Hi rcruz!
In order to answer most of the question, we'll need to make use of
formulas that return the future and present value of annuities. You
can find these formulas and a very good explanation of what an annuity
is at the following link:
Calculating the Present and Future Value of Annuities
http://www.investopedia.com/articles/03/101503.asp
So now, here are the answers to your questions.
Question 1
a. In order to solve these problems, we'll make use of the formula
stated in the aforementioned link. I'll assume here that the flow of
payments start one year from now, and thus this is like an ordinary
annuity (you can use the other formula shown in the link if the first
payment actually happens today). The formula for the present value is
then:
C * [1-(1+i)^(-n)]
------------------
i
where C is the payment per period ($100 in all cases), i is the
interest rate (0.08 for point a) and n is the number of payments (10
for point a). So, the present value of this flow of payments is
100*(1-(1+0.08)^(-10))
---------------------- = 671
0.08
So the present value is $671.
b. We use the same formula, with i=0.08 and n=20; getting that the
present value is $981.81
c. Same formula, with i=0.04, n=10. Present value: $811.08
d. Same formula, with i=0.04, n=20. Present value: $1359.03
Question 2
This question can be answered in the same fashion as the previous one,
using the appropiate formula. You can find in the link that the
formula for the future value of an ordinary annuity is:
C * [(1+i)^n - 1]
------------------
i
So filling C, i and n just as we did in the previous question, we find
that the future value in each case is:
a. $1448.65
b. $4576.19
c. $1200.61
d. $2977.80
which makes sense intuitively, as the future value increases with both
an increase in the number of payments and an increase in the itnerest
rates.
Question 6
In this question, we don't have an annuity anymore. The formula for
the future value of a single payment today (as in this question) is
the following:
FV = PV*(1+i)^n
For example, if the interest rate is 10%, if we invest $100 today,
we'll have 100*(1.10)=$110 in one year. In the second year,
reinvesting the interest payment (compunding), we'll have
100*(1.10)*(1.10)=100*(1.10)^2=$121; and so on. That is the rationale
for the above formula.
So in this question we're given PV, FV and n, and we must find i. In
the first case we have:
684 = 400*(1+i)^11
So we isolate i from this equation:
684/400 = (1+i)^11
1.71 = (1+i)^11
(1.71)^(1/11) = 1+i
(1.71)^(1/11) - 1 = i
1.0499 - 1 = i
i = 0.0499
So the interest rate is about 5%.
The following cases can be solved using the exact same procedure:
249 = 183*(1+i)^4
i = 0.08 (so the interest rate is 8%)
300 = 300*(1+i)^7
i = 0 (so the interest rate is 0%)
Question 10
This question can be solved using the same equation we used in the
previous one. The only difference is that now the unknown variable is
n rather than i.
a. 1000 = 400*(1.04)^n
2.5 = 1.04^n
log(2.5) = log(1.04^n)
log(2.5) = n*log(1.04)
n = log(2.5)/log(1.04)
n = 23.36
So, rounding up, we get that we'll need 24 years to get from $400 to
$1000 at a 4% interest rate. Note that since n=23.36, after 24 years
we'll have a bit more than $1000, but if we only wait 23 years, we'll
end up with less than $1000.
b. Same rationale here:
1000 = 400*(1.08)^n
n = 11.9 (so we need 12 years to get from $400 to $1000)
c. 1000 = 400*(1.016)^n
n = 6.17 (so we need 7 years)
Question 20
This question can be solved again using the same formula as in
question 1. We need to find which present value is greater.
In the first case, we have that the quarterback will receive $3
million a year for 5 years. Let's assume that the first payment comes
one year from now. Using the same formula as in question 1, with the
interest rate at 10%, we get that the present value of this contract
is $11.37 million.
In the second case, the quarterback receives $5 now and $2 for 5
years. The present value of this contract will be the present value of
a $2 annuity for years plus $5. Notice that we are not discounting the
$5 with the interest rate, because that payment is happening right
now. So, the present value of receiving $2 per year for 5 years is,
using the same formula as before, $7.58. Adding the $5 he receives
now, we get that the present value of this contract is $12.58 million.
