Separate bond pricing, money measures and an identity

Separate bond pricing, money measures and an identity

Question and evidence

Does a larger money stock by itself identify the cause of inflation?

A fictional one-year zero-coupon bond pays 105 credits with certainty. At price 100 its yield is 5%; at price 105 its yield is 0%. In a separate quantity-equation scenario M=200,V=5 and real output Y=100, so the price index P satisfies MV=PY.

The numerical case is a teaching model, not a measured real-world effect.

Mechanism

Bond price and yield move inversely for a fixed promised payment. Money performs exchange, accounting and value-storage roles, with narrow and broad aggregates defined by statistical institutions. Under constant velocity and output, proportional money growth implies proportional price growth in the quantity equation. Relaxing those assumptions changes the prediction.

Worked calculation

Yield=(105−100)/100=5%. In the separate scenario P=200×5/100=10. If money rises to 220 but output rises to 110 at the same velocity, P stays 10; if output stays 100, P becomes 11.

Challenge and revision

The equation MV=PY is an accounting relationship once terms are consistently defined; a causal inflation forecast requires assumptions about velocity and output. Money definitions include different instruments and can change over time. A bond is an asset, but not every asset is counted as money.

Plot bond price against yield for fixed payment 105. Build two money-growth scenarios and explicitly identify which variables are held fixed before making any inflation claim.

Revise your answer to “Does a larger money stock by itself identify the cause of inflation?” using both source IDs. Separate the checked calculation from a judgment, and identify evidence that could reverse your judgment.

Vocabulary

Velocity of money: Nominal expenditure relative to a consistently defined money stock in the quantity equation.

Assumption: A condition accepted for a model or argument; it needs justification before applying the result elsewhere.

Independent extension

Independently extend the investigation “Separate bond pricing, money measures and an identity” with a documented real dataset or scholarly source. Preserve definitions, recreate one calculation or evidence table, compare a rival explanation and report why the teaching example cannot establish the real-world conclusion.

Worked graph

Points are not interpolated; check their observation order and definitions. Axes start at zero.

Series	Bond price (credits)	One-year yield (%)
Fixed payment105	95	10.5263
Fixed payment105	100	5
Fixed payment105	105	0

Separate the promise, price and outcome

Change the price, then test the payment

Fictional one-year zero-coupon bond: pay the purchase price now, receive one payment after exactly one year. No interim coupons, fees or inflation adjustment. Predict the return before changing one control.

Purchase price80 credits90 credits100 credits110 credits120 credits Promised payment after one year100 credits105 credits110 credits Fraction of promise actually paid0% paid80% paid100% paid

The last control selects an observed outcome, not its probability. Keeping the promise fixed isolates the price/yield relationship. Changing the promise compares a different contract.

Circles: promised yield. Outlined squares: realized return under the selected payment outcome. Larger/bolder markers identify the selected price. Points are separate scenarios, not a time series or a straight-line interpolation. At full payment the markers overlap.

Selected price 100; promise 105; actual payment 105.00 credits. Promised yield 5.00%; realized return 5.00%.

Promised gain = 105 − 100 = 5.00 credits. Realized gain = 105.00 − 100 = 5.00 credits. Divide each gain by the original price, then multiply by 100 for its percentage.

Same promise and payment fraction, five purchase prices

Price	Payment received	Promised yield	Realized return
80	105.00	31.25%	31.25%
90	105.00	16.67%	16.67%
100	105.00	5.00%	5.00%
110	105.00	-4.55%	-4.55%
120	105.00	-12.50%	-12.50%

A promised yield is conditional on the payment occurring. No probabilities are supplied, so this panel does not calculate an expected return. A negative yield is possible when the price exceeds the final payment. A zero price would make the return ratio undefined and is not an allowed purchase here.

Restore bond comparison

Controls are temporary and reset on reload. Record the price, promise, observed payment and explanation in the source notebook. This one-year calculation does not price a multi-year coupon bond or estimate an actual investment's risk.

