Answer:
c) 9.5% compounded annually
Explanation:
effective rate for a)
[tex]e^{0.0875}=1+r\\[/tex]
1.091442264 = 1+r
r = 0.09144= 9.14%
effective rate for b)
[tex](1+0.09/4)^4 = 1+r_e[/tex]
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: 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)
