Question

Eric deposits $100 into a savings account at time 0, which pays interest at a nominal rate of i, compounded semiannually. Mike deposits $200 into a different savings account at time 0, which pays simple interest, at an annual rate of i. Eric and Mike earn the same amount of interest during the
last 6 months of the 8th year. Calculate i. Choose one of the following. Source: Society of Actuaries.

(a) 9.06%
(b) 9.26%
(c) 9.46%
(d) 9.66%
(e) 9.86%

Answer 1

Answer:

Option (c) 9.46%

Step-by-step explanation:

Amount deposited by Eric = $100

Amount deposited by Mike = $200

Interest rate = i

Now,

For Eric

Amount after 7.5 years, A = Principle × $( 1 +\frac{i}{2})^{2\times15}$

=  $100 × $( 1 +\frac{i}{2})^{2\times15}$

thus,

for the last 6 months i.e 0.5 year =  A × $( 1 +\frac{i}{2})^{2\times0.5}$

= A × $( 1 +\frac{i}{2})^{1}$

therefore,

Interest earned = A × $( 1 +\frac{i}{2})^{1}$ - A

= A × $( \frac{i}{2})$

= $100 × $( 1 +\frac{i}{2})^{2\times15}\times( \frac{i}{2})$

= $50i $\times( 1 +\frac{i}{2})^{2\times15}$ ......(1)

For Mike

Interest  = Principle × i × Time

= $200 × i × 0.5

= $100i .........(2)

Equating (1) and (2)

$50i $\times( 1 +\frac{i}{2})^{2\times15}$ = $100i

or

$( 1 +\frac{i}{2})^{2\times15}$ = 2

or

$( 1 +\frac{i}{2})$  = 1.0473

or

$\frac{i}{2}$   =  1.0473 - 1

or

$\frac{i}{2}$  = 0.0473

or

i = 0.0946

or

i = 0.0946 × 100% = 9.46%

Hence,

Option (c) 9.46%