All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.

Description

QF2101 1112S1 Tutorial 2

Tags

Transcript

AY2011-12 Sem1 QF2101 Tut 2
NG Wee SengEmail:matnws@nus.edu.sg Tel: 65164673
1
QF2101 Basic Financial MathematicsTutorial 2Basic Problems
1
Calculate the present value (at time
t
= 0) of a 5-year annuity that pays $100 yearly,with the first payment made at
end
of the tenth year. Take the effective annualinterest rate to be 10%. Give your answer to the nearest dollar.[161]
2
Calculate the
future value,
at the end of the tenth year, of a 10-year annuityimmediate of $100
per month
when the effective
annual
interest rate is 8.5%. Giveyour answer to the nearest dollar
[
18485]
3
A man wants to accumulate $10, 000 by the end of two years by making a deposit of $300 at the end of each month during the first year and $(300
+ x
) at the end of eachmonth of the following year. Given that the account pays 12% compounded monthly,find the value of
x
.[150.44]
4
A 20-year annuity is such that the first payment of $
x
is made at the end of the firstyear, and that subsequently, payments increase in such a way that each payment is104% of the preceding one. If the present value of this annuity is $7229.50, find thevalue of
x
to the nearest integer. Take the effective annual rate of interest to be 7%.[500]
5
A perpetual annuity that pays $1 for the first 5 years, with the first payment made
t
=1, and pays $
A
thereafter, has present value
P
. If the effective annual rate of interestis 10%, express
A
in terms of
P
.
[
11.11.1
10
55
P A
]
AY2011-12 Sem1 QF2101 Tut 2
NG Wee SengEmail:matnws@nus.edu.sg Tel: 65164673
2
6
A man takes out a loan of $35000 to be repaid by annual installments over 20 years.The first payment will be made at the end of the first year. Take the interest rate to be7% per annum.(i) Calculate the annual installment.(ii) Suppose the man asks for the term of the loan to be extended by 5 years after making the 13
th
payment. Find the outstanding loan after the 13
th
paymentand hence, calculate the new installment to the nearest dollar.[(i) $3303.7524 (ii) $17804.878; $2242]
7
I take out a $50,000 mortgage on a home at 12.5% interest convertible semi-annually.I will pay off the mortgage with monthly instalments for 20 years. Suppose that after making the 60
th
payment, I decide to renegotiate the loan so that I will repay theoutstanding amount by making a lump sum payment of $10000, and clearing the balance by means of monthly instalments of $
x
for 10 years at 11% interestconvertible semi-annually. Find
x
to 2 decimal places.[490.32]
8
A loan of $ 8,000 is to be repaid by annual instalments of $600. The effective annualinterest rate is 6% . Determine the total number of payments to be made and theamount of the last payment.[ 28 payments; 376.627 ]
AY2011-12 Sem1 QF2101 Tut 2
NG Wee SengEmail:matnws@nus.edu.sg Tel: 65164673
3
9
A 30-year loan of $10,000 is to be repaid by annual repayments. The interest rate isguaranteed at 8% for the first 5 years of the loan. Prevailing interest rates applythereafter.(i) Calculate the annual repayment for the first five years based on a 8% interestrate for a 30-year term. (That is, find
A
such that $10000 can be fully repaidwith 30 annual repayments of $
A
at 8%)(ii ) Calculate the loan balance immediately after the fifth payment has beenmade.The interest rate is increased to 9% between year 6 and year 30
inclusive.
(iii) Calculate the new annual repayment if the outstanding loan is to be fully paid at the end of year 30.(iv) If instead, the same annual repayment (i.e. your answer to (i)) is madethroughout the term of the loan, how many years does it take to fully repaythe loan? How much is the last payment?[ (i) $888.27 (ii) 9482.13 (iii) 965.34 (iv) Further 37.565 years
total of 5 + 38= 43 years. Full repayment for 42 years; last payment= 511.53 ]
10 (a)
It is given that the force of interest over the time interval [1, 3] is given by
1
)(
t t
. If $100 invested at
t
= 1 grows to $120.74 at time
t
= 2 and $100invested at
t
= 2 accumulates to $114.00 at
t
= 3. find
and
.
(b)
A bank account credits interests using a force of interest
23)(
32
t t t
. A deposit of $1000 is placed in the account at time
0
t
. Calculate the amount of interest theaccount earns from the end of the forth year to the end of the eighth year.[
;2/1)(
3
t t a
interest = 1000(
a
(8)
–
a
(4)) = 224,000]
AY2011-12 Sem1 QF2101 Tut 2
NG Wee SengEmail:matnws@nus.edu.sg Tel: 65164673
4
Discussion Problems
1
Consider projects A and B with cash flows (
),...,,,
0
aaaa
and (
),...,,,
0
bbbb
respectively, both having the same length. The numbers
baba
,,,
00
are positive andsuch that
bbaa
00
.Prove that IRR
A
< IRR
B
.
2
Consider two projects A and B with cash flows (
),...,,,
0
aaaa
and(
),...,,,
0
bbbb
both having the same length. The numbers
baba
,,,
00
are positive.It is given that
00
ba
bbaa
00
and
0)(
00
abnba
. We have shown inQuestion 1 that IRR
A
< IRR
B
.Let NPV
A
(
d
) and NPV
B
(
d
) be the present values of the cash flows of project A and project B respectively, where
r d
11
. We assume that
r
0.(i) Show that the difference, PV
A
(
d
) - PV
B
(
d
) has different signs at
0
d
and
1
d
.(ii) Deduce that there is an unique interest rate
r
* for which the two projects have equal NPV. We call
r
* the
crossover rate
of the two projects.(iii) Show further that NPV
A
< NPV
B
if and only if
r
>
r
*.(iv)
Sketch the graphs of NPV
A
and NPV
B
as a function of the rate of interest
r
, indicating
r
*
and the IRR of each project on your graphs.

Related Search

We Need Your Support

Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks