Present Value Theory: - Cal State LA

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PV Problem Set
1. Assume a .05 (5%) time value of money. We have a debt to pay and are given a
choice of paying $1,000 now or some amount X five years from now. What is the
maximum amount that X can be for us to be willing to defer payment for 5 years?
Loan = $1,000, time value of money 5%. We have two options: pay it now or defer the
payment for 5 years.
(i)
if we pay it now, we should pay a $1,000.
(ii)
If we defer the payment 5 years, we must
calculate the value of this $1,000 in 5 years from now.
Using your calculator:
PV=1000
I/YR=5
N=5
Press FV to compute
FV  PV  (1  r ) n , Future value of a single value
1,000
0
1
2
3
4
5
Same ThAng!!!
FV  $1,000  (1.05) 5  1,276.28
Answer: The maximum amount for us to be willing to defer the payment
for 5 years would be $1,276.28.
1
2. How much are you willing to pay today for the following cash-flows (CF’s):
100$ in year one, $200 in years two and three, and 400 in years four and five if the
current interest rate is 7%? How about in year 6? How about in year 3?
First, draw a time line and enter the cash-flows:
0
1
2
3
4
5
100
200
200
400
400
6
(Now, after all that work, take a break and have a beer—like most MBA’s, your probably
exhausted at this point. Once you have had a couple MGD’s, and maybe a shot of mad
dog, you will be ready do the hard part.)
To find the price today, you compute the present value of the CF’s:
PV 
100
200
200
400
400




 1021.76
1.07 1.07 2 1.07 3 1.07 4 1.07 5
Another way of writing this equation is:
PV 
5
100 3 200
400



t
t
1.07 t  2 1.07
t  4 1.07
It is the same thing—study how the sum sign works. Payments 2 and 3 are 200 dollars,
and payments 3 and 4 are 400 dollars. So this is the range of the indices on the
respective sum signs.
To compute the price in year 6—that is the future value in year 6—we simply future value
the PV at the rate at which interest can be earned:
FV6  1021.76  1.07  1533.38
6
Alternatively, we could compute it directly:
4
2
t 3
t 1
PV  100 1.07 5   200 1.07 t   400 1.07 t  1533.38
Which is the same as:
4
2
t 3
t 1
PV  100 1.07 5  200   1.07 t  400   1.07 t  1533.38
2
or:
PV  100 1.07 5  200 1.07 4  200 1.07 3  400 1.07 2  400 1.07
(This is probably the way that makes the most sense to you. However, it is the same
thing.)—Remember, you multiply and divide before adding and subtracting.
And finally, to compute the value in year 3, we either FV the value from time zero,
discount the value from time 6, or we do it the hard way, and start from scratch. Either
way, done correctly, we still get the same answer. You should do it all three ways so you
convince yourself.
i.
FV 3 1021.76  1.07 3  1251.70
ii. PV 3 1533.38  1.07 3  1251.70
iii. PV  100 2 1.07  200 1.07  200 
400 400

 1251.70
1.07 1.07 2
Thus, the answers are 1021.76, 1533.38, and 1251.70, respectively.
Note: I have gone to a lot of trouble in this problem to show you how the sum sign
functions. Study how it works so that you are fluent in it. Additionally, you should carry
through the multiplication and compute answers two and three using the alternative
methods, not just the quickest method. This will reinforce the concept of time value of
money and how we are able to move money to different points in time. Finally, after this
very rough evening, you deserve another beer! Join your brethren down at the local
tavern, Gentleman’s (or Lady’s) club, or whatever haunt piques your interest. Remember,
you’re an MBA, or at least you getting there. So go behave like one, and take the evening
off—after all, this was a lot of work and you deserve a treat!
3
3. Suppose that you are considering an investment with the following cash flows
that are being guaranteed by your kid brother: $100 in years 1-3, $300 in years 5-8.
If the return that you want on the investment is 9%, what price should you pay
today? Suppose that you are asked to make a single payment in year 4 for the entire
set of cash flows, how much would you pay?
0
1
2
3
100 100 100
3
4
5
6
7
8
300 300 300 300
8
PV   1100
  1300
 941.66
.09t
.09t
t 1
t 5
Using your calculator – use the cash-flow function:
CF0 = 0
CF1 = 100, (Number of times) N = 3
CF2 = 0, Nj = 1
CF3 = 300, N j= 4
I% or I%/YR = 9
Press NPV to compute the PV.
To solve for the amount in period 4:
3
4
FV4  100 1.09  
t
t 1
t 1
300
1.09t
, or,
FV4  941.66 1.09 4  1,329.23
Using your calculator – use the TVM function:
PV=941.66
I%/YR = 09
N=4
PMT = 0
Press FV.
4
4. Compute the present value for a coupon bond that promises to pay a coupon
(interest payment) of $50 a year for 30 years and has a face value of $1,000. The first
interest payment is one year from now. Use a rate of discount of .05.
30 years bond with coupon of $50 (5%), face value of $1,000 and 5% of discount rate.
30
PV  
i 1
50
1000

