Central Texas College Money and Banking Questions
ANSWER
We can use the formula for the present Value of an annuity to determine the lump sum that needs to be invested today in order to make annual payments of $2000 for seven years at a 5% annual interest rate compounded annually:
PV=PMT×(1−(1+r)−n)rPV=rPMT×(1−(1+r)−n)
Where:
PVPV = Present Value, which refers to the upfront payment you must make.
PMTPMT stands for Payment Amount Per Period (in this case, $2000).
(5% or 0.05) rr = Interest rate per period
nn = Number of periods (in this instance, seven years).
Inserting the values:
PV=2000×(1−(1+0.05)−7)0.05PV=0.052000×(1−(1+0.05)−7)PV=2000×(1−0.71178)0.05PV=0.052000×(1−0.71178)PV=2000×0.288220.05PV=0.052000×0.28822PV=576.440.05PV=0.05576.44PV=11528.8PV=11528.8
Therefore, a one-time contribution of roughly $11,528.80 is required today.
(a) Continuous Compounding: The following formula can be used to estimate the future Value of an investment that uses continuous compounding:
FV=P×ertFV=P×ert
Where:
Future Value = FVFV
PP = $20k initial principal
Euler’s Number, ee, is around 2.71828.
rr = Periodic Interest Rate (4.75% or 0.0475)
The Number of years, tt, is five.
Inserting the values:
FV=20000×e0.0475×5FV=20000×e0.0475×5 FV=20000×e0.2375FV=20000×e0.2375 FV = 20000 1.26763 FV = 20000 1.26763 FV = 25352.6 FV = 25352.6
(b) Quarterly Compounding: Using quarterly compounding, the formula for determining the future Value is as follows:
FV=P×(1+rn)ntFV=P×(1+nr)nt
Where:
Future Value = FVFV
PP = $20k initial principal
(5.25% or 0.0525) = Interest rate per period
nn is the Number of annual compounding periods (four for quarterly).
(10 years) tt = Number of years
Inserting the values:
FV=20000×(1+0.05254)4×10FV=20000×(1+40.0525)4×10 FV=20000×(1.013125)40FV=20000×(1.013125)40 FV 20 000 1, 60 578 1 FV 32 115 62 FV 32 115 62
3. (A) Distinctly Monthly Compounding
The following equation can be used to determine the future value using monthly compounding:
FV=P×(1+rn)ntFV=P×(1+nr)nt
Where:
Future Value = FVFV
PP = $1,200 initial principal
(8% or 0.08) rr = Interest rate per period
nn is the Number of annual compounding periods (12 for monthly).
Four years is the Number of years in it.
Inserting the values:
FV=1200×(1+0.0812)12×4FV=1200×(1+120.08)12×4 FV=1200×(1.00666667)48FV=1200×(1.00666667)48 FV 1200 1.328.6 FV 1200 1.328.6 FV 1594.04 FV 1594.04
(B) Continuous Compounding: The same formula as in Question 2a can be used for continuous compounding.
FV=P×ertFV=P×ert
Where:
Future Value = FVFV
PP = $1,200 initial principal
Euler’s Number, ee, is around 2.71828.
(8% or 0.08) rr = Interest rate per period
Four years is the Number of years in it.
Inserting the values:
FV=1200×e0.08×4FV=1200×e0.08×4 FV=1200×e0.32FV=1200×e0.32 FV 1200.378811 FV 1200.378811 FV 165.47 FV 165.47
Please be aware that the values have been rounded for convenience.
Question Description
I’m working on a economics multi-part question and need the explanation and answer to help me learn.
1.) What lump sum deposited today would allow payments $2000/ a year, for 7 years, at 5% compounded annually?
2.) If you invest $20,000 at an annual interest rate of 4.75%, continuously, calculate the future value (FV) of your investment over a 5-year period. Then, go back and calculate the future-value (FV) of your initial $20,000 investment with a discrete-quarterly compounded annual interest rate of 5.25%, over a 10-year period.
3.) Find the accumulated amount (future-value [FV]) after 4 years of $1,200 invested at an 8% annual interest rate compounded:
(A) Discretely monthly
(B) Continuously
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