Verified Solution

Step 1: The cash flow diagram for the given problem is shown in Fig. 5.2.
Step 2: The P.W. of the net receipts at an interest rate of $i^{\prime }$ is calculated as:

$P.W.=₹53,000\left(P/A,i^{\prime}\% ,5\right)+₹30,000\left(P/F,i^{\prime}\% ,5\right)$

Step 3: The P.W. of the net expenditures at an interest rate of $i^{\prime }$ is calculated as:

$P.W.=₹1,05,815.4+₹30,000\left(P/A,i^{\prime}\% ,5\right)$

Step 4: The net present worth N.P.W. is obtained as:

$N.P.W.=₹53,000\left(P/A,i^{\prime}\% ,5\right)+₹30,000\left(P/F,i^{\prime}\% ,5\right)-₹1,05,815.4-₹30,000\left(P/A,i^{\prime}\% ,5\right)$

Step 5: $0 =₹53,000\left(P/A,i^{\prime}\% ,5\right)+₹30,000\left(P/F,i^{\prime}\% ,5\right)-₹1,05,815.4-₹30,000\left(P/A,i^{\prime}\% ,5\right)$

The equation given at Step 5 normally involves trial-and-error calculations until the $i^{\prime }\%$ is found. However, since we do not know the exact value of $i^{\prime }\%$, we will probably try a relatively low $i^{\prime }\%$, such as $5\%$, and a relatively high $i^{\prime }\%$, such as $12\%$.

At $i^{\prime }\% = 5\%$:

$₹53,000\left(P/A,5\%,5\right)+₹30,000\left(P/F,5\%,5\right)-₹1,05,815.4-₹30,000\left(P/A,5\% ,5\right)$

$₹53,000\left(4.3295\right)+₹30,000\left(0.7835\right)-₹1,05,815.4-₹30,000\left(4.3295\right)=+₹17,268.1$

At $i^{\prime }\% = 12\%$:

$₹53,000\left(P/A,25\%,5\right)+₹30,000\left(P/F,25\%,5\right)-₹1,05,815.4-₹30,000\left(P/A,25\% ,5\right)$

$₹53,000\left(3.6048\right)+₹30,000\left(0.5674\right)-₹1,05,815.4-₹30,000\left(3.6048\right)=-₹5,883$

Since we have both a positive and a negative P.W. of net cash flows, linear interpolation can be used as given below to find an approximate value of $i^{\prime }\%$

$\frac{12\%-5\%}{₹17,268.1-\left(-₹5,883\right)}=\frac{i^{\prime }\%-5\%}{₹17,268.1-₹0}$

$i^{\prime}\%=5\%+\frac{₹17,268.1}{₹17,268.1-\left(-₹5,883\right)}\left(12\%-5\%\right)$

$i^{\prime }\% = 10.22\%$, which is approximately equal to $10\%$.

Step 6: Since the value of $i^{\prime }\% = M.A.R.R.$, the investment in the project is economically barely justified.

Note: Let us check whether the value of N.P.W. at $i^{\prime } = 10\%$ is 0.

$N.P.W.=₹53,000\left(P/A,i^{\prime}\%,5\right)+₹30,000\left(P/F,i^{\prime}\%,5\right)-₹1,05,815.4-₹30,000\left(P/A,i^{\prime}\%,5\right)$

At $i^{\prime }\% = 10\%$:

$N.P.W.=₹53,000\left(P/A,10\%,5\right)+₹30,000\left(P/F,10\%,5\right)-₹1,05,815.4-₹30,000\left(P/A,10\%,5\right)$

$=₹53,000\left(3.7908\right)+₹30,000\left(0.6209\right)-₹1,05,815.4-₹30,000\left(3.7908\right)$

$N.P.W.=0$

Thus $i^{\prime } = 10\%$ which is equal to the given $M.A.R.R.$ and therefore, the investment in the project is economically barely justified.