(1.05) i (1.05) 30
Using your calculator:
PMT= 50
FV=1,000
I/YR=5
N=30
Press PV to compute
Answer: The present value (price) for this bond is $1,000
5.
A 20-year $1,000 coupon bond promises to pay a 4.5% coupon rate annually.
The current interest rate is 5%. How much is the bond worth now? How much
would the bond be worth if the current interest rate were 4%?
20 years bond with coupon of 4.5%, face value of $1,000 and 5% of discount rate.
20
PV  
i 1
45
1000

i
(1.05)
(1.05) n
Using your calculator:
PMT= 45
FV=1,000
I/YR=5
N=20
Press PV to compute
Answer: The present value for this bond is $937.689
if the inters rate fall to 4%
20
PV  
i 1
45
1000

(1.04) i (1.04) n
Using your calculator:
PMT= 45
FV=1,000
I/YR=4
N=20
Press PV to compute
Answer: The present value for this bond is $1,067.95
5
6. The newspaper headline states “Baseball Player Signs for $14 Millions.” A
reading of the article revealed that the player will receive $1 Million per year for the
next 6 years. He will then receive $400,000 per year for 20 years (i.e.: 6 * $1,000,000
+ 20 * $400,000 = $14,000,000). Assuming that the player can borrow funds at 10%
per year, what is the present value of this contract?
Contract: 1 million for 6 years and then 0.4 millions for the next 20 years. This is the
same as 0.6 million for 6 years plus 0.4 million for 26 years. The rate of return is 10%.
0
1
1
1
1
1
1
.4
.4
.4
…
1
2
3
4
5
6
7
8
9
…
.4
.4
.4
.4
.4
.4
.4
.4
.4
.4
6
1,000,000 26 400,000
PV  

i
(1.10) i
i 1
i 7 (1.10)
or, rewriting the equation to make it more calculator friendly:
6
600,000 26 400,000
PV  

i
i
i 1 (1.10)
i 1 (1.10)
.4
.4
.4
.4
.4
.4
25
26
Using your calculator:
PMT= 600,000
I/YR=10
N=6
Press PV to compute first
sum sign
PMT= 400,000
I/YR=10
N=26
Press PV to compute and
add the results together
Answer: The present value of this contract is $6,277,534.61
6
7. It is your lucky day: you have been fired from your job. But when your stupid
boss kicked you out of the office, he accidentally handed you two $100,000 dollar
bearer bonds (non-registered bonds – you only need to present the coupons for
payment) with the stack of papers that he cleared from your desk. The bonds have
a 10% annual coupon and maturity of five years. If the current market rate on the
bonds is 8.7%, what is this severance package worth?
0
1
2
10,000
10,000
3
10,000
4
5
10,000
110,000
P/YR = 1
I/yr =8. 7
N=5
PMT = 10,000
FV = 100,000
Press PV
5
PV   1.087t 
10, 000
t 1
100, 000
1.0875
Answer: Package  2  PV  $210,192.31
8. Suppose that you decide to hold the bonds in the previous question for a year,
and then sell them. The next year, the interest rate on comparable bonds increases
to 9.5%. How much will you be able to sell them for at that time? If you received a
5% rate of return on your money and you stick the coupons in the bank, in
retrospect, would you have been better off selling the bonds when you first got them,
or, holding them for a year, and selling them at the higher yield (9.5%)?
In one year, the future CF’s from the bonds will be:
0
1
2
10,000
10,000
3
4
10,000 100,000
4
, 000
, 000
PV   10
 100
 101,602.24
1.095t
1.0954
t 1
Package  2 * PV  2 * Coupons
 2 * 101,602.24  2 * 10,000
 223,204.48
If you sold the bonds when they were received, you would have gotten $210,192.31 (See #8). Then
re-investing the proceeds at 5% for 1 yr:
210,192.31(1.05) = 220,701.93
Answer: You are better off retaining them – getting 10% interest on your money.
7
9. Assume that you have just purchased a $75,000 house. One bank will give you a
9% mortgage with repayment in equal annual installments over 20 years with
$15,000 down payment. Another bank wants a 10% rate of interest but will give you
a 25-year equal-annual-installment mortgage with a $15,000 down payment.
Assuming that you have that $15,000, which of the two deals will minimize the
annual payments?
Purchase a house for $75,000, bank (a) 9% on equal payment over 20 years with $15,000
down payment; bank (b) 10% on equal payment over 25 years with $15,000 down
payment.
(a) The $15,000 of down payment will decrease the amount borrowed to $60,000
(amount borrowed=$75,000-15,000)
20
PMT
 60,000  PMT  6,572.79

