Why a firm's investment decisions are settled in the bond market: replication, the Law of One Price, where NPV actually comes from, and how bond prices turn into a yield curve.
Corporate finance decisions are not made in a boardroom vacuum — they are disciplined by what investors are already doing in asset markets. That sentence is the spine of the whole unit, and Week 1 is where it gets its first proper demonstration. A firm deciding whether to build a factory is, whether it realises it or not, competing with every bond a pension fund could buy instead. If the firm cannot beat what the market already offers, it should not be spending the money.
The machinery that makes this precise turns out to be simpler than the machinery you met in first-year corporate finance, and it comes in three steps. First, work out what an asset is worth by finding other traded assets that deliver exactly the same future cash. Second, notice that a firm's internal project can be valued the same way. Third — and only at the end — convert prices into interest rates, because that is how practitioners talk, not because the rates were doing any work.
The order matters. Nearly every finance course starts with a discount rate and asks you to accept it. This one earns the discount rate from prices you can observe, which means that by the time you write down a present-value formula you will know exactly where every number in it came from and what would have to be true for it to be wrong.
Given a set of zero-coupon bond prices, build the portfolio that reproduces any fixed payoff stream and read off its value.
Apply the same construction to an internal project, and say precisely why a positive NPV is worth having.
Convert zero prices into annualised yields and back, and know what a yield curve is and is not telling you.
Before any valuation, it helps to be concrete about who is in the room. At any moment some economic actors are holding more cash than they currently want to spend, and others want to spend more than they currently hold. The financial system exists to move purchasing power between the two, across time.
The two sides could in principle find each other directly, but searching is expensive and the terms each wants rarely match. So intermediaries appear. A broker matches buyers with sellers and takes a fee without holding the asset itself. Other intermediaries take the asset onto their own books, transforming its size, its maturity or its risk before passing a differently shaped claim to the saver. Alongside the general public sit specialist investors — funds and institutions who trade in size, watch prices constantly, and whose activity is the reason market prices mean anything at all.
A bond is a transferable promise to pay stated amounts on stated dates. That is the whole definition, and almost everything in fixed income is a variation on it. What makes bonds the natural starting point for a finance course is that the promise is fixed in advance: you do not have to forecast anything to know what the holder is owed.
Whether the holder actually receives it is a separate question. Default risk is the possibility that the issuer fails to honour the promise. Bonds issued by a sovereign government in its own currency are conventionally treated as free of default risk, because the state can raise the money to pay. Corporate bonds carry real default risk, and their prices reflect it — which is exactly why we set them aside this week and work only with default-free promises.
One further point is easy to skim past and worth dwelling on. A bond is not an object — it is a relationship between two parties, one of whom owes and one of whom is owed. The certificate is just the evidence. When the bond changes hands, the debt does not disappear; the identity of the creditor changes while the debtor stays put. Holding that picture in mind makes the trading mechanics in the final section much easier to follow.
A trade is an exchange of ownership for payment. In a spot trade, agreement and settlement happen together — you pay now and the asset is yours now. In a forward trade, the two parties agree today on a price and a quantity, but the exchange itself happens on a specified later date. The distinction matters because a forward agreement is itself a promise, and therefore itself something that can be valued.
The price is what the buyer gives up to obtain ownership. When an asset trades in a competitive market, its price aggregates the views of everyone who could have traded it and chose to or chose not to. That is the sense in which a market price is informative: it is not one person's opinion but the price at which the marginal buyer and marginal seller were both content.
Secondary trading is what makes primary issuance possible. An investor who knows they can sell a thirty-year bond next Tuesday if circumstances change will accept a lower yield today than one who must hold it to maturity. Liquidity in the secondary market is therefore paid for by the issuer, in the form of a better price at issue.
An investor holding an asset has a long position: they own the future inflows. An investor who has sold an asset they do not own, and must therefore deliver those payments, has a short position: the same cash flows, with the sign reversed. A short position is not merely the absence of a long one — it is an active obligation, and it is exactly as real as ownership.
Suppose you are convinced an asset is about to fall in price, and you do not own any. You would like to sell it now and buy it back cheaper. To do that you must first get hold of one, and the mechanism is a borrowing contract.
Two investors are involved. The lender owns a unit of the asset and intends to keep holding it — they have no plans to sell in the near term. The borrower takes temporary possession, with a contractual obligation to return an identical unit at an agreed later date, plus a fee.
During the borrowing period the lender does not hold the asset, so on paper their long position has gone. But the contract is written so that they are economically unaffected: they are owed an identical unit back, and any payments the asset makes in the interim are passed through to them by the borrower. Their exposure to the asset's price is unchanged. In return for a position they were not using anyway, they collect a fee.
