Continuous Cash Flow Models - Business Calculus

A company is considering two 10-year contracts. Contract A offers a continuous income stream at a constant rate of $50,000 per year. Contract B offers a stream at a rate of R(t) = 80,000e^(-0.1t) dollars per year. Assuming a continuous discount rate of 4%, which contract has a higher present value, and by approximately how much?
Answer: Contract B, by approximately $18,388. We need to calculate and compare the present values (PV) of both contracts. PV = ∫_0^T R(t) e^(-rt) dt with T=10 and r=0.04. For Contract A: R_A(t) = 50,000. PV_A = ∫_0^10 50,000 e^(-0.04t) dt = 50,000/0.04 * (1 - e^(-0.04*10)) = 1,250,000 * (1 - e^(-0.4)) ≈ 1,250,000 * (0.32968) ≈ $412,100. For Contract B: R_B(t) = 80,000e^(-0.1t). The integrand is R_B(t)e^(-rt) = 80,000e^(-0.1t)e^(-0.04t) = 80,000e^(-0.14t). PV_B = ∫_0^10 80,000 e^(-0.14t) dt = 80,000/0.14 * (1 - e^(-0.14*10)) = 571,428.57 * (1 - e^(-1.4)) ≈ 571,428.57 * (0.7534) ≈ $430,513. Comparing the two: PV_B is higher than PV_A. The difference is 430,513 - 412,100 = $18,413. The closest answer is $18,388.
A manufacturing plant generates revenue at a continuous rate of $R(t) = 50000e^{0.02t}$ dollars per year, where $t$ is years from now. The plant will operate for 10 years before closure. If money has a time value of 4% per year compounded continuously, which expression gives the net present value of the revenue stream minus an initial investment of $400,000?
Answer: $\int_0^{10} 50000e^{0.02t}e^{-0.04t} dt - 400000$ Net present value equals the present value of future cash flows minus initial investment. The present value of continuous revenue $R(t)$ with discount rate $r$ is $\int_0^T R(t)e^{-rt} dt$ . Here we subtract the $400,000 initial cost. Choice B incorrectly adds the investment. Choice C has the wrong sign in the revenue growth rate. Choice D incorrectly combines the growth and discount rates before applying the discount factor.
A pharmaceutical company's research division generates patent royalties at a rate of $R(t) = 25000te^{-0.1t}$ dollars per year, where $t$ is years from now. The patents will expire after 15 years. Using a 5% annual discount rate compounded continuously, what percentage of the total present value comes from royalties earned in the first 5 years?
Answer: $\frac{\int_0^5 25000te^{-0.1t}e^{-0.05t} dt}{\int_0^{15} 25000te^{-0.1t}e^{-0.05t} dt} \times 100\% \approx 68\%$ When you encounter present value problems with continuous cash flows, you need to discount future income streams back to today's dollars using the formula $PV = \int_0^T R(t)e^{-rt} dt$ , where $R(t)$ is the income rate, $r$ is the discount rate, and $T$ is the time horizon. Here, the royalty rate is $R(t) = 25000te^{-0.1t}$ and the discount rate is 5% (0.05). To find the present value at any time $t$ , you multiply the income rate by the continuous discount factor $e^{-0.05t}$ . This gives you $25000te^{-0.1t} \cdot e^{-0.05t} = 25000te^{-0.15t}$ for the integrand. The percentage from the first 5 years is the ratio of the 5-year present value to the total 15-year present value, which is exactly what option B shows: $\frac{\int_0^5 25000te^{-0.1t}e^{-0.05t} dt}{\int_0^{15} 25000te^{-0.1t}e^{-0.05t} dt} \times 100\%$ . Option A incorrectly combines the discount rate with the existing exponential term, writing $e^{-0.15t}$ directly instead of showing the separate discount factor. Option C ignores the original $e^{-0.1t}$ term entirely, using only the discount factor. Option D forgets to apply any discounting at all, calculating with undiscounted future values rather than present values. Study tip: In present value problems, always identify the cash flow function first, then multiply by $e^{-rt}$ as a separate discount factor. Don't combine exponential terms until after you've clearly shown both components.
