31:T56d,Plot.plot({ style: "overflow: visible;", color: { range: ["#77b6b2", "#4f79a7", "#e25759", "#f28e2c"] }, y: { grid: true, label: "↑ price (ref)" }, marks: [ Plot.line(prices, { x: "time", y: "price", stroke: "design" }), Plot.text(prices, { x: "time", y: "price", z: "design", text: "design", textAnchor: "start", fontWeight: 600, dy: -4, dx: 4, filter: (d) => d.time === math.round(priceGraphInputs.term / 2, 1) && d.design === "Collect" }), Plot.text(prices, { x: "time", y: "price", z: "design", text: "design", textAnchor: "start", fontWeight: 600, dy: 0, dx: 4, filter: (d) => d.time === priceGraphInputs.term && d.design === "Pure FY" }), Plot.text(prices, { x: "time", y: "price", z: "design", text: "design", textAnchor: "start", fontWeight: 600, dy: 6, dx: 4, filter: (d) => d.time === priceGraphInputs.term && d.design === "Recycle" }), Plot.text(prices, { x: "time", y: "price", z: "design", text: "design", textAnchor: "start", fontWeight: 600, dy: 20, dx: 4, filter: (d) => d.time === priceGraphInputs.term && d.design === "Drag" }), Plot.ruleY([0]) ] })32:T84d,We assume: * The current Target interest rate is nominal, compounds continuously, and is stable through maturity * Historical rates have been flat, and the rate is expected to stay the same at all points in the future * ie, the spot and forward curves are straight horizontal lines We use the definitions: * ${tex`r`} = Target APY * ${tex`t`} = years until maturity * ${tex`x`} = years since issuance ${tex`<= t`} And a reminder: * PY is the realized past yield * FY is the projected future yield ### Drag *Discounted (PY + FY)* Both the PY and FY are delivered at maturity so that cash flow is discounted back to the present (i.e. x). ${tex`PV = \frac{(e^{rx} - 1) + e^{rx} (e^{r(t - x)} - 1)}{e^{r (t - x)}}`}
### Collect *PY + Discounted FY stream* PY is extracted continuously and can be valued in today's terms. FY is delivered as it's accrued, so each cash flow is discounted based on when it will be delivered. The valuation is iterative and adds PY to the FY discounted back as daily cash flows until maturity. We define a new variable ${tex`dr`}, as the continuous rate over the summation interval. ${tex`PV = e^{rx} + \sum_{n=x}^{t} \frac{e^{dr (t - x)} - 1}{e^{dr (t - n)}}`}
### Recycle *PY morphed into FY at market price + Discounted FY* PY is extracted continuously and morphed into FY at market price (i.e. YT / Target's Underlying), then is added to the YT position. However, a continuous discharge schedule is not a requirement, nor is it practical, so let's define another variable that improves the accuracy of this valuation. *Cadence* is an additional variable that represents the frequency of PY → FY transformations. In between each recycling event, the valuation model should mirror Drag. The valuation for Recycle is piecewise and can be viewed in the latter half of the `prices` cell in the appendix. ### Pure FY *Discounted FY* PY is discharged, so FY is the only determinant of price. The FY component is delivered at maturity and needs to be discounted back to the present day. ${tex`PV = \frac{e^{r(t - x)} - 1}{e^{r (t - x)}}`}
33:T714,prices = { const day = 1 / 365; const inc = day; const term = priceGraphInputs.term; const times = d3.ticks(0, term, math.divide(term, inc)); const rate = priceGraphInputs.rate; const data = []; for (let i = 0; i < times.length; i++) { const time = times[i]; const yieldDragPV = math.divide( py(rate, time) - 1 + py(rate, time) * (fy(term, rate, time) - 1), fy(term, rate, time) ); const fyPurePV = math.divide( fy(term, rate, time) - 1, fy(term, rate, time) ); const autoCollectedPV = pvCollect(term, rate, time); data.push({ price: yieldDragPV, time, design: "Drag" }); data.push({ price: fyPurePV, time, design: "Pure FY" }); data.push({ price: autoCollectedPV, time, design: "Collect" }); } let issuance = 0; let multiplier = 1; const recycleFreq = 13; const waypoints = distributedCopy(times, recycleFreq); const freezeWindow = 1 / (term * 12); for (let time of times) { const pv = multiplier * discount( py(rate, time - issuance) - 1 + math.multiply(py(rate, time - issuance), fy(term, rate, time) - 1), term, rate, time ); if (time === waypoints[0]) { if (time < term - freezeWindow) { const newYield = multiplier * (py(rate, time - issuance) - 1); const valueOfFutureYield = discount( multiplier * (fy(term, rate, time) - 1), term, rate, time ); const efficiency = priceGraphInputs.efficiency; const kMultiplier = efficiency * (newYield / valueOfFutureYield); multiplier = multiplier * (kMultiplier + 1); issuance = waypoints.shift(); } } data.push({ price: pv, time, design: "Recycle" }); } return data; }34:[] 2f:{"title":"Rudimentary YT Pricing","cells":[{"id":0,"value":"# Rudimentary YT Pricing","pinned":false,"hidden":false,"mode":"md"},{"id":214,"value":"Here we've outlined very basic pricing models for **Collect**, **Drag**, and **Recycle** YTs. In addition, we've modeled **Pure FY ** for reference. This notebook leads with the visual demonstrating the models, and then sketches them out in more detail in the next section.","pinned":false,"hidden":false,"mode":"md"},{"id":199,"value":"## Graphed Prices","pinned":false,"hidden":false,"mode":"md"},{"id":125,"value":"The following is a graph of the basic valuation models and it shows the price of different YT positions over time.","pinned":false,"hidden":false,"mode":"md"},{"id":119,"value":"* **Rate**: Average Target interest rate\n* **Term**: Term length\n* **Efficiency**: Dampening factor to account for PY -> YT market inefficiencies in the Recycle YT's design (0 is saying that the recycling mechanism fails to buy any YTs from the market)","pinned":false,"hidden":false,"mode":"md"},{"id":16,"value":"viewof priceGraphInputs = view`

\n\n \n \n \n \n \n \n \n \n \n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":26,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":140,"value":"## The Models","pinned":false,"hidden":false,"mode":"md"},{"id":142,"value":"$32","pinned":false,"hidden":false,"mode":"md"},{"id":22,"value":"---\n_**Disclaimer:** This post is for general information purposes only. It does not constitute investment advice or a recommendation or solicitation to buy or sell any investment and should not be used in the evaluation of the merits of making any investment decision. It should not be relied upon for accounting, legal or tax advice or investment recommendations. The opinions reflected herein are subject to change without being updated._\n\n\n---\n## Appendix","pinned":false,"hidden":false,"mode":"md"},{"id":14,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":11,"value":"import {\n math,\n distributedCopy,\n fy,\n py,\n discount,\n pvCollect,\n view\n} from \"@sense/claim-sensitivity-analysis\"","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W34","stdlib":"1","titling":"auto"}

${[\n \"rate\",\n Inputs.range([0, 1], {\n value: 0.2,\n step: 0.01,\n label: \"Rate\"\n })\n ]}

${[\n \"term\",\n Inputs.range([0.5, 5], { value: 1, step: 0.25, label: \"Term (yrs)\" })\n ]}

${[\n \"efficiency\",\n Inputs.range([0, 1], {\n value: 1,\n step: 0.05,\n label: \"Efficiency\"\n })\n ]}