Question

Consider the case of the following annuities, and the need to compute either their expected rate of return or duration.

1. Matthew needed money for some unexpected expenses, so he borrowed $2,587.09 from a friend and agreed to repay the loan in three equal installments of $950 at the end of each year. The agreement is offering an implied interest rate of   .

2. Matthew’s friend, Gregory, has hired a financial planner for advice on retirement. Considering Gregory’s current expenses and expected future lifestyle changes, the financial planner has stated that once Gregory crosses a threshold of $920,925 in savings, he will have enough money for retirement. Gregory has nothing saved for his retirement yet, so he plans to start depositing $40,000 in a retirement fund at a fixed rate of 5.00% at the end of each year. It will take   for Gregory to reach his retirement goal.

3.

A series of cash flows may not always necessarily be an annuity. Cash flows can also be uneven and variable in amount, but the concept of the time value of money will continue to apply.

Consider the following case:

The Purple Lion Beverage Company expects the following cash flows from its manufacturing plant in Palau over the next six years:

Annual Cash Flows
Year 1	Year 2	Year 3	Year 4	Year 5	Year 6
$400,000	$37,500	$480,000	$450,000	$550,000	$375,000

3. The CFO of the company believes that an appropriate annual interest rate on this investment is 4%. What is the present value of this uneven cash flow stream, rounded to the nearest whole dollar?

$1,775,000

$917,500

$1,979,094

$2,292,500

4. Identify whether the situations described in the following table are examples of uneven cash flows or annuity payments:

Description	Uneven Cash Flows	Annuity Payments
You recently moved to a new apartment and signed a contract to pay monthly rent to your landlord for a year.
SOE Corp. hires an average of 10 people every year and matches the contribution of each employee toward his or her retirement fund.
Franklinia Venture Capital (FVC) invested in a budding entrepreneur’s restaurant. The restaurant owner promises to pay FVC 10% of the profit each month for the next 10 years.
You have committed to deposit $600 in a fixed interest–bearing account every quarter for four years.

Answer 1

Let the interest rate be x

2,587.09 = 950*PVAF(x%, 3 years)

PVAF(x%, 3 years) = 2.7232

Using present value annuity table,

X = 5%

Hence, Implied interest rate = 5%

Let it takes x years

40,000*[{(1.05)^x – 1}/0.05] = 920,925

(1.05)^x = 2.15115625

X lies between 15 and 16 years

Present value = 400,000/(1.04) + 37,500/(1.04)² + 480,000/(1.04)³ + 450,000/(1.04)⁴ + 550,000/(1.04)⁵ + 375,000/(1.04)⁶

= $1,979,094

Annuity

Uneven

Uneven since profits will be different

Annuity