The borrower's position is the mirror image. Having sold the borrowed unit, they hold cash and owe an asset. If the price falls, buying it back costs less than they received and the difference is theirs. If the price rises, they must still return the unit and the loss is real and uncapped — there is no ceiling on how far a price can rise.
Two things are worth carrying forward. First, short-selling is what lets an investor hold a negative quantity of an asset, and every replication argument in this unit relies on being able to do that. Second, the practice is not exotic: it is a routine, collateralised, fee-bearing transaction, and it is what allows prices to reflect pessimism as well as optimism.
To value bonds we need a shared vocabulary for their shapes. Time runs in periods. Period 1 runs from date 0 to date 1, period 2 from date 1 to date 2, and so on — dates are instants, periods are the gaps between them. Payments land on dates.
A coupon bond pays a fixed amount at the end of each period until maturity, and returns its face value (or par value) alongside the final coupon. The coupon rate expresses the periodic payment as a percentage of face: a £1,000 bond with a 7% annual coupon pays £70 a year. Note carefully that the coupon rate is a feature of the contract, fixed at issue and never changing. It is not a return, and it will generally differ from anything the market is currently offering.
Compare a bond's price to its face value. Trading at par means price equals face. A discount bond trades below face; a premium bond above. The reason is always the same: the coupon rate was fixed at issue, market conditions have moved since, and the price adjusts to compensate. A generous coupon relative to what is currently available commands a premium; a stingy one sells at a discount.
The zero-coupon bond deserves special attention, because the entire valuation method in this week rests on it. One price, one payment, one date — nothing else. Give me a zero for every date at which some asset pays, and I can assemble that asset out of zeros. That is the whole trick, and the next section does it.
Here is the central move of the week. We want the fair price of a coupon bond. Instead of discounting anything, we build a portfolio of other traded bonds that pays exactly what our bond pays, on exactly the same dates. If two things deliver identical cash, nobody can prefer one to the other, so they must cost the same. The portfolio's price is observable. Therefore so is our bond's.
Take three default-free zero-coupon bonds, one maturing at each of the next three dates, each with a face value of £100. Their market prices are given — they are the outcome of competitive trading, and we take them as accurate.
| Asset | Price at date 0 | Date 1 | Date 2 | Date 3 |
|---|---|---|---|---|
| Z1 | (96.30) | 100 | — | — |
| Z2 | (91.20) | — | 100 | — |
| Z3 | (85.60) | — | — | 100 |
| Bond B | (?) | 70 | 70 | 1,070 |
Prices are shown in brackets to mark them as outflows to the buyer. Bond B is our 3-year, £1,000 face, 7% coupon bond.
Because each zero pays on one date only, the portfolio can be found date by date — there is nothing to solve simultaneously. Bond B pays £70 at date 1 and one Z1 pays £100, so we need 70/100 = 0.7 units of it. The same reasoning gives 0.7 units of Z2. At date 3 bond B pays £1,070, so we need 1,070/100 = 10.7 units of Z3.
The portfolio's market value is price times quantity, summed:
So bond B must be priced at £1,047.17. Note that this sits above the £1,000 face value: B is a premium bond, because its 7% coupon is generous relative to what the zero prices imply. We will confirm that interpretation once we have the rates, in From Prices to Interest Rates below.
Notice what did not happen here. We never chose a discount rate, never formed a view about the future, and never needed to know who was buying. The price came entirely from other prices. That is a much stronger foundation than a discount rate someone handed you, and it is the reason this construction comes first.
The Law of One Price is not a convention that markets have politely agreed to follow. It is enforced, and the enforcement mechanism is that anyone violating it is offering a profit that costs nothing to take.
Suppose bond B were quoted at £1,020 while the replicating portfolio still costs £1,047.17. B is cheap relative to its own ingredients. A trader does two things at once: buy B for £1,020, and short-sell portfolio Q for £1,047.17 — borrowing 0.7, 0.7 and 10.7 units of the three zeros from a willing lender and selling them.
At date 0 the trader has taken in £1,047.17 and paid out £1,020, pocketing £27.17. At every date afterwards, nothing happens to them. The long position in B pays £70, £70 and £1,070; the short position in Q owes exactly £70, £70 and £1,070. Every future date nets to zero by construction — that is what replication means.
A positive amount today and nothing owed ever again is not a clever trade — it is an error in the price. Traders pile in, buying B and selling the zeros, which pushes B's price up and the zeros' prices down until the gap closes. If the mispricing ran the other way, with B above £1,047.17, the whole strategy reverses: sell B, buy Q, and collect the difference.
Now change the question. Your firm's engineers have developed a project nobody else can copy. It costs money today and produces cash at future dates. Should the firm