t
1
.
09
t 1
(b)
25
PMT
 1.10
t 1
t
 60,000  PMT  6,610.08
PV=60,000
FV=0
I/YR=9
N=20
Press PMT
PV=60,000
FV=0
I/YR=10
N=25
Press PMT
Answer: Bank (a) at 9% interest minimizes the annual payment
8
10. Assume that a bank charges .01 interest per month. You borrow $50,000, to be
paid by equal payment over a 35-month period, first payment to be due one month
from now. How much will you have to pay each month? What is the annual effective
interest cost?
Loan $50,000, 1% month, over 35 month
35
PMT
 1.01
t
t 1
 50,000  PMT  1,700.18
PV=50,000
FV=0
I/YR=1
N=35
Press PMT to compute
EAR  1  rmonth   1  (1.01)12  1  12.68%
12
Answer: The annual payment will be $1,700.18 and the effective annual rate EAR is
12.68%.
11. Suppose the tooth fairy wants to buy a car, the cost of the car is $14,000. If you
agree to finance it at an interest rate of 9% per annum (that is an EAR of 9%), and
she agrees to make monthly payments at the end of each month for the next 48
months, how much will her payments be?
0
1
2
3
-14,000
PMT
PMT
PMT
r
 1.09
monthly
48
PMT
 1.0072
t 1
t
1
12
 1  .0072
 14,000
…
…
48
PMT
Using your calculator – use the TVM
function:
P/YR =1
PV=14,000
I%/YR = .72%
N = 48
FV = 0
Press PMT
Answer: 346.00 per month.
9
12. If the time value of the money is .10 how much do you have to save per year for
20 years to have $50,000 per year for perpetuity? Assume that the first deposit is
immediate and that the first payment will be at the beginning of the 21st year.
Time value of money is 10%, the period is 20 years, $50,000 per year as a perpetuity,
deposits are made at the beginning of the period, the first payment of the perpetuity will
be made at the beginning of the 21st year or at the end of 20th year (same thing).
0
-X
1
-X
2
-X
3
-X
4
-X
5
…
… 19
-X -X
20
0
21
50
22
50
…
…
Step 1: Because the perpetuity begins in year 21, it’s PV using the perpetuity formula
gives a PV in YEAR 20. Therefore, the PV of the perpetuity in YEAR 20 is:
50,000
 $500,000
0.10
PV 
In other words, we will need $500,000 in YEAR 20 to support a 50K annual perpetuity
thereafter.
.
Step 2: Now we PV the $500,000 to year zero:
PV 
500,000
 74,321.814
(1.10) 20
PV= 0
PMT= 0
FV=500,000
I/YR=10
N=20
Press PV to compute
Step 3: Now we compute the corresponding annuity due for the first 20 years:
19
20
PMT
PMT
74,321.814  
 1.10  
t
t
t 0 1.10
t 1 1.10
PMT  7,936.19
Set at BEG
PV=74,321.814
FV= 0
I/YR=10
N=20
Press PMT to compute
Answer: The savings per year will be $7,936.19 made at the beginning of
each year.
10
13. If an investor can earn 8% per year, how much will the investor have at the end
of 10 years, if he contributes $100 each year? What is the present value if cost of
money is 8%?
0
1
2
3
4
5
6
7
8
9
10
100 100 100 100 100 100 100 100 100 100
Cash flow $100 per year, rate of return r=8%, period of time 10 years.
th
(i) The future value at the end of the 10 year the investor will be:
10
FV  100   (1.08) i  1,448.66
PV= 0
PMT= 100
I/YR=8
N=10
Press FV to compute
i 1
Answer: The investor will have $1,448.66 at the end of the 10th year.
(ii) The present value of this investment is:
10
PV  
i 1
100
 671.01
(1.08) i
PMT= 100
FV= 0
I/YR=8
N=10
Press PV to compute
Answer: The present value of this investment is $671.01
11
14. Determine the annual payments and amortization schedule for a $1000, 8% per
year installment loan, to be repaid over 3 years.
3
$1000  
t 1
PMT
1.08t
PV=1,000
FV= 0
I/YR=8
N=3
Press PMT
PMT  388.034
Amortization Schedule
Year1
Year2
Principal
1,000.00
Interest (8%)
80.00
Amortization
308.03
Payment
388.03
Year3
Total
691.97
359.29
0.00
55.36
28.74
164.10
332.68
359.29 1,000.00
388.03
388.03 1,164.10
15. Now suppose that payments are to be made semi-annually, what are the
payments, and is the total amount in interest and principle less or more than the
annual loan? Explain.
Payment made semiannually as a follow, 8% annually—APR (4% semiannually):
6
1,000  
t 1
PMT
1.04t
PMT  190.762
PV=1,000
FV= 0
I/YR=4
N=6
Press PMT
The total of the payments for the annual loan is $1,164.10, for the semi-annual loan,
$1,144.56. The total of the payments is less than for the semi-annual payment loan
