Answer:
a. $42,572 at 2%,
b. $35,559 at 5%
c. $29,702 at 8%
Step-by-step explanation:
The formula used for FV calculation for Continuous Compounding is as under:
[tex]FV = PV e^{i * t}[/tex]
Where,
FV = Future Value = $8000 each year (At the end of 6 years = $8000 x 6 = $48,000)
PV = Present Value
e = Mathematical Constant = 2.713
i = Interest Rate
t= time in years
a) For 2%:
[tex]FV = PV e^{i * t}\\48000 = PV e^{0.02 * 6}\\48000 = 1.1275 (PV)\\PV = 42,572[/tex]
b) For 5%:;
[tex]FV = PV e^{i * t}\\48000 = PV e^{0.05 * 6}\\48000 = 1.35 (PV)\\PV = 35,559\\[/tex]
c) For 8%:
[tex]FV = PV e^{i * t}\\48000 = PV e^{0.08 * 6}\\48000 = 1.616 (PV)\\PV = 29,702[/tex]
Note: Investing $42,572 at 2%, $35,559 at 5% and $29,702 at 8% today will get $48,000 at the end of 6 years.
Answer:
To determine the present value at the rate r, compounded continuously we have
P= Ae-rt
where
P = Present value
A=amount = $8000
r=interest rate
t=time taken = 6years
dP = 8000[tex]\int\limits^6_0 {e-rt} \, dt[/tex]
a) So now for r = 2% = 0.02, we have
dP = -8000(e-0.02t[tex]\left \{ {{t=6} \atop {t=0}} \right.[/tex])
dP = -8000 (e-3/25 - 1) = -8000 (-0.1131)
dP = $905 at 2%
b) for r = 5% = 0.05, we have
dP = -8000(e-0.05t[tex]\left \{ {{t=6} \atop {t=0}} \right.[/tex])
dP = -8000 (e-3/10 - 1) = -8000 (-0.2592)
dP = $2074 at 5%
c) for r = 8% = 0.08, we have
dP = -8000(e-0.08t[tex]\left \{ {{t=6} \atop {t=0}} \right.[/tex])
dP = -8000 (e-12/25 - 1) = -8000 (-0.3812)
dP = $3050 at 8%
