Question

Many academic institutions offer a sabbatical policy. Every seventh year a professor is given a year free of teaching and other administrative responsibilities at full pay. For a professor earning $ 50 comma 000 per year who works for a total of 42​ years, what is the present value of the amount she will earn while on sabbatical if the interest rate is 4 % ​(EAR)? Note: Assume that the sabbatical annual salary is paid in one lump sum every 7 years.

Answer 1

$ 127,773.36

Step-by-step explanation:

The professor will be in sabbatical in years 7, 14, 21, 28, 35 and 42

In each of these years, he receives full pay amounting to=50,000

The PV of the sabbatical full pay

=
`(50,000)/(1.04^7) + (50,000)/(1.04^14) + (50,000)/(1.04^21) + (50,000)/(1.04^28) + (50,000)/(1.04^35) +(50,000)/(1.04^42) = 84,101.22`

=50,000/(1+4%)^7+ 50,000/(1+4%)^14+ 50,000/(1+4%)^21+ 50,000/(1+4%)^28+50,000

/(1+4%)^35+ 50,000/(1+4%)^42

==50,000/(1+4%)^7+ 50,000/(1+4%)^14+ 50,000/(1+4%)^21+ 50,000/(1+4%)^28+50,000

/(1+4%)^35+ 50,000/(1+4%)^42

=
`(50,000)/(1.316) + (50,000)/(1.732) + (50,000)/(2.279)  +(50,000)/(2.999) +(50,000)/(3.946) + (50,000)/(5.193)`

=37,993.92 + 28,868.36 + 21,939.45 + 16,672.22 + 12,671.06 + 9,628.35

= $ 127,773.36

Thus, at an interest rate of 4%, the present value of all the sabbatical earnings amount to $ 127,773.36