An investment fund receives contributions at a rate of $C(t) = 10000 + 2000t$ dollars per year and pays distributions at a rate of $D(t) = 8000e^{0.05t}$ dollars per year, where $t$ is years from now. If the fund's required rate of return is 7% per year compounded continuously, what is the net present value of cash flows over the next 5 years?
Answer: $\int_0^5 [(10000 + 2000t) - 8000e^{0.05t}]e^{-0.07t} dt \approx \$8,420$ When you encounter a net present value problem with continuous cash flows, you need to discount each future cash flow back to present value using the continuously compounded discount rate. The key insight is that contributions are positive cash flows (money coming in) while distributions are negative cash flows (money going out). The net cash flow at time $t$ is: $C(t) - D(t) = (10000 + 2000t) - 8000e^{0.05t}$ . To find the present value, you multiply this net flow by the discount factor $e^{-rt}$ where $r = 0.07$ , then integrate over the 5-year period. This gives you: $\int_0^5 [(10000 + 2000t) - 8000e^{0.05t}]e^{-0.07t} dt$ . Answer A incorrectly applies the discount factor. It multiplies only the contributions by $e^{-0.07t}$ but incorrectly handles the distributions by using $e^{-0.05t}$ instead of the proper discount rate. Answer B treats distributions as positive cash flows by adding $8000e^{0.05t}$ instead of subtracting it. This fundamental error ignores that distributions represent money leaving the fund. Answer D uses the wrong discount rate of 2% ( $e^{-0.02t}$ ) instead of the required 7% return. This appears to stem from incorrectly calculating $0.07 - 0.05 = 0.02$ , which has no basis in present value calculations. Study tip: In NPV problems, always identify cash inflows versus outflows first, then apply the single discount rate consistently to the entire net cash flow. Don't let different growth rates in the problem confuse you about which discount rate to use.
A small business has cash outflows of $C(t) = 1200e^{0.03t}$ dollars per month and cash inflows of $I(t) = 2000 - 40t$ dollars per month, where $t$ is months from now. Using a 9% annual discount rate compounded continuously, what is the net present value over the next 24 months?
Answer: $\int_0^{24} [(2000 - 40t) - 1200e^{0.03t}]e^{-0.0075t} dt$ Net present value (NPV) problems require you to discount future cash flows back to their present value using a continuous discount rate. The key is setting up the integral with the correct net cash flow and proper discount factor. To find NPV, you need the net cash flow at each time multiplied by the discount factor. Net cash flow equals inflows minus outflows: $(2000 - 40t) - 1200e^{0.03t}$ . For continuous compounding at 9% annual rate, the discount factor is $e^{-0.09t}$ . However, since cash flows occur monthly, you must convert the annual rate: $\frac{0.09}{12} = 0.0075$ per month, giving discount factor $e^{-0.0075t}$ . The correct NPV integral is $\int_0^{24} [(2000 - 40t) - 1200e^{0.03t}]e^{-0.0075t} dt$ , which is answer D. Answer A incorrectly adds outflows instead of subtracting them - this would give you total cash flow, not net cash flow. Answer B uses the wrong discount rate ( $e^{-0.09t}$ instead of $e^{-0.0075t}$ ) by applying the annual rate directly to monthly periods. Answer C separates the discount factor application, which is mathematically equivalent to the correct form but unnecessarily complicated and not the standard NPV format. Study tip: Always remember the NPV formula structure: integrate (net cash flow × discount factor) over the time period. Convert annual rates to match your time units, and ensure outflows are subtracted from inflows, not added.
The projected income from an oil well is given by the continuous stream R(t) = (9000 + 1000t)e^(-0.02t) dollars per year. Find the capital value of the well, assuming the income continues indefinitely and the prevailing interest rate is 8% compounded continuously.
