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## Borrowing at I%

Working out the NPV of an investment tells you how good it is compared to making a safe investment at I%.

You might be wondering if there is any relationship between making the safe investment at I% and borrowing the money to fund the risky investment at I%?

You can see that this question is raised by the idea of using the market rate for capital as I% in the calculation of NPV.

The reasoning goes something along the lines of:

as the cash sums are discounted by the interest rate I% a positive NPV means that there is a profit to be made over and above the market rate.

However to make this more precise we need to work out exactly how borrowing the money needed for the investment effects the situation.

If we assume that the cash sums received are used to pay off the loan then we can keep a current balance as each time period passes. Suppose that the cashflow starts with a sum \$S1.at the end of the first time period.

This means that the at the end of the first time period the loan account stands at \$S1. At the end of the second time period the cashflow received \$S2 is added to the loan account along with the interest due to the debt:

`S1*(1+I)+S2`

At the end of the third time period the loan account stands at

`S1*(1+I)^2+S2^(1+I)+S3`

and so on.. You should recognise this as nothing more than the Net Future Value of the entire cashflow calculated using I%.

In other words, the NFV is the profit that you would make on the investment if you borrowed the negative sums in the cashflow at I%.

This gives us another interpretation of the NPV investment decision rule. If the NPV is positive and you can borrow money at I% to make the investment then you will make a profit of NPV*(1+I)^n where n is the duration of the investment.

This interpretation takes our understanding of the NPV and its related FPV one step further and makes the NPV investment rule even more reasonable. If an investment has a positive NPV then it not only outperforms a safe investment at I%, it gives a positive return even if the investment has to be funded by borrowing an I%.

## NPV and NFV - a summary

We now have several good reasons for using the NPV as a measure of the ‘goodness’ of an investment.

A positive NPV calculated at I% implies that:

• the investment outperforms a compound interest investment based on the negative part of the cashflows at the same rate of interest.
• the investment is still profitable if the negative cashflows have to be borrowed at the same rate of interest.

In both cases it is assumed that it is possible to reinvest at I%.

The NFV also gives the profit made if the investment is funded by borrowing at I%.

## Rate differentials

There is no doubt that at this point you may have noticed that there is an unrealistic assumption built into the calculation of the NFV. In calculating the interest paid on the loan and the interest earned on any positive deposits we have used the same rate of interest!

It is a well known fact that the rate for borrowing isn’t the same as the rate for a saving. In most cases this difference can be ignored but if you want a completely accurate answer then you really do need to calculate a ‘modifed’ NFV using appropriate rates for the positive and negative sums.

If you do this in for the example earlier and assume that the rate for borrowing is 10% and that for a deposit is 8% then the appropriate calculation is

`=NPV(8%,B3:B6)*(1+8%)^3+B2*(1+10%)^4`

and the result is -\$462.72.

In other words, if you take into account a 2% differential in interest rates for borrowing and saving then the investment that previously showed a profit now shows a loss.

Notice that in this case the NPV is unaffected by an interest rate differential because the negative sum is borrowed at the start of the investment - but this is not generally true.

Also notice that although the investment now makes a loss if the money has to be borrowed at 10% it still makes a better return than investing the same sum of money at 8%.

In conclusion:

• if you have to borrow at more than I% to fund an investment then a positive NPV isn’t a guarantee that you will make a profit.

#### Coming next:

IRR, Internal Rate of Return, and MIRR, Modified Internal Rate of Return.

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