# Many academic institutions offer a sabbatical policy. Every seven years a professor is given a year free of teaching. A professor earning 70,000 per year who works for a total of 42 years what is the present value of the amount they will earn while on sabbatical, if the interest rate is 6% EAR?

### Solution:

- In his seventh year, the professor receives $70,000. Since the question does not state when he receives this amount, let’s assume it is received at the end of the seventh year as a lump sum. The present value of this is $46,554 (i.e. $70,000/(1.06^7))
- Similarly, the professor receives $70,000 at the end of year 14, and the present value of this is $30,961.07
- Similarly, the professor receives $70,000 at the end of year 21, and the present value of this is $20,590.88
- Similarly, the professor receives $70,000 at the end of year 28, and the present value of this is $13,694.11
- Similarly, the professor receives $70,000 at the end of year 35, and the present value of this is $9,107.37
- Similarly, the professor receives $70,000 at the end of year 42, and the present value of this is $6,056.92

Now we can add up each of the six present values, to arrive at the value of all the amounts received during the sabbaticals: $126,964.34 (if you add up the numbers above you get $126,964.35, but that’s due to rounding error).

If the professor receives the money at the middle of each year, you can re-calculate the above (e.g. by discounting the seventh year by 6.5 years rather than by 7 years). You might also want to recalculate this assuming that the salary is paid monthly, and see how it compares to the above result.

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