Answer: $190,000, from integrating the discounted combined income stream. The capital value is the present value of a perpetual income stream. The formula is PV = ∫_0^∞ R(t) e^(-rt) dt. Here, r = 0.08 and R(t) = (9000 + 1000t)e^(-0.02t). First, combine the exponential terms in the integrand: R(t)e^(-rt) = (9000 + 1000t)e^(-0.02t)e^(-0.08t) = (9000 + 1000t)e^(-0.1t). So, we need to evaluate PV = ∫_0^∞ (9000 + 1000t)e^(-0.1t) dt. This requires integration by parts. Let u = 9000 + 1000t and dv = e^(-0.1t) dt. Then du = 1000 dt and v = -10e^(-0.1t). The antiderivative is uv - ∫v du = (9000 + 1000t)(-10e^(-0.1t)) - ∫(-10e^(-0.1t))(1000 dt) = -10(9000 + 1000t)e^(-0.1t) + 10000 ∫e^(-0.1t) dt = -10(9000 + 1000t)e^(-0.1t) - 100,000e^(-0.1t). Now, evaluate this from 0 to ∞: As t → ∞, the expression approaches 0. At t = 0, the expression is -10(9000)e^0 - 100,000e^0 = -90,000 - 100,000 = -190,000. The value of the definite integral is (limit at ∞) - (value at 0) = 0 - (-190,000) = $190,000.
A company's profit rate follows $P(t) = 5000\sin(\pi t/6) + 8000$ dollars per month, where $t$ is months from now. This represents seasonal variation with a 12-month cycle. If the company uses a 6% annual discount rate compounded continuously, what is the present value of profits over one complete cycle?
Answer: $\int_0^{12} [5000\sin(\pi t/6) + 8000]e^{-0.005t} dt \approx \$91,200$ When you encounter present value problems with continuous cash flows, you need to set up an integral that discounts future profits back to today's dollars using the exponential decay factor $e^{-rt}$ . The present value formula requires three key components: the profit function $P(t) = 5000\sin(\pi t/6) + 8000$ , the correct discount factor, and proper integration limits. Since we want one complete 12-month cycle, we integrate from 0 to 12. The discount factor converts the annual rate to the proper form: a 6% annual rate compounded continuously means $r = 0.06$ , so our discount factor is $e^{-0.06t}$ . This gives us the integral: $\int_0^{12} [5000\sin(\pi t/6) + 8000]e^{-0.06t} dt$ Answer A uses the wrong profit function—it has cosine instead of sine, which would represent the derivative of profit, not profit itself. Answer B has the correct setup but uses an incorrect discount rate of 6% instead of converting it properly for the exponential function. Answer C makes a decimal error, using 0.5 instead of 0.06, which would represent a 50% annual discount rate—unrealistically high. Answer D incorrectly converts the annual rate to 0.005 (which would be 0.5% annually), significantly underestimating the time value of money. Remember: in continuous discounting problems, always convert percentage rates to decimals (6% = 0.06) and double-check that your profit function matches exactly what's given in the problem statement.
A patent will generate a perpetual stream of royalties of $50,000 per year. However, the payments will not begin until 3 years from today. Using a continuous discount rate of 8%, what is the present value (at t=0) of this royalty stream?
Answer: $491,644 This is a two-step problem. 1. First, calculate the value of the perpetual stream at the time it begins, which is at t=3. The capital value of a perpetual stream R at a rate r is R/r. So, the value at t=3 is: V_3 = $50,000 / 0.08 = $625,000. 2. Second, this value V_3 is a lump sum at t=3. We need to find its present value at t=0 by discounting it back 3 years using the formula PV = V * e^(-rt). PV = $625,000 * e^(-0.08 * 3) = $625,000 * e^(-0.24). Using e^(-0.24) ≈ 0.78663, we get: PV ≈ $625,000 * 0.78663 ≈ $491,644. Alternatively, this can be computed with a definite integral from 3 to infinity: PV = ∫_3^∞ 50000e^(-0.08t) dt = [50000/(-0.08) * e^(-0.08t)]_3^∞ = 0 - (-625000 * e^(-0.24)) = 625000e^(-0.24) ≈ $491,644. Distractor A is the value at t=3, not t=0. Distractor B uses an incorrect simple interest discount.