because we amortize the principal sooner, so the interest expense will be lower.
12
16. A has borrowed $100,000. Repayment is over four years, and the interest rate is
20% per year (first payment at the end of the year). Determine an amortization
schedule. Then compute payments for a five year loan with the first payment to be
made at the beginning of the year.
4
$100,000  
t 1
PV=100,000
FV= 0
I/YR=20
N=4
Press PMT to compute
PMT
1.20t
Answer: The annual payment will be $38,628.91
Amortization Schedule
Year1
100,000.000
Principal
Interest (20%) 20,000.000
Amortization 18,628.912
38,628.912
Payment
Year2
81,371.088
16,274.218
22,354.694
38,628.912
Year3
59,016.394
11,803.279
26,825.633
38,628.912
Year3
32,190.760
6,438.152
32,190.760
38,628.912
Total
32,190.760
54,515.648
100,000.000
154,515.648
The first payment of a five payment loan with the first payment at the beginning is
essentially an annuity due:
Using your calculator:
4
100,000  
t 0
PMT
1.20t
Set at BEG
PV=100,000
I/YR=20
N=5
Press PMT to compute
Answer: The annual payment will be $27,864.975
13
17. Assume a 10% discount rate (assume you can invest and borrow at 10%).
Compute:
(a) $10,000 in cash or $1,000 per year for perpetuity (first payment at the end of
the first period).
(b) $10,000 in cash or $1,100 per year for perpetuity (first payment at the end of
the first period).
(c) $10,000 in cash or $900 per year for perpetuity (first payment at the
beginning of the first period).
a) $1,000 per years as a perpetuity:
PV 
1,000
 10,000
0.10
b) $1,100 per years as a perpetuity:
PV 
1,100
 11,000
0.10
c) $900 per years as a perpetuity, but first payment received at the first year.
PV  900 
900
 9,900
0.10
14
18. Convert 6% APR to the corresponding EAR, assuming a) Annual compounding,
b) quarterly compounding, c) monthly compounding, and d) daily compounding:
a. Because compounding is Annual, the EAR and APR are 6%.
b. The corresponding monthly rate is: 6%  1.5%
4
4
4
EAR  1  rQuarterly   1  1.015  1  6.13636%
 .5%
12
 1  1.00512  1  6.16778%
c. The corresponding monthly rate is: 6%
EAR  1  rMonthly 
12
 .016438%
365
365
EAR  1  rDaily   1  1.0001643812  1  6.18313%
d. The corresponding daily rate is: 6%
19. Convert 6% EAR to the corresponding EAR, assuming a) Annual compounding,
b) quarterly compounding, c) monthly compounding, and d) daily compounding:
a. Again, for annual compounding, the period rate is the annual rate, so the
APR equals the EAR.
b. The corresponding PERIOD RATE is:
rQuarterly  1  EAR 4 1  1.06 4 1  1.467385%
APR  4 1.467385%  5.86954%
1
1
c. The corresponding period rate is:
1
rMonthly  1.06 12  1  .486755%
APR  12 .486755%  5.84106%
d. The corresponding period rate is:
1
rDaily  1.06 365  1  .015965%
1
APR  1.06 365 1  5.82736%
15
20. Suppose that you just turned 25, and you decide to contribute $100 a month at
the end of each month until age 60. Compute how much you will have, also compute
how much you will have if you leave the money in the account until age 67. Assume
that at age 67, you plan to withdraw equal payments for the next 30 years—how
much can you withdraw at the end of each month? Finally, realizing that you may
live past the ripe old age of 97 (your hoping that won’t be the case—at that point
what do you have to live for except your morning constitutional and the one day
every four years that you put on a suit, and vote with a bunch of other senile
codgers), so you also compute what can be withdrawn on a monthly basis in
perpetuity. Assume that you can get 12% APR with monthly compounding.
This seems like a complex problem, but it is not, its verbose—deal with it. If you read
carefully, the problem has already been broken into smaller chunks to illustrate the
steps of converting one annuity to another (or, in the second part a perpetuity). The
initial problem looks like this:
25-0 25-1 25-2 …
-100 -100
60-0 60-1 …
-100
FV60
67-0 67-1 …
FV67
97-12
Where, FV60 is the amount in the bank at the beginning of age 60 (60-0), and FV67
is the amount at age 67 (67-0). (Note that 60-0 is 59-12, the end of one year is the
beginning of the next.)
Step 1: Figure out what your annuity is worth in present value terms at age 25.
PV 
420
100
100
100
100


