CHAPTER 6
B-93
With fee and annual rate = 6.20%:
PVA = $10,200 = $200{ [1 – (1/1.005167)
t
] / .005167 } where .005167 = .082/12
Solving for
t
, we get:
1/1.005167
t
= 1 – ($10,200/$200)(.005167)
1/1.005167
t
= .7365
t
= ln (1/.7365) / ln 1.005167
t
= 59.35 months
68.
We need to find the FV of the premiums to compare with the cash payment promised at age 65. We have to
find the value of the premiums at year 6 first since the interest rate changes at that time. So:
FV
1
= $900(1.12)
5
= $1,586.11
FV
2
= $900(1.12)
4
= $1,416.17
FV
3
= $1,000(1.12)
3
= $1,404.93
FV
4
= $1,000(1.12)
2
= $1,254.40
FV
5
= $1,100(1.12)
1
= $1,232.00
Value at year six = $1,586.11 + 1,416.17 + 1,404.93 + 1,254.40 + 1,232.00 + 1,100
Value at year six = $7,993.60
Finding the FV of this lump sum at the child’s 65
th
birthday:
FV = $7,993.60(1.08)
59
= $749,452.56
The policy is not worth buying; the future value of the deposits is $749,452.56, but the policy contract will
pay off $500,000. The premiums are worth $249,452.56 more than the policy payoff.
Note, we could also compare the PV of the two cash flows. The PV of the premiums is:
PV = $900/1.12 + $900/1.12
2
+ $1,000/1.12
3
+ $1,000/1.12
4
+ $1,100/1.12
5
+ $1,100/1.12
6
PV = $4,049.81
And the value today of the $500,000 at age 65 is:
PV = $500,000/1.08
59
= $5,332.96
PV = $5,332.96/1.12
6
= $2,701.84
The premiums still have the higher cash flow. At time zero, the difference is $1,347.97. Whenever you are
comparing two or more cash flow streams, the cash flow with the highest value at one time will have the
highest value at any other time.
Here is a question for you: Suppose you invest $1,347.97, the difference in the cash flows at time zero, for