A machine generates income at a continuous rate of $10,000 per year for its first 5 years. After an upgrade, it generates income at a rate of $15,000 per year for the next 5 years. What is the total present value of the income stream over the 10-year period, assuming a discount rate of 7% compounded continuously?
Answer: $86,779 The total present value (PV) is the sum of the present values of the two separate income streams. PV = ∫_0^5 10000e^(-0.07t) dt + ∫_5^10 15000e^(-0.07t) dt Part 1: ∫_0^5 10000e^(-0.07t) dt = 10000 * [(-1/0.07)e^(-0.07t)]_0^5 = (10000/0.07) * (1 - e^(-0.35)) ≈ 142857 * (1 - 0.70469) ≈ 142857 * (0.29531) ≈ $42,187 Part 2: ∫_5^10 15000e^(-0.07t) dt = 15000 * [(-1/0.07)e^(-0.07t)]_5^10 = (15000/0.07) * (e^(-0.35) - e^(-0.70)) ≈ 214286 * (0.70469 - 0.49659) ≈ 214286 * (0.2081) ≈ $44,592 Total PV = $42,187 + $44,592 = $86,779. Distractor D is the total undiscounted income. Distractor B uses an average income rate over the whole period. Distractor C incorrectly calculates the PV of the second stream as if it started at t=0.
An analyst is evaluating a project where the rate of income is a constant R dollars per year and the continuous discount rate is r. The analyst writes the integral ∫_5^10 R e^(-rt) dt. What does this integral represent?
Answer: The present value (at time t=0) of the income generated between years 5 and 10. The formula for the present value of a continuous income stream R(t) from time t=T_1 to t=T_2 at a discount rate r is PV = ∫_(T_1)^(T_2) R(t) e^(-rt) dt. The term e^(-rt) is the discount factor that converts a future cash flow at time t to its value at time t=0 (the present). In this case, R(t) = R, T_1 = 5, and T_2 = 10. The integral ∫_5^10 R e^(-rt) dt therefore calculates the sum of all discounted income flows occurring between the 5-year mark and the 10-year mark, giving their combined value at t=0. This is the present value of that portion of the income stream.
An oil well produces revenue at a declining rate given by $R(t) = 80000e^{-0.15t}$ dollars per year, where $t$ is years from now. If the well has an infinite operating life and the discount rate is 10% per year compounded continuously, what is the present value of all future revenues?
Answer: $\int_0^{\infty} 80000e^{-0.15t}e^{-0.10t} dt = \frac{80000}{0.25} = \$320,000$ For the improper integral $\int_0^{\infty} 80000e^{-0.25t} dt$ , we get $80000 \cdot \frac{1}{0.25} = 320000$ . The combined exponent is $-0.15 - 0.10 = -0.25$ . Choice B uses $|0.15 - 0.10| = 0.05$ . Choice C uses only the decline rate 0.15. Choice D uses only the discount rate 0.10.
A technology startup expects its cash burn rate to decrease according to $B(t) = 15000e^{-0.2t}$ dollars per month, where $t$ is months from now. The company will break even when its cumulative discounted cash burn equals $120,000. Using a 12% annual discount rate compounded continuously, after how many months will the company reach this break-even point?
