Question

You just bought a car and plan on driving to campus everyday. The university no longer offers free parking because of
the increase of parking needs. You have two options to buy a parking permit. A monthly permit costs $30 (due at the
end of each month), which is a pay-as-you-go plan and you pay every month. However, the auxiliary services offers
a discounted annual permit which only costs $250 (upfront). You may assume that the academic year (12 months)
starts from Sep 1st, and ends on Aug 31st.
(1) If you plan to use the permit for all 12 months of the year, what is your internal rate of return (implicit interest
rate)?
(2) If you are not going to be on campus during the summer break (two months, from Jun 1st to Aug 31st), is
your internal rate of return going to change? If so, how much?
(3) You heard that a partially used annual permit can be sold easily on the black market for $15×number of
remaining months unused, because the parking permit is not associated with the car plate and can be used by any car
in the same class. How this is going to change your internal rate of return?

Answer 1

Solution 1
The scenario becomes like pay 250 upfront and save 30 per month
Year	Cash Flow	PV fatctor@5%	Amount	PV fatctor@7%	Amount
0	(250)	1.000	(250)	1.000	(250)
1	30	0.952	29	0.935	28
2	30	0.907	27	0.873	26
3	30	0.864	26	0.816	24
4	30	0.823	25	0.763	23
5	30	0.784	24	0.713	21
6	30	0.746	22	0.666	20
7	30	0.711	21	0.623	19
8	30	0.677	20	0.582	17
9	30	0.645	19	0.544	16
10	30	0.614	18	0.508	15
11	30	0.585	18	0.475	14
12	30	0.557	17	0.444	13
			15.90		(11.72)
NPV @ 5%	15.90
NPV @ 7%	(11.72)
Difference in both	27.62
YTM	=Lower rate + Difference in rates*(NPV at lower rate)/(Lower rate NPV-Higher rate NPV)
	=5%+2%*(15.9/27.62)
	6.15%
So his effective monthly return	6.15%
Solution 2
The scenario becomes like pay 250 upfront and save 30 per month
Year	Cash Flow	PV fatctor@1%	Amount	PV fatctor@2%	Amount
0	(250)	1.000	(250)	1.000	(250)
1	30	0.990	30	0.980	29
2	30	0.980	29	0.961	29
3	30	0.971	29	0.942	28
4	30	0.961	29	0.924	28
5	30	0.951	29	0.906	27
6	30	0.942	28	0.888	27
7	30	0.933	28	0.871	26
8	30	0.923	28	0.853	26
9	30	0.914	27	0.837	25
10	-	0.905	-	0.820	-
11	-	0.896	-	0.804	-
12	-	0.887	-	0.788	-
			6.98		(5.13)
NPV @ 5%	6.98
NPV @ 7%	(5.13)
Difference in both	12.11
YTM	=Lower rate + Difference in rates*(NPV at lower rate)/(Lower rate NPV-Higher rate NPV)
	=1%+1%*(6.98/12.11)
	1.58%
So his effective monthly return	1.58%
Solution 3
The scenario becomes like pay 250 upfront and save 30 per month
Year	Cash Flow	PV fatctor@4%	Amount	PV fatctor@5%	Amount
0	(250)	1.000	(250)	1.000	(250)
1	30	0.962	29	0.952	29
2	30	0.925	28	0.907	27
3	30	0.889	27	0.864	26
4	30	0.855	26	0.823	25
5	30	0.822	25	0.784	24
6	30	0.790	24	0.746	22
7	30	0.760	23	0.711	21
8	30	0.731	22	0.677	20
9	30	0.703	21	0.645	19
10	15	0.676	10	0.614	9
11	15	0.650	10	0.585	9
12	15	0.625	9	0.557	8
			2.31		(10.43)
NPV @ 4%	2.31
NPV @ 5%	(10.43)
Difference in both	12.74
YTM	=Lower rate + Difference in rates*(NPV at lower rate)/(Lower rate NPV-Higher rate NPV)
	=4%+1%*(2.31/12.74)
	4.18%
So his effective monthly return	4.18%