Calculating Present Value for Investment Decisions Finance

This assignment focuses on determining the most valuable prize among several options using present value calculations and addresses an investment scenario involving semi-annual withdrawals. It calculates the present value of a perpetuity, an annuity, a lump sum payment, and a deferred annuity to determine the most valuable prize, concluding that $15,000 per year for ten years, with the first amount paid at year 5, is the most valuable due to its highest present value. Additionally, it determines the amount that must be invested today to allow a university student to withdraw $1,500 every six months for four years, starting ten years from today, considering a 6.5% interest rate, concluding that $25919.87 should be invested.

e tionQu s 7
A morning radio show offered the following four prizes as alternative prizes for one of its
competitions. Which prize is the most valuable if your discount rate is 6%? (Show working to
justify your answer.)
(a) $5,000 per year, forever, with the first amount paid after 1 year.
¿ t h is case ,t h e winner is offered a perpetuity .
PV of a perpeuity = D
r ,
w h ere D is t h eannual payment w hile r is t h e discount rate
PV of t h e perpeuity= 5000
0.06 =$ 83,333.33
(b) $12,000 per year for eight years, with the first amount paid today.
PV =P ( 1+ r )−n
PV =12000 ( 1.06 )0 +12000 ( 1.06 )−1 +…+12000 ( 1.06 )−7
¿ 12000 ( 1+1.06+ 1
1.06 + 1
1.062 + 1
1.063 + …+ 1
1.067 )
¿ 12000 ( 6.58238 )=$ 78988.56
(c) $20,000 today and $80,000 six years from now.
PV =P ( 1+r )−n
PV =20000 ( 1+r ) 0 +80000 ( 1.06 ) −6
¿ 12000+56396.84
¿ $ 76396.84
(d) $15,000 per year for ten years, with the first amount paid at year 5.
PV = ( 1+r )−n
PV = 15000
1.065 + 15000
1.066 + …+ 15000
1.069 + 15000
1.0610
¿ 15000 ( 1
1.065 + …+ 1
1.0610 )
¿ 15000∗6.247143
¿ $ 93,707.14
n
i co ntinD s u g
actorf ( 1.06 ) −n
5 1.06
0.74725
8
6 1.06
0.70496
1

7 1.06
0.66505
7
8 1.06
0.62741
2
9 1.06
0.59189
8
10 1.06
0.55839
5
11 1.06
0.52678
8
12 1.06
0.49696
9
13 1.06
0.46883
9
14 1.06
0.44230
1
15 1.06
0.41726
5
mSu
6.24714
3
(e) State the most valuable prize. (Give a reason for your choice.)
The most valuable is prize is :$ 15,000 per year for ten years , with the first
amount paid at year 5.This is becausehas the highest Present Value .
e tionQu s 12
How much must be invested today, so that a university student can withdraw $1,500 every 6
months for 4 years? The first amount is to be withdrawn 10 years from today, and the appropriate
interest rate is 6.5%.
Assuming that the givenappropriate interest rate is discounted semi−annually ,then
Given that thecash is withdrawn semi−annually ,
The number of periods , n=4∗2=8 periods
While 10 years=10∗2=20 periods
r =6.5∧¿

C=1500
Let the amount ¿ be invested today be P .
After ten years ¿ now , the amount ∈theaccount=P ( 1+i ) n
¿ P ( 1.065 ) 20=3.5236 P
Consequently , 3.5236 P should be enough ¿ pay 1500 every six months for 4 years .
At year 10th year ( when annuity Payment should start ) ,
PV =1500 ( 1.065+1.065−2 +…+ 1.065−7 +1.065−8 )
¿ 15000∗6.088751
¿ 91331.27
Thus 3.5236 P=91331.27
P= 91331.27
3.5236
¿ $ 25919.87
Hence $ 25919.87 should be invested .