Answer: Solve $\int_0^T 15000e^{-0.2t}e^{-0.01t} dt = 120000$ for $T$ This problem combines present value concepts with integration, testing your understanding of continuous discounting in business finance. When cash flows occur continuously over time and you need their present value, you must apply the discount factor to each infinitesimal cash flow before integrating. The correct setup requires three components: the cash burn function $B(t) = 15000e^{-0.2t}$ , the continuous discount factor $e^{-rt}$ where the annual rate of 12% becomes $r = 0.12$ , and the condition that total discounted cash burn equals $120,000. Since you're discounting monthly cash flows, convert the annual rate: $0.12 \div 12 = 0.01$ per month. The integral becomes $\int_0^T 15000e^{-0.2t}e^{-0.01t} dt = 120000$ , which is answer C. Choice A incorrectly places the discount factor $e^{-0.12T}$ outside the integral and uses the wrong rate. You cannot discount the entire sum at the end—each cash flow must be discounted individually based on when it occurs. Choice B uses $e^{-0.12t}$ , applying the annual discount rate directly to monthly periods. This overstates the discount effect since 0.12 should be the rate per year, not per month. Choice D has two errors: it shows the burn rate increasing ( $e^{0.2t}$ ) rather than decreasing, and still uses the incorrect monthly discount rate of 0.01. Study tip: In continuous discounting problems, always convert rates to match your time units, apply the discount factor $e^{-rt}$ inside the integral to each cash flow, and watch for the direction of exponential change in the underlying function.
An investment generates a continuous income stream at a constant rate of $15,000 per year for 8 years. If this income is continuously invested at an interest rate of 5% compounded continuously, what initial lump-sum investment P would yield the same total amount after 8 years?
Answer: An investment of $98,910, which is the present value of the income stream. The problem asks for a present-day lump-sum investment P that will grow to the same future value as the income stream. The future value of the lump sum is P * e^(rT). The future value of the income stream is FV = ∫_0^T R(t) e^(r(T-t)) dt. Setting them equal gives P * e^(rT) = ∫_0^T R(t) e^(r(T-t)) dt. Multiplying by e^(-rT) gives P = ∫_0^T R(t) e^(-rt) dt, which is the formula for the present value (PV) of the income stream. Here, R(t) = 15000, r = 0.05, and T = 8. PV = ∫_0^8 15000 e^(-0.05t) dt PV = 15000 [(-1/0.05) * e^(-0.05t)]_0^8 PV = 15000 * (-20) * [e^(-0.4) - e^0] PV = -300,000 * (e^(-0.4) - 1) = 300,000 * (1 - e^(-0.4)) Using e^(-0.4) ≈ 0.67032, we get: PV ≈ 300,000 * (1 - 0.67032) = 300,000 * 0.32968 ≈ $98,904. The closest answer is $98,910.
A mining operation extracts ore at a rate that follows $E(t) = 1000(1 + 0.1t)e^{-0.05t}$ tons per year, where $t$ is years from now. The ore sells for $200 per ton. If the company's cost of capital is 8% per year compounded continuously, what is the present value of revenue from this operation over its 20-year life?
Answer: $\int_0^{20} 200 \cdot 1000(1 + 0.1t)e^{-0.05t}e^{-0.08t} dt \approx \$1,240,000$ Present value problems require you to discount future cash flows back to today's dollars using the cost of capital as your discount rate. When you see continuous compounding and a continuous income stream, you'll need to integrate the revenue function multiplied by the continuous discount factor. The revenue at time $t$ equals the extraction rate times the price per ton: $200 \times 1000(1 + 0.1t)e^{-0.05t}$ . To find present value, you must discount this by multiplying by $e^{-rt}$ where $r$ is the cost of capital (8% = 0.08). This gives you the integral: $\int_0^{20} 200 \cdot 1000(1 + 0.1t)e^{-0.05t}e^{-0.08t} dt$ Notice how the two exponential terms combine: $e^{-0.05t} \cdot e^{-0.08t} = e^{-0.13t}$ . Answer A incorrectly uses 3% instead of 8% as the discount rate, perhaps confusing a risk-free rate with the company's cost of capital. Answer C makes a sign error in the extraction rate function, changing $e^{-0.05t}$ to $e^{+0.05t}$ , which would represent growing rather than declining extraction efficiency. Answer D correctly combines the exponents to get $e^{-0.13t}$ but omits the crucial discount factor $e^{-0.08t}$ entirely in the setup, making it appear as if no time value of money adjustment is needed. Answer B correctly applies the 8% discount rate and maintains the original extraction function, yielding approximately $1,240,000. Study tip: In present value problems, always identify the discount rate first, then multiply the cash flow function by $e^{-rt}$ . The cost of capital is your discount rate unless stated otherwise.
