Normal Distribution, Time Value of Money

1.Consider the normal distributions drawn below (with different scales)

They both have m = 20 and the area of the shaded region of each is 0.90. Which of the following holds?
a. x1 < x2
b. x1 > x2
c. x1 = x2
d. There is not enough information
e. x1 >= x2

Answer: a. x1 < x2

The standard deviation for b) s= 8 is greater than that of a) s= 5.
This means that the spread around the mean is more for b) than for a)
Thus x2 is farther from mean = M = 20.00
Thus x1 < x2

We can also calculate the numerical values of x1 and x2

For x1
Mean=M = 20.00
Standard deviation =s= 5.00
Z corresponding to 90% is 1.2816
z=(x-M )/s or x=M +z*s
or x1=20+1.2816*5= 26.41

For x2
Mean=M = 20.00
Standard deviation =s= 8.00
Z corresponding to 90% is 1.2816
z=(x-M )/s or x=M +z*s
or x2=20+1.2816*8= 30.25

Therefore a. x1 < x2

2.If you deposit $5000 into a fund paying 6% interest compounded quarterly, how much can you withdraw at the end of each quarter for 5 years?
a. $135.92
b. $970.45
c. $291.23
d. $215.17
e. none of the above

Answer: c. $291.23

To calculate the amount that can be withdrawn each quarter we need to know PVIFA (Present Value Interest Factor for an Annuity)

Withdrawal Q Quarterly
No of years= 5
No of Periods= 20 (=5 x 4 quarters per year)
Discount rate annually= 6.00% annual
Discount rate per period= 1.5% (= 6%/4)
n= 20
r= 1.5%

PVIFA (20 periods, 1.5% rate=) 17.1686 (From the tables)

Withdrawal (equal amount for each quarter):
Present Value= 5,000 (The amount deposited at time t=0)
Therefore
withdrawal: 291.23 =5000/17.1686

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