So the less famous quarterback actually got the best deal
The intuition behind what's happening here is that, although the less
famous one is receiving less money in total, he's receiving it much
earlier than the famous one, so the present value of his contract
becomes higher.
Question 22
a. We use the same formula as in question 1. The present value of this
annuity is thus $267.30
b. We can think of this question in the following way. We're now at
the beginning of year 0. In one year (at the beginning of year 1), the
annuity payments will start in exactly one more year. Therefore, one
year from today (at the beginning of year 1), this annuity will be
identical to the one in point a. Therefore, the value of the annuity
one year from today will be $267.30. Now, in order to find the value
of the annuity today, we simply discount this value by the interest
rate one year. So the present value is 267.30/(1.06)^1=252.16.
Question 37
a. In order to solve this problem, we use a the formula for the
Outstanding Loan Balance (OLB), which is:
C*(1-(1/1+i)^(n-q))
OLB = --------------------
i
where i is the interest rate (0.06 in this case), C is the periodic
payment, n is the number of periods the loan lasts (4 in this case)
and q is the number of payments that have already been made.
We'll use this equation at year 0, where we know the OLB (it's simply
$1000) and no payments have been done yet. Therefore, we set OLB=1000
and q=0. The unknown is the periodic payment, C.
C*(1-(1/1.08)^4)
1000 = --------------------
0.08
1000 = C*3.3121...
C = 301.92
So we've found that the yearly payment must be $301.92
b. The table can be easily filled.
The loan balance at year x+1 is simply the loan balance at year x
minus the amortization of loan at year x.
The year-end interest due on balance is the interest rate (0.08 in
this case) multiplied by the loan balance.
Finally, the amortization of loan is the year-end payment (301.92 in
this case) minus the year-end interest due on balance.
So the table becomes:
Loan Year-End Interest Year-End Amortization
Time Balance Due on Balance Payment of Loan
0 $1,000 $80 $301.92 $221.92
1 $778.08 $62.24 301.92 $239.68
2 $538.40 $43.07 301.92 $258.84
3 $279.56 $22.36 301.92 $279.56
4 0 0 ? ?
c. The annuity factor is defined as the present value of $1 paid for
each of t periods.
Annuity factor
http://itlocus.com/glossary/annuity_factor.html
We can calculate the 3-year annuity factor using the present value
formula we used in question 1, setting C=1, i=0.08 and n=3. Thus we
get that the annuity factor is 2.577. Finally, 2.577*301.92=778.077,
which is the same as the loan balance in year 1 (the small diffrence
is due to rounding the numbers throughout this question)
Question 39
If the first payment comes in one year, we use, once again, the same
formula as in question 1, setting C=2, i=0.08 and n=20. We thus get
that the present value of this lottery prize is $19.63 million.
If the first payment comes today, we use the other formula shown in
the first link I gave you, the one under the title of "Calculating the
Present Value of an Annuity Due". An annuity due is an annuity in
which the first payment happens today instead of happening in one
period. The formula is the same formula as before, multiplied by
(1+i).
So we get that in the second case, the present value of the lottery
prize is 19.63*(1.08)= $21.20 million.
Question 59
This question can be solved once again using one of the future value
formulas provided in the first link I mentioned. Let's assume that the
first deposit in the savings account will be made today. So we use the
formula shown in the link under the title of "Calculating the Future
Value of an Annuity Due". The formula is:
FV = C*((1+i)^n - 1)*(1+i)/i
In this case, we must find C, knowing the FV=500,000, i=0.06 and n=40. So we have
500000 = C*((1.06)^40 - 1)*(1.06)/0.06
500000 = C*164.04
C = 500000/164.04
C = 3047.89
So you must make 40 yearly payments of $3,047.89 (starting today) in
order to have $500,000 in 40 years.
Google search terms
present future value formula
://www.google.com/search?hl=en&q=present+future+value+formula
"amortizing loan" formula
://www.google.com/search?hl=en&q=%22amortizing+loan%22+formula
"annuity factor" definition
://www.google.com/search?hl=en&lr=&q=%22annuity+factor%22+definition
I hope this helps! If you have any doubts regarding my answer, please
don't hesitate to request clarification before rating it; otherwise I
await your rating and final comments.
Best wishes!
elmarto |