Hold the promise at 105 and payment at 100%. Compare prices 100 and 120. Then return price to 100 and select 80% payment. Explain why the first comparison changes the purchase terms while the second changes what actually happened after purchase.

Check price changes and payment shortfalls

Price 100 gives a promised gain of 5 and yield 5%. Price 120 gives a loss of 15 even with full payment: −15/120=−12.5%. With price 100 and only 80% of the promise paid, the receipt is 84 and the realized return is −16%. The promised yield remains 5%; it was not a guarantee of the observed result. A 20% shortfall in the promised payment is not the same denominator as a 20% loss on the purchase price.

Compare access to money, not just the final number

Consider three fictional claims, each acquired for 100 credits. A payment balance can settle a 60-credit bill today and pays no interest. A one-year debt claim promises 105 but cannot be sold or used to pay that bill before maturity in this case. An ownership claim also cannot be sold or used for this bill today and has no promised payment; next year’s combined distribution and sale proceeds could be 80 or 120, with no probabilities supplied. Ignore fees and inflation for this comparison.

Create columns for payment access today, contractual claim, possible final proceeds, and missing evidence. Which claim can settle the bill as described? Can you calculate the ownership claim’s expected return? Explain why a claim that stores value is not necessarily usable as the means of payment for this transaction.

Check liquidity and ownership claims

Only the payment balance can settle the bill under the supplied rules. The debt claim has a promised payment; the ownership claim has no fixed repayment promise. Its two possible returns are −20% and +20%, but their simple average is not an expected return unless equal probabilities are justified. Payment access and terminal proceeds answer different questions. The no-sale restriction is a teaching assumption, not a claim that all bonds are illiquid. An asset can be valuable yet require sale, conversion or waiting before it pays a bill.

Calculate an opportunity cost with the assumptions visible

In a separate comparison, assume a certain one-year alternative pays 4% only if held for the full year, both balances use the same currency, and neither has fees. What nominal return is forgone by keeping 250 credits in a zero-interest balance for the entire year? Why does that calculation alone not settle how much transaction money someone needs?

Check the forgone return

The alternative pays 250×0.04=10 credits, ending at 260 instead of 250. Ten credits is the stipulated one-year nominal opportunity cost. It is not a monthly cost, a guaranteed real purchasing-power gain or a measured cost of every cash holding. Access needs, risk, actual contract terms and the horizon matter. If the alternative is risky, its promised rate alone does not establish the expected forgone return.

Transfer: repair a misleading return caption

A different one-year claim costs 200, promises 220 and actually pays 198. A caption says “The investor earned 10%, because 90% of the promise was paid.” Calculate the promised yield, payment fraction and realized return separately. Then state whether any observed payment alone tells you what probability should have been assigned beforehand.

Check the unfamiliar return caption

Promised yield is (220−200)/200=10%. Payment fraction is 198/220=90%. Realized return is (198−200)/200=−1%. The caption combines two correct ratios into an incorrect outcome claim. One realization does not identify its prior probability; no expected return follows from the supplied record alone.

The SEC’s bond explanation distinguishes credit, interest-rate and liquidity risks. Use it to classify a missed promised payment, a changed resale price and difficulty finding a buyer. These can coexist; the workbench changes only acquisition price and final payment. Its invented numbers do not measure any issuer’s creditworthiness. For college extension, compare a dated actual instrument’s payment terms with this one-payment simplification; identify every omitted cash flow before using the formula.

Keep your classification table and repaired caption in the notebook. These public checks support guided practice. Independent assessment needs unfamiliar terms and a learner explanation without opening the solution; money-aggregate calculation remains a separate investigation.