...


 $9846.88

1
2
3512
t
(1.01) (1.01)
(1.01)
t 1 (1.01)
Step 2: Compute the future value for age 60 and 67. This can be done two different
ways. Either compute the future value based on compounding for 420 monthly
periods using the monthly rate of interest of 1%,
FV 60  $9846.88  (1.01) 420  $643,095.95
FV 67  $9846.88  (1.01) 504  $1,483,444.05
Or, compute the EAR—the corresponding annual interest rate—and use annual
compounding:
EAR  1.0112  1  12.683%
FV 60  $9,846.88  (1.12683) 35  $643,095.95
FV 67  $9,846.88  (1.12683) 42  $1,483,444.05
16
Now you are ready to solve the more complex problem regarding the size of the
annuity payments you can withdraw. The revised problem looks like this:
25-0 25-1 25-2 …
-100 -100
60-0 60-1 …
-100
67-0 67-1 …
PMT…
97-12
PMT
Step 3: This is just another straight-forward annuity problem that my dead mother
with dementia could solve. Because we have already computed the value of the
account at time 67-0, we can re-write the problem as follows:
67-0
67-1
-1,483,444
…
PMT
97-12
…
PMT
So now you just have to compute what the payments will be:
1,483,444 
3012
PMT
 (1.01)
t 1
t
PMT  $15,258.89
If you wish to compute the PMT corresponding to a perpetuity:
PMT
.01
PMT  $14,834.44
1,483,444 
17
21. Let’s try something a little simpler. Suppose that you are again a viral 25 years
old, and wish to make annual contributions of $2000 at the end of each year until
age 60—you will be making annual contributions at the end of each year age 25
through 59—a total of 35 contributions. Suppose again that you want to know what
you’ll have in the bank when you turn 60 and 67, and plan to make annual
withdrawals at the beginning of each year starting a the beginning of age 67 (an
annuity due). You want to compute the size of the withdrawal so that your account
will carry you for 35 years. Assume that your money compounds at an APR rate of
9% compounded monthly.
This is ultimately what you want to do (remember, the end of age 59 is the beginning
of age 60):
25
26
27 …
-2000
60 61
-2000
…
67 68
-2000
…
96
PMTPMT…
PMT
Step 1: compute the present value of the CF’s:
There is a little trick here. You have annual CF’s and a monthly compounding rate of
.75% (9% divided by 12 months). So, you must compute the EAR or annual rate of
interest to conduct your analysis.
EAR  (1.0075)12  1  9.381%
35
PV  
t 1
2000
 20,395.38
(1.09381) t
Step 2: Compute the FV at ages 60 and 67:
FV 60  20,395.38  (1.09381) 35  $470,433.28
FV 67  20.395.38  (1.09381) 42  $881,234.05
Step 3: Using FV67 as the PV at age 67, compute the value of an annuity due:
35
881,234  1.09381
t 1
PMT
(1.09381) t
PMT  $79,003.71
If you wanted to know what the payments for an ordinary annuity would be:
35
PMT
t
t 1 (1.09381)
PMT  $86,415.04
881,234  
18
22. Let’s work it the other way. Suppose that you would like to receive $10,000 per
month at the end of each month for the rest of your life (a perpetuity), beginning at
age 67. How much would you have to contribute in terms of an ordinary annuity
