Question

6) 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) 7) Your company is considering purchasing a new machine. at the end of the third year it will increase company income by $10,000, increasing by $1,000 per year for the following five years. At the end of the eighth year it will be worn out and have no scrap value. If the cost of capital to the company is 15%, what is the most it is worth paying for the machine now? (5 pts) It will take three years to put into operation, but

Answer 1

Question 6

We need to find how long the annuities need to be paid so that PV of them add upto ≥ $ 1,500

We can find this using NPER function of excel. Inputs are:

Rate = 5%, PMT = $ 100, PV = -1500, FV = 0

Hence, NPER (Rate, PMT, PV, FV) = NPER (5%, 100, -1500, 0) = 28.41339817 years

Hence,  Mrs Henry Dashwood have to live for 29 years more, to make John Dashwood regret choosing the annuity over the lump sum, if he can invest his money at 5%.

If rate = 10%, the break even can never occur. That means, even if Mrs. Henry Dashwood live for infinite number of years, the PV of $ 100 at 10% rate = PV of $ 100 in perpetuity = 100 / 10% = $ 1,000 < $ 1,500, the lumpsum payment.

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Question 7

If i denotes the period from now and C_i is the cash flow occurring at the end of period i, then

$NPV = -C_{0}&plus;\sum_{i=1}^{n}\frac{C_{i}}{(1&plus;R)^{i}}$

where R is the cost of capital = 15%, C₀ = Cost paid in year 0. The project should have NPV ≥ 0

Hence,

$NPV = -C_{0}&plus;\sum_{i=1}^{n}\frac{C_{i}}{(1&plus;R)^{i}}\geq 0\Rightarrow \sum_{i=1}^{n}\frac{C_{i}}{(1&plus;R)^{i}}\geq C_{0}$

Hence, the most it is worth paying for the machine now

$=\sum_{i=1}^{n}\frac{C_{i}}{(1&plus;R)^{i}}$

C₁ = C₂ = 0; C₃ = 10,000; C₄ = 11,000; C₅ = 12,000; C₆ = 13,000; C₇ = 14,000; C₈ = 15,000

Hence, the most it is worth paying for the machine now

$=\sum_{i=3}^{8}\frac{C_{i}}{(1&plus;R)^{i}} = \frac{10,000}{(1&plus;0.15)^{3}}&plus;\frac{11,000}{(1&plus;0.15)^{4}}&plus;...&plus;\frac{15,000}{(1&plus;0.15)^{8}}$

=$ 34,617.47