Build money measures from a dated ledger

A money measure counts specified claims, not everything someone owns. Use the Federal Reserve H.6 release dated September 22, 2026, footnotes to Tables 1–2, as the definition vintage. In this version, M1 combines public currency, eligible demand deposits and other liquid deposits, including savings. M2 adds small time deposits and retail money-market funds, then subtracts IRA/Keogh balances. The base combines currency in circulation, including bank vault cash, with reserve balances at the Fed. These are different measures, not three independent piles to add together.

Classify before adding

The following is an invented ledger in millions of dollars, not U.S. statistics. All customer deposits are eligible domestic balances after collection/float adjustments. There are no government/interbank/foreign-official deposits, IRA/Keogh balances or other exclusions. Small time deposits meet the release’s size definition. Each row is separate; savings is not already included in the demand-deposit row.

Record	Balance
Currency held by the public	40
Currency in bank vaults	10
Banks’ reserve balances at the Fed	30
Eligible demand deposits	100
Savings deposits	200
Other checkable deposits	20
Small time deposits	50
Retail money-market fund balances	25
Household corporate bonds	60
Household corporate shares	90

Before opening feedback, create three inclusion columns: M1, M2 and base. Mark each row and calculate the three totals. Add a fourth column naming the holder and the claim: a customer’s deposit and a bank’s reserve account are not the same relationship. Do not classify an item merely because its name contains “money.”

Check the three money totals

M1=40+100+200+20=360. M2=360+50+25−0=435. Base=40+10+30=80. The 40 of public currency appears in all three definitions; adding 360+435+80 would count overlapping entries repeatedly. Bonds and shares are financial assets but do not enter these selected aggregates. Vault cash belongs in this base calculation but not the public-currency part of M1. Reserve balances belong to banks, while customer deposits are claims on banks.

This ledger omits many real reporting adjustments. It is a classification exercise under the stated assumptions, not a complete bank balance sheet or a measurement of national wealth.

Trace three changes from the same starting ledger

For each case, start again from the original table; do not accumulate the cases. Draw an arrow between the two affected rows and show both entries before recalculating totals.

1. A customer transfers 15 from savings to a demand account at the same bank.
2. A customer withdraws 8 from a demand account, paid from the bank’s existing vault cash. No new cash delivery from the Fed occurs.
3. A customer converts 12 from a demand deposit into a qualifying small time deposit at the same bank.

Predict which measure changes and why. Does a change in the composition of someone’s assets necessarily change every aggregate?

Check the account transfers

In case 1, demand deposits become 115 and savings 185. M1 remains 360, M2 435 and base 80. Both customer accounts are included in this version of M1.

In case 2, public currency becomes 48, vault cash 2 and demand deposits 92. M1 remains 360 because the eight-million-dollar currency increase offsets the deposit decrease. M2 remains 435. Base remains 48+2+30=80: currency moved between holders, while total currency in circulation and Fed reserve balances stayed fixed.

In case 3, demand deposits become 88 and small time deposits 62. M1 falls to 348; M2 remains 348+62+25=435; base stays 80. This is an account conversion, not evidence that wealth fell by twelve or that twelve of physical currency disappeared.

Audit a real published base total

Table 1 of the dated release reports August 2026, not seasonally adjusted, in billions of dollars: currency in circulation 2,475.6; reserve balances 2,936.0; monetary base 5,411.6. Recreate the sum and retain the month, release date, units and adjustment label. Explain why adding the ledger’s fictional millions to these observed billions would be invalid.

Check the dated base reconciliation

2,475.6+2,936.0=5,411.6 billion dollars. The month is the reference period; September 22 is the publication date. The fictional ledger illustrates the rule but is not a component of this release. A release-vintage reconciliation is not a claim about today’s balances or the policy cause of a change. Other rounded components may not sum exactly to a published total; inspect the stated rounding convention before calling a discrepancy an error.

Recognize a definition change

The Fed’s technical Q&A explains that savings deposits entered M1 retroactively from May 2020; this produced a series break without changing M2. It also explains the July 2026 move of IRA/Keogh netting from components to the M2 aggregate. Nonseasonally adjusted M2 was unchanged by that netting change, while seasonal treatment caused minor revisions to adjusted M2.

