Value a continuously paid income stream

A bank is arranging a 20-year loan to a solar farm, secured on its power sales. The farm sells power around the clock, so its income arrives continuously. It earns $2.4 million a year now, indexed to grow at a continuous 2% a year.

The bank's model values that income as a continuous stream. The buyer, though, pays at the end of each month for the power delivered in it. The credit team wants both values, and the cost of the monthly timing.

Your task

1. Write pv_continuous(c0, g, r, T): the present value of income paid continuously at $c_0 e^{gt}$ a year for $T$ years, discounted at the continuous rate $r$. Raise ValueError if T is negative.
2. Write monthly_payments(c0, g, T): the list of monthly payments, first month first. Payment $m$ is the income earned during month $m$, paid at its end. Raise ValueError if T is negative or is not a whole number of months.
3. Write pv_monthly(c0, g, r, T): the present value of those payments, each discounted at $r$ from the end of its month.

Keep full precision when $g$ equals $r$ or nearly does, and when $g$ is 0 or tiny.

The data

INCOME, GROWTH, RATE and YEARS in the code hold the farm's terms. Rates are annual decimals, continuously compounded, and times are in years.

Hint: Where to start

Lesson 2.5.5 values a growing stream as the integral of $c_0 e^{gt} e^{-rt}$ over $[0, T]$. Write the closed form with math.expm1, as that lesson does, so that a tiny $r - g$ keeps its digits.

Hint: The monthly payments

The income earned between times $a$ and $b$ is the integral of its rate from $a$ to $b$ (lesson 2.5.3). An antiderivative of $c_0 e^{gt}$ is $\frac{c_0}{g} e^{gt}$ (lesson 2.5.2), so the payment is $\frac{c_0}{g}\left(e^{gb} - e^{ga}\right)$. Factor out $e^{ga}$ and use math.expm1 for the rest.

Hint: Checking your answer

Paid at month ends, each month's income arrives on average half a month late. So the monthly value should be about $e^{-r/24}$ times the continuous one. In the same way, lesson 2.5.5 found mid-year payments worth almost as much as the continuous stream.