tom is comparing two investments. investment a pays an annual $10000 stream of cash flows that last for 25 years. each year the investment pays out at the beginning of the year. investment b pays a similar $10000 stream of cash flows, however, the payments are made at the end of the year. also, investment b does not make a payment in year 17. both investments pay a 4.25% rate of return. what is the difference in present value between the two investments?

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To find the difference in present value (PV) between Investment A and Investment B, we analyze their cash flow timing and amounts, given:

Both pay $10,000 annually for 17 years.
Investment A pays at the beginning of each year (annuity due).
Investment B pays at the end of each year (ordinary annuity), but skips the payment in year 17.
Both have a 4% annual discount rate.

Step 1: Define the cash flows

Investment A: $10,000 at the start of each year for 17 years.
Investment B: $10,000 at the end of each year for 16 years (no payment in year 17).

Step 2: Present Value formulas

Annuity due (payments at beginning):

PV=P×1−(1+r)−nr×(1+r)PV=P\times \frac{1-(1+r)^{-n}}{r}\times (1+r)PV=P×r1−(1+r)−n×(1+r)

Ordinary annuity (payments at end):

PV=P×1−(1+r)−nrPV=P\times \frac{1-(1+r)^{-n}}{r}PV=P×r1−(1+r)−n

Where:
P=10,000P=10,000P=10,000,
r=0.04r=0.04r=0.04,
n=n=n= number of payments.

Step 3: Calculate PV for Investment A

Since payments are at the beginning for 17 years:

PVA=10,000×1−(1+0.04)−170.04×1.04PV_A=10,000\times \frac{1-(1+0.04)^{-17}}{0.04}\times 1.04PVA=10,000×0.041−(1+0.04)−17×1.04

Calculate the factor:

1−(1.04)−170.04=1−1/1.93870.04=1−0.51570.04=0.48430.04=12.1075\frac{1-(1.04)^{-17}}{0.04}=\frac{1-1/1.9387}{0.04}=\frac{1-0.5157}{0.04}=\frac{0.4843}{0.04}=12.10750.041−(1.04)−17=0.041−1/1.9387=0.041−0.5157=0.040.4843=12.1075

Multiply by 1.04:

12.1075×1.04=12.591812.1075\times 1.04=12.591812.1075×1.04=12.5918

So,

PVA=10,000×12.5918=125,918PV_A=10,000\times 12.5918=125,918PVA=10,000×12.5918=125,918

Step 4: Calculate PV for Investment B

Payments at the end of each year for 16 years (since no payment in year 17):

PVB=10,000×1−(1+0.04)−160.04PV_B=10,000\times \frac{1-(1+0.04)^{-16}}{0.04}PVB=10,000×0.041−(1+0.04)−16

Calculate the factor:

1−(1.04)−160.04=1−1/1.8720.04=1−0.53440.04=0.46560.04=11.64\frac{1-(1.04)^{-16}}{0.04}=\frac{1-1/1.872}{0.04}=\frac{1-0.5344}{0.04}=\frac{0.4656}{0.04}=11.640.041−(1.04)−16=0.041−1/1.872=0.041−0.5344=0.040.4656=11.64

So,

PVB=10,000×11.64=116,400PV_B=10,000\times 11.64=116,400PVB=10,000×11.64=116,400

Step 5: Find the difference in present value

Difference=PVA−PVB=125,918−116,400=9,518Difference=PV_A- PV_B=125,918-116,400=9,518Difference=PVA−PVB=125,918−116,400=9,518

Answer:

The difference in present value between Investment A and Investment B is approximately $9,518 , with Investment A being more valuable due to earlier payments and the full 17-year stream compared to Investment B's 16 payments at year-end

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