To isolate classification, hold the fictional ledger fixed. Compare a teaching definition that excludes savings from M1 with the September 2026 definition. Do not interpret this fixed-ledger exercise as the actual 2020 growth rate. Separately, suppose raw component totals are M1=360, small time=50 and retail funds=25, with an aggregate retirement-account deduction of 30. Calculate M2 once. Diagnose a spreadsheet that first subtracts 30 from small time deposits and then subtracts it again from the aggregate.

Check reclassification and netting

The fixed-ledger M1 comparison is 160 before including savings and 360 after, a rise of 200 entirely from the teaching definition change. No transaction, income gain or inflation is supplied. It is not a reconstruction of actual May 2020 data.

For the separate netting exercise, M2=360+50+25−30=405. The faulty spreadsheet produces 375 by deducting the same 30 twice. The input record must say whether components are raw or already net of exclusions. Neither simply adding all raw components nor subtracting a correction twice follows the supplied convention.

Transfer without the starting answer

A new fictional ledger has public currency 25, vault cash 5, Fed reserve balances 15, demand deposits 60, savings 90, other checkable deposits 10, small time deposits 20 and retail funds 5. All eligibility/exclusion assumptions match the original zero-retirement-balance ledger. Reconstruct the totals, then withdraw 3 from demand deposits using vault cash. Explain the changed rows and unchanged aggregates before opening the check.

Check the unfamiliar money ledger

Initially M1=185, M2=210 and base=45. After withdrawal, public currency is 28, vault cash 2 and demand deposits 57. M1=28+57+90+10=185; M2=210; base=28+2+15=45. The customer exchanged one included form for another. This does not model a new loan, a central-bank asset purchase or the later use of those balances.

Save your inclusion table, transaction arrows and definition-vintage note in the notebook. AP practice should explain both changed and unchanged totals; college extension should reconcile another dated release with its own footnotes and adjustment labels. Independent assessment needs an unseen ledger and a correction task without public feedback. A money-stock increase alone still does not identify an inflation mechanism.

Money growth over time: level, rate and assumptions

Predict the price path, then change an assumption

Original fictional annual records. Year zero starts with money 200 credits, velocity 5 per year and real output 100 units per year. Price is 10 credits per output unit; its displayed index is 100. The identity M × V = P × Y reconciles the selected assumptions; it does not prove what causes inflation.

Money-change timingOne change in year 1, then fixed moneyRepeat the money growth every year Money growth when applied0%10%20% Annual real-output growth0%5% Annual velocity growth0%-5%

Predict whether the price index will flatten after year one. Output and velocity keep their selected annual growth rates even when the money change occurs only once. No economy-wide adjustment speed is estimated.

Dots are annual observations; connecting segments guide the eye, not within-year measurements. The dotted horizontal line is the starting index. Axes stay fixed at years 0–4 and index 0–220.

Year 4 price index 110.00; year 4 inflation 0.00%. These are a level and a growth rate, respectively.

Exact calculations use unrounded inputs; displayed values have two decimals

Year	Money (credits)	Velocity (/year)	Real output (units/year)	Price index	Inflation (%)	Growth approximation (%)
0	200.00	5.00	100.00	100.00	No prior year	No prior year
1	220.00	5.00	100.00	110.00	10.00	10.00
2	220.00	5.00	100.00	110.00	0.00	0.00
3	220.00	5.00	100.00	110.00	0.00	0.00
4	220.00	5.00	100.00	110.00	0.00	0.00

Restore money-growth comparison

Controls are temporary. Save the assumptions, two annual calculations and a revised claim in the notebook. A balanced identity is not independent evidence for the chosen growth assumptions.

One price increase is not continuing inflation

Keep output and velocity fixed. Predict the price index in years one through four after a one-time 10% money increase. Then switch to repeated 10% growth. Sketch both paths on the same axes and label the annual inflation rate, not just the last index. Explain why a price index remaining above 100 is not proof of positive inflation every year.

