Question

Consider 3 time periods (now, a bit later, a lot later).

• The “Now” period:  If a student goes to school now, they will have to pay $100 for books/transportation/fees/etc.  In addition, they will not be able to work, hence the family income is reduced by $600.
• The “a bit later” period:  If a person goes to school in the “now” period, they will earn $500 more than a person who does not go to school.
• The “a lot later” period:  If a person goes to school in the “now” period, they will earn $1000 more than a person who does not go to school.

1. Suppose that the discount rate is 25%.  Calculate the discounted net present value of going to school in the “now” period.

2. Calculate the internal rate of return.  Use quadratic formula.

Answer 1

(a)

Time period	Now ( time period 0 )	A bit later ( time period 1 )	A lot later ( time period 2 )
Cashflow	-700	500	1000
Present value of the cash flows	-700	400	640

Cashflow in the "now period" = - 100 - 600 = - 700 ( - ve sign indicates that it is a cash outflow )

Cashflow in the "a bit later period" = 500 ( + ve sign indicates that it is a cash in flow )

Cashflow in the "a later later" period = 1000 ( + ve sign indicates that it is a cash in flow )

Discounted value of cash flow in particular time period ( t = n ) is given by = ( cash flow in the period n ) / ( 1 + ( discount rate) )ⁿ

Discounted value of cash flow in "now" period ( t = 0 ) = - 700 / ( 1 + 0.25 )⁰ = - 700

Discounted value of cash flow in "a bit later" period ( t = 1 ) = 500 / ( 1 + 0.25 )¹ = 400

Discounted value of cash flow in "a lot later period" ( t = 2 ) = 1000 / ( 1 + 0.25 )² = 640

Discounted net present value of going to school in the “now” period = - 700 + 400 + 640 = $ 340

(b)

Internal rate of return = discount rate at which the net present value of the cashflows become zero

The NPV of going to school in the " now " period at a discount rate " r " is

- 700 / ( 1 + r )⁰ + 500 / ( 1 + r )¹ + 1000 / ( 1 + r )²

Equating the NPV to zero, we get

- 700 / ( 1 + r )⁰ + 500 / ( 1 + r )¹ + 1000 / ( 1 + r )² = 0

Multiplying both sides with ( 1 + r )² , we get

=> - 700 ( 1 + r )² + 500 ( 1 + r ) + 1000 = 0

=> - 700 ( 1 + 2r + r² ) + 500 + 500r + 1000 = 0

=> - 700 - 1400r - 700r² + 500 + 500r + 1000 = 0

=> - 700r² - 900r + 800 = 0

=> 700r² + 900r - 800 = 0

=> 7r² + 9r - 8 = 0 ( ax² + bx + c ) ----- ( 1 )

Using the solution for quadratic equation formula, we get

r = - b + ( b² - 4ac )^0.5 / 2a ( and ) r = - b - ( b² - 4ac )^0.5 / 2a

Since r has to be positive, we get

r = - b + ( b² - 4ac )^0.5 / 2a

b = 9, a = 7, c - 8 ( from ( 1 ) )

Substituting these values, we get

r = ( - 9 +  ( 9² - 4*7*(-8) )^0.5 ) / (2*7)

=> r = ( - 9 + ( 81 + 224)^0.5 ) / 14

=> r = ( 17.46 - 9 ) / 14

=> r = 0.6045 ( or ) 60.45 %

Therefore, the internal rate of return for the student will be 60.45 %