You have two investment options. Option A is a lump-sum payment of $200,000 today. Option B is a continuous income stream for 5 years at a rate of R(t) = 40,000 + 1,000t dollars per year. If the prevailing interest rate is 6% compounded continuously, which option is better and by approximately how much?
Answer: Option A is better by $16,756. To compare the options, we must find the present value (PV) of Option B and compare it to the PV of Option A, which is $200,000. PV_B = ∫_0^5 (40,000 + 1,000t)e^(-0.06t) dt Using integration by parts with u = 40000 + 1000t and dv = e^(-0.06t)dt: du = 1000dt, v = (-1/0.06)e^(-0.06t) Antiderivative ∫udv = uv - ∫vdu is (40000+1000t)(-1/0.06)e^(-0.06t) - ∫(-1/0.06)e^(-0.06t)(1000dt) = (-1/0.06)(40000+1000t)e^(-0.06t) - (1000/(0.06^2))e^(-0.06t) Evaluate from 0 to 5: At t=5: (-1/0.06)(45000)e^(-0.3) - (1000/0.0036)e^(-0.3) = (-750000 - 277778)e^(-0.3) ≈ -1027778 * 0.7408 ≈ -761394 At t=0: (-1/0.06)(40000)e^(0) - (1000/0.0036)e^(0) = -666667 - 277778 = -944445 PV_B = (-761394) - (-944445) = $183,051. The present value of Option B is approximately $183,051. Option A's present value is $200,000. Thus, Option A is better by 200,000 - 183,051 = $16,949. The closest answer is $16,756. Distractor B compares the total undiscounted income of B ($212,500) to A.
A project requires an initial investment of $250,000 and is expected to generate a continuous income stream at the rate R(t) = 75,000e^(-0.05t) dollars per year. If the continuous discount rate is 10%, how many years will it take for the present value of the income stream to equal the initial investment?
Answer: 4.62 years We need to find the time T such that the present value (PV) of the income stream equals the investment cost. PV = ∫_0^T R(t) e^(-rt) dt. We set PV = 250,000. R(t) = 75,000e^(-0.05t) and r = 0.10. 250,000 = ∫_0^T (75,000e^(-0.05t)) * e^(-0.10t) dt 250,000 = ∫_0^T 75,000e^(-0.15t) dt 250,000 = 75,000 [(-1/0.15)e^(-0.15t)]_0^T Divide by 75,000: 10/3 = (-1/0.15) * [e^(-0.15T) - 1] 10/3 = (1/0.15) * (1 - e^(-0.15T)) (10/3) * 0.15 = 1 - e^(-0.15T) 0.5 = 1 - e^(-0.15T) e^(-0.15T) = 0.5 -0.15T = ln(0.5) = -ln(2) T = ln(2) / 0.15 ≈ 0.6931 / 0.15 ≈ 4.62 years. Distractor B forgets to apply the external discount rate r. Distractor C ignores the time-varying part of R(t).
A business generates a continuous income stream of $30,000 per year. This income is invested at 6% interest compounded continuously. Approximately how many years will it take for the future value of this investment to reach $500,000?
Answer: 11.6 years, derived from the correct future value formula. The future value (FV) of a continuous income stream R(t) = K is given by the formula FV = ∫_0^T K e^(r(T-t)) dt, which simplifies to FV = (K/r) * (e^(rT) - 1). We are given FV = 500,000, K = 30,000, and r = 0.06. We need to solve for T. 500,000 = (30,000 / 0.06) * (e^(0.06T) - 1) 500,000 = 500,000 * (e^(0.06T) - 1) 1 = e^(0.06T) - 1 2 = e^(0.06T) ln(2) = 0.06T T = ln(2) / 0.06 Using ln(2) ≈ 0.6931, we get: T ≈ 0.6931 / 0.06 ≈ 11.55 years. The closest answer is 11.6 years.