Check one-time and repeated growth

The one-time path is 100, 110, 110, 110, 110. Inflation is 10% in year one and zero afterward. The repeated path is 100, 110, 121, 133.1, 146.41; inflation is 10% each year. Year four’s cumulative increase of 46.41% is not its annual inflation rate. No prior-year value is supplied for year zero, so its inflation is unavailable rather than zero.

Derive the growth calculation

The quantity equation is M × V = P × Y. Here M is a consistently defined money stock in credits, V is annual spending relative to that stock, Y is annual real output in units and P is credits per real-output unit. Divide the new year’s equation by the previous year’s equation. Each ratio is a growth factor: 10% growth means multiplying by 1.10, not by 10.

New price ÷ old price = (new money ÷ old money) × (new velocity ÷ old velocity) ÷ (new output ÷ old output). Subtract one and multiply by 100 to express inflation as a percent. The shortcut “money growth + velocity growth − output growth” is an approximation for small changes, not the exact finite-change rule.

Set repeated money growth to 10%, output growth to 5% and velocity growth to zero. Predict whether inflation is exactly 5%. Next change velocity growth to −5%. Explain why the shortcut now misses even the direction relative to zero.

Check exact growth and the approximation

With constant velocity, the price factor is 1.10/1.05 = 1.047619…, so inflation is about 4.76%, compared with the shortcut’s 5%. With velocity falling 5%, the factor is 1.10×0.95/1.05 = 0.995238…, so inflation is about −0.48%, although the shortcut gives 0%. Negative inflation means the price level falls; it does not imply the price level is negative. Rounding the factors before dividing can erase this difference.

In the one-time-money mode, output and velocity continue to change after year one. Even fixed money then need not give a flat price path. State all three assumptions before attributing a result to money alone.

An identity does not determine the adjustment path

In a separate thought experiment, start again at M=200, V=5, Y=100 and P=10. Double M to 400, keep V fixed and suppose long-run capacity remains Y=100. Calculate the conditional long-run price. A different short-run assumption sets Y=110 after the monetary change; calculate its price too. Do not infer that Y=110 follows from the identity.

Check conditional short-run and long-run outcomes

At fixed long-run output, P=400×5/100=20: nominal expenditure doubles, real output does not. Under the separate short-run output assumption, P=2000/110≈18.18. The identity permits that combination but does not explain why output rose, how fast prices changed, or whether velocity stays fixed. An AD/SRAS account requires spending behavior, price adjustment and capacity assumptions in addition to this equation. Monetary neutrality here describes the stipulated long-run model, not a claim that every monetary change has no real effects at any horizon.

Unfamiliar evidence: calculate velocity before choosing a story

A fictional annual record gives M=300 credits, nominal expenditure=1800 credits per year and real output=150 units per year. Next year M=330, nominal expenditure=1980 and real output=165. Reconstruct V and P in both years, then evaluate “Money grew 10%, so inflation must have been 10%.” Propose what additional evidence would be needed to argue that a monetary change caused a price change in an actual economy.

Check the unfamiliar quantity records

Velocity is 1800/300=6 per year and then 1980/330=6. Price is 1800/150=12 credits per unit and then 1980/165=12. Money and real output both grow 10%; inflation is zero. Nominal expenditure growth is not automatically inflation. In an actual comparison, consistent money definitions, price/output measurement, timing, expectations and plausible alternative shocks matter. Defining velocity from nominal expenditure and money makes the equation hold; it does not independently validate a causal theory.

For high-school practice, retain the two price paths and explain one changed assumption. AP preparation should additionally reconstruct the growth calculation and distinguish the short-run versus long-run assumptions. College extension should retrieve a dated, consistently defined series and discuss measurement revisions and causal identification. These are practice tasks with public feedback, not automatic evidence of independent proficiency.