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Find the range of interest rates for which each of the contracts in (question below) has a higher present value then the other two. The contracts are as follows

$(a)$ $3,200,000$ per year for the next five years, payable at the end of each year

$(b)$ $3,000,000$ per year for the next five years, payable at the beginning of each year

$(c)$ $1,800,000$ per year for the next ten years, payable at the end of each year.

I do not think I am doing this right at all. I was think that I would set them equal to each other to come out with a quadratic formula at which I could find which rates are better but that isn’t working for me. I know the answers are supposed to be (a) is greatest for $0.05155 \lt i \lt 0.06667$ (b) is greatest for $i \gt 0.06667$ and c is greatest for $i \lt 0.05155$ I am not sure how to get these answers. Please help. Thank you.

callculus42
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Harper Jeffrey
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1 Answers1

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In general there are two formulas for present value for n annuities:

a) payable at the end of the years:

$$PV=r\cdot \frac{1-q^n}{1-q}\cdot \frac1{q^n}$$

$q=1+i$

b) payable at the beginning of the years:

$$PV=r\cdot q\cdot \frac{1-q^n}{1-q}\cdot \frac1{q^n}$$

For instance, contract (a) has a higher present value than contract (b) if the following inequality holds:

$$3,200,000\cdot \frac{1-q^5}{1-q}\cdot \frac1{q^5}>3,000,000\cdot q\cdot \frac{1-q^5}{1-q}\cdot \frac1{q^5}$$

Now we can divide the inequality by $\frac{1-q^5}{1-q}\cdot \frac1{q^5}$ and $3,000,000$ and we get

$\frac{32}{30}>q$

$\frac{32}{30}>1+i$

$\frac{32}{30}-\frac{30}{30}>i$

$i<\frac2{30}\approx 0.06666$

I think you can manage the rest.

callculus42
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