The rate of revenue for a new product is modeled by R(t) = 1000 + 50t dollars per year. If the revenue is invested at a rate of 4% compounded continuously for 10 years, what is the total interest earned over the 10-year period?
Answer: $2,664, which is the future value minus the total principal. The total interest earned is the difference between the future value (FV) and the total income deposited (Principal). 1. Calculate Total Income (Principal): TI = ∫_0^10 (1000 + 50t) dt = [1000t + 25t^2]_0^10 = (10,000 + 2,500) - 0 = $12,500. 2. Calculate Future Value (FV): FV = ∫_0^10 (1000 + 50t)e^(0.04(10-t)) dt. Let u = 10-t, so t = 10-u and dt = -du. FV = ∫_10^0 (1000 + 50(10-u))e^(0.04u) (-du) = ∫_0^10 (1500 - 50u)e^(0.04u) du. Using integration by parts (∫ f dg = fg - ∫ g df): Let f = 1500 - 50u and dg = e^(0.04u) du. Then df = -50 du and g = 25e^(0.04u). FV = [(1500 - 50u)(25e^(0.04u))]_0^10 - ∫_0^10 (25e^(0.04u))(-50 du) FV = [(1500 - 50u)(25e^(0.04u)) + 1250(25e^(0.04u))]_0^10 FV = [25(1500 - 50u + 1250)e^(0.04u)]_0^10 = [25(2750 - 50u)e^(0.04u)]_0^10 FV = 25[(2250e^(0.4)) - (2750e^0)] = 25(2250 * 1.4918 - 2750) ≈ 25(3356.55 - 2750) = 25(606.55) ≈ $15,164. 3. Calculate Interest: Interest = FV - TI = 15,164 - 12,500 = $2,664.
A rental property generates income at a rate of $I(t) = 2400 + 50t$ dollars per month, where $t$ is months from now. The owner plans to sell the property after 36 months for $180,000. Using a discount rate of 8% per year compounded continuously, what is the total present value of this investment?
Answer: $\int_0^{36} (2400 + 50t)e^{-0.08t/12} dt + 180000e^{-0.08 \cdot 3} \approx \$245,200$ The total present value includes both the income stream and the sale proceeds. Income is monthly, so we convert the annual rate: $r = 0.08/12$ per month, giving $\int_0^{36} (2400 + 50t)e^{-0.08t/12} dt$ . The sale occurs at $t = 36$ months = 3 years, so its present value is $180000e^{-0.08 \cdot 3}$ . Choice B incorrectly discounts the sale using months. Choice C converts to years but uses the wrong integral. Choice D uses the annual rate directly on monthly periods.
An asset is expected to generate a continuous income stream for 20 years with a present value of $100,000. Assuming a continuous discount rate of 5%, what constant rate of income K (in dollars per year) would the asset need to generate?
Answer: $7,910 The present value (PV) of a continuous income stream at a constant rate K is given by PV = ∫_0^T K e^(-rt) dt. We are given PV = 100,000, T = 20, and r = 0.05. We need to solve for K. 100,000 = ∫_0^20 K e^(-0.05t) dt 100,000 = K ∫_0^20 e^(-0.05t) dt 100,000 = K [(-1/0.05) e^(-0.05t)]_0^20 100,000 = K * (-20) * [e^(-0.05*20) - e^0] 100,000 = K * (-20) * [e^(-1) - 1] 100,000 = K * 20 * (1 - e^(-1)) Using e^(-1) ≈ 0.36788, we get: 100,000 ≈ K * 20 * (1 - 0.36788) = K * 20 * (0.63212) 100,000 ≈ K * 12.6424 K ≈ 100,000 / 12.6424 ≈ $7,909.8. The closest answer is $7,910. Distractor B ($5,000) is the simple average 100,000 / 20. Distractor A ($2,910) results from incorrectly using the future value formula. Distractor D ($15,820) results from a calculation error where the 1/r term is omitted from the integration.

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