Ecns 201
Capital Value Problems: Extra Credit Homework Due Monday, December 3, 2012
1. What is the present value of winning "$425 million" on Powerball if you select the "annuity option" for
payment? Suppose the "lump sum" cash option would pay you $278.3 million. (These are the actual
estimated values of the jackpot for the Nov. 28, 2012 drawing.) If you could invest the money from the
cash option a in "sure" savings plan that would return you 6%, would you be better off taking the cash
option or the annuity option? Would your answer stay the same if the interest rate was 2%? Show all
work; assume the choice is made solely on financial grounds and that your age, life expectancy, or
personal time preference are not factors. If you choose the annuity, assume that there will be 30 annual
payments of $14.167 million per year starting right now with the last payment coming in 29 years. The
lump sum cash option amount is paid once...right now. [Yes, we know there are tax implications involved
in this choice, but for this problem, ignore them.] Note: Be careful. Your annuity tables are for annuities
that start in one year; this one starts right now. So, make the necessary adjustment.
Time
0
1
2
3
…
28
29
Payment
$14.167
$14.167
$14.167
$14.167
…
$14.167
$14.167
The cash option is right now, so has a present value of $278.3 million.
The 30 payments can be broken down into two parts: the $14.167 million that is paid right now (and has
a present value of $ 14.167 million) PLUS a 29 year annuity of $14.167 million per year that starts in one
year. The table 18-1 value for a 29 year $1 annuity at 6% is 13.5907, so the present value of the 29
future payments from the lottery would be 13.5907 x $14.167 million = $192.5 million. Adding to that
the $14.167 million received right now give a value of $206.7 million. If you took the prize from the
lottery with the cash option, you would get $278.3 million. Making the wrong choice on that would
result in a loss of $71.6 million. I know, chump change! LOL.
If you could only get 2% on your money (which is totally unrealistic for this much money…bank savings
accounts will not be used!), the PV of the annuity option is $323.64 million. (Your answer may differ
slightly due to round-off error; I did it on Excel.)
2. Suppose you borrow $1,000 and are given the option regarding how you want to repay the loan. Option
1 will require you to repay a single amount of $1,628.90 in 10 years. Option 2 will require 10 annual
payments (with the first payment due in 1 year) of $162.70. The bank will inform you that the total
interest paid with option 1 is $628.90 (= $1,628.90 - $1,000) while the total interest paid with option 2 is
$627.00 (= [10 x 162.70] - $1,000). Which is the better option to take? Why? Use an interest rate of 5%
to render your calculations. (Remember, you are calculating the present value of what you pay back on
the loan, so lower numbers mean you are paying less back)
Time
0
1
2
3
…
9
10
Option 1
$1,000
$0
$0
$0
…
$0
$1,628.90
Option 2
$1,000
$162.70
…
$162.70
$162.70
$162.70 $162.70
Present value of payments: Option 1: Since this is a single future amount in 10 years, use table 17-2. At
5%, the present value of $1 received in 10 years is $0.6139, so the present value of $1,628.90 received in
10 years would be 0.6139 x $1,628.90 = $999.98.
Present value of payments: Option 2: The payments are an annuity of $162.70 per year for 10 years
(starting in one year). Use table 18-1. The present value of a $1 annuity for 10 years at 5% is $7.7217, so
the present value of the loan payments will be 7.7217 x $162.70 = $1,256.32
Clearly, you pay back LESS with option 1. By the way, another way of thinking about this is: Choose
Option 1, but make payments to YOUR savings account according to Option 2. If your savings account
gets any decent interest on it, you will accumulate interest on the payments. In 10 years, you will have
more than 10 x $162.70 by the amount of the accumulated interest. 10 x $162.70 = $1,627…which is
virtually that payoff to the loan from Option 1. Then you get to keep the interest! Pretty smart!
Yet another way of looking at this is to look at Tables 17-1 and 19-1. Notice that the implied interest rate
on Option 1 is 5% (Table 17-1), but the implied interest rate of Option 2 is 10% (table 19-1). Would you
rather take out a loan that charges 5% or 10%?
This problem points out that the Federal requirement that lenders tell you how much interest you are
going to pay on the loan is potentially misleading information! Option 2 is slightly lower by that
measure, but clearly is the worse loan to take out! Unless the payment structures of the loans are
identical, the amount of interest to be paid is COMPLETELY USELESS…and POSSIBLY MISLEADING
INFORMATION!!!
3. Suppose you can borrow $10,000 right now and can keep the loan with 0 interest and no payments for 4
years before you have to start repaying it (meaning you will owe $10,000 in 4 years). Then suppose the
lender requires you to repay the loan in annual installments for 10 years (the first payment coming 5
years from now). If the interest rate charged on the loan is 6%, the annual payments will be $1,359
(verify this from table 19-1). What is the present value of obtaining this loan if you can invest the money
at 8%? [You should be able to show that this loan...which is a simplified version of Stafford Student
Loans...represents a gift to the recipient whose present value is almost $3,300!]
Time
0
Payment $10,000
1
$0
…
There are two ways you can do this:
4
$0
5
-$1,359
6
-$1,359
…
13
-$1,359
14
-$1,359
Process 1: Find the PV of each individual payment (table 17-2).
Time
5
6
7
8
9
10
11
12
13
14
FV
$1,359 $1,359 $1,359 $1,359 $1,359 $1,359 $1,359 $1,359 $1,359 $1,359
Factor 0.6806 0.6302 0.5835 0.5403 0.5002 0.4632 0.4289 0.3971 0.3677 0.3405
PV
$924.94 $856.44 $792.98 $734.27 $679.77 $629.49 $582.88 $539.66 $499.70 $462.74
The present value of all of the payments is the sum of the PV’s of the payments. That is
$6,702.87 (approximately due to round-off error). This means you received a loan of $10,000
but only have to pay back a PV of $6,702.87. That is essentially a gift to you of almost $3,300!
Process 2: Annuity method. The payments of $1,359 represent a 10 annuity. The trick here is
to recognize that the first payment is in 5 years, to the annuity table will tell us the value of
those payments at the time you graduate, i.e., in year 4 (the annuity table gives the value one
year before the payments start!) The factor for 8% and 10 payments from Table 18-1 is 6.7101.
Therefore in 4 years, the value of the loan payments will be 6.7101 x $1,359 = $9,119.03. But
that is an amount in 4 years. The present value (right now in time 0) of the $9,119.03 is found
using table 17-2. The factor at 4 years and 8% is .7350. Therefore the PV of the $9,110.03 is
.7350 x $9,119.03 = $6,702.49.
(The difference between the answers using the different processes is entirely due to round-off
error; the more precise method of calculating (and MUCH FASTER) is done using Excel. The
more precise answer is $6,701.16.)