Question

At the beginning of the film Sense and Sensibility, Mr John Dashwood and his avaricious wife, Mrs John Dashwood, are debating how he can most cheaply discharge his obligations towards his stepmother, the recently widowed Mrs Henry Dashwood and her three daughters, Elinor, Marianne and Margaret. He first considers giving her a lump sum of £1 500; then, since £1 500 seems a lot to part with in one lump, he considers paying her an annual sum of £100 for as long as she lives. But, objects his wife, although Mrs Dashwood is old, such an arrangement will encourage her to cling to life for an unreasonable time; and should she survive for more than 15 years, her stepson will lose money by the arrangement. To see if Mrs Dashwood is correct, how long would Mrs Henry Dashwood have to live to make John Dashwood regret choosing the annuity over the lump sum, if he can invest his money at 5%? How about if the interest rate were 10%? (5 pts) (Hint: look at capitalized values, aka N-infinity, factors, right at the beginning of the Appendix)

Answer 1

The data given in the question can be collated as below:

Annuity amount = 100 pounds

Rate = 5%

Mr. John Dashwood chooses to consider paying 100 pounds each year, instead of a wholesome 1,500 pounds.

To compare the annuity of 100 pounds each year with the lump sum amount of 1,500 pounds, we use the following formula:

Present Value of Annuity = Annuity Amount* {(1/r) - [(1/r)*(1+r)^n)]}

PVA = 1,500, we have to find the value on 'n'

1,500 = 100* {(1/0.05) - [(1/0.05)*(1.05)^n)]}

15 = (1/0.05) - {1/[(0.05)*(1.05^n)]}

15 = 20 - {1/[0.05*(1.05)^n]}

5 = 1/[0.05*(1.05^n)]

1 = 0.25 * (1.05)^n

4 = (1.05)^n

Taking log on both sides, we get:

ln 4 = n* ln(1.05)

1.3863 = n * 0.04879

n = 28.41

Any value of at and over 29 years would make John Dashwood regret choosing the annuity over the lumpsum amount.

Similarly, we can solves this for r = 10%.