Comparison of alternative prizes with discount rate of 6%

The present value of four alternative prizes is calculated using the discount rate of 6%. The most valuable prize is determined based on the highest present value. In question 7, the prize of $15,000 per year for ten years, with the first amount paid at year 5, is the most valuable. In question 12, $25,919.87 should be invested today to withdraw $1,500 every 6 months for 4 years starting 10 years from today.

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 The first amount is to be withdrawn 10 years from today, and the appropriate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Given that thecash is withdrawn semi−annually ,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 .