today until age 65, if you just turned 30? Assume that you can get an EAR of 11.5%
on your money.
The problem looks like this:
30-0 30-1 30-2 …
PMTPMT
65-0 65-1 …
PMT
67-0 67-1 …
-10,000
forever
… -10,000
Step 1: Since your are getting an effective annual rate of 11.5%, the real annual
rate, you must compute a monthly rate to deal with the monthly cash-flows:
1
rmonthly  (1.115) 12  1  .911%
Step 2: Compute the value of your annuity at 67-0:
PV 67 
10,000
 $1,097,397.49
.00911
Step 3: Now you just PV the amount back to 30-0:
PV  1,097,397.49  1.00911
3712
 19,573.37
Step 4: Compute the corresponding monthly annuity:
19,573.37 
3512
PMT
 (1.00911)
t 1
t
PMT  $182.36
19
23. Lets redo problem 3, but this time assume that you want a 30-year monthly
annuity starting at age 67 as opposed to a perpetuity.
Because we have already done some of the work, we can start at Step 2.
Step 2: Compute the value of your annuity at 67-0:
PV 67 
3012
10,000
 1.00911
t
t 1
 1,055,755.84
Notice, when you get out to 30 years, there is not a great deal of difference between
having an annuity and a perpetuity. The reason is that the more distant payments
add very little to the present value.
Step 3: Now you just PV the amount back to 30-0:
PV  1,055,755.84  1.00911
3712
 18,830.64
Step 4: Compute the corresponding monthly annuity:
18,830.64 
3512
PMT
 (1.00911)
t 1
t
PMT  $175.44
20
Appendix:
For those of you interested in old formulas in the pre-calculator days, the following
identity holds:
1  (11r ) N
PMT
PVA  
 PMT 
t
t 1 (1  r )
 r
N



It is known as the annuity formula. And what it says is that for a series of equal
payments, also known as an “annuity”, there is a reduced form expression that does away
with the sum sign to compute the present value. In fact, to compute the expression, all
that you need do is carry through the algebra. This is the one formula that I ask you to
remember, and it is handy. But it is only for an annuity. Specifically, the present value
of the annuity (PVA) that falls in period zero, is the PV of the subsequent equal payments
in periods 1 through N. So by definition, the PVA is in the period immediately prior to
the first payment. In other words, if you have payments in periods 3 though 7, this
formula computes the value of those payments in period 2, not period zero. To find the
true present value of those payments using this formula, you must first compute the value
in period 2, then, discount the amount you computed back two periods, to time zero. The
formula is handy because most annuities start in time one. But, you must understand
exactly what the formula is giving you. You may also use it as an alternative to the sum
sign, where appropriate. Moreover, unlike the sum sign, the formula can actually be used
to compute the value of an annuity by simply plugging in the payment, number of periods
(n), and the discount rate, r.
21
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