Question Compounding frequency and time value: You plan to invest $2,000 in an i
ID: 2776988 • Letter: Q
Question
Question
Compounding frequency and time value: You plan to invest $2,000 in an individual retirement arrangement (IRA) today at a nominal annual rate of 8%, which is expected to apply to all future years.
a. How much will you have in the account at the end of 10 years if interest is compounded (1) annually, (2) semiannually, (3) daily (assume 365-day year), and (4) continuously?
b. What is the effective annual rate (EAR) for each compoundings period in part a?
c. How much greater will your IRA balance be at the end of 10 years if interest is compounded continuously rather than annually?
d. How does the compounding frequency affect the future value and effective annual rate for a given deposit? Explain in terms of your findings in part a throughc.
Explanation / Answer
Calculation of Future value at the end of 10 years:
1) If interest is compounded annually:
Future value = Present value * (1+r)^n
Present value =2000
r= rate of interest = 8% = 0.08
n = number of year = 10
Hence ,
Future value = 2000 * (1+0.08)^10
= 2000 * 2.158925
= $4317.85
2) If interest is compounded Semi-annually:
Future value = Present value * (1+r)^n
Present value =2000
r= rate of interest = 8%/2 = 0.04
n = number of semiannual = 10*2= 20
Hence ,
Future value = 2000 * (1+0.04)^20
= 2000 * 2.1911231
= $ 4,382.25
3) If interest is compounded Daily:
Future value = Present value * (1+r)^n
Present value =2000
r= rate of interest = 8%/365 = 0.0002192
n = number of days = 10*365= 3650
Hence ,
Future value = 2000 * (1+0.0002192)^3650
= 2000 * 2.22552
= $ 4,451.05
4) If interest is compounded continuously:
Future value = Pe ^rt
P = Present value = 2000
e = constant value= 2.718
r= rate of interest = 8% = 0.08
t= years = 10
Future value = 2000*(2.718 ^(0.08*10))
= 2000*2.2253563
= $ 4,450.71
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