Question

18. If you have $1,000 to deposit in a savings account for 1 year, which of the following should you choose: 8.75% compounded continuously, 9% compounded quarterly, or 9.5% compounded annually. a) 8.75% compounded continuously d) a and b are equivalent and are best. b) 9% compounded quarterly e) a, b and c are all exactly equivalent. c) 9.5% compounded annually

Answer 1

Answer:

c) 9.5% compounded annually

Explanation:

effective rate for a)

$e^{0.0875}=1+r$

1.091442264 = 1+r

r = 0.09144= 9.14%

effective rate for b)

$(1+0.09/4)^4 = 1+r_e$

1.093083319 = 1+ re

re = 0.0931 = 9.31%

effective rate for c)

as it comounds annuity it is the effective rate already 9.5%

As we are capitalizing the interest we want the higher rate thus 9.5 percent compounding annually

Answer 2

Answer: 9.5% compounded annually (option c)

Explanation:

Formula for calculating continuous compounding:

Fv = Pv × [(e)^(i × t)]

Where Fv = future value

Pv = present value

e = mathematical constant approximated as 2.7183

Now, in the first case ---- 8.75% compounded continuously

Fv = Pv × [(e)^(i × t)]

Here "Pv" is $1,000. "i" is 0.0875 (divide 8.75 by 100) and "t" is 1

Therefore Fv = 1000 × (2.7183)^(0.0875 × 1)

= 1000 × [(2.7183)^(0.0875)]

= 1000 × 1.091443

= $ 1091.44

Subtracting $1000 from $1091.44:

1091.44 - 1000

This means that I will gain $91.44 over this period (1 year).

If it was 9% compounded quarterly:-

We apply the formula

Fv = Pv × [1 + (i/n)^(n×t)]

Where Pv = present value of investment

i = stated interest rate

n = number of compounding periods

t = time in years

Here, Pv is $1,000. "i" is 9% or 0.09, "t" is 1year and "n" is 4(since it will be compounded quarterly).

Fv = 1000 × [1 + (0.09/4)^(4×1)]

= 1000 × [(1.0225)^4]

= $1,093.08

Subtracting the pv from 1,093.08 it is clear that I'll gain $93.08 over this period

For 9.5% compounded annually (n is 1 in this case):

Fv = 1000 × [1 + ((0.095/1)^(1×1)]

= 1000 × [(1.095)^1]

= $1,095

Subtracting the Pv from the Fv, it shows that I'll gain $95 if it's compounded this way over the same period.

Since I will always prefer the system that can yield the most return, it's only logical that I go for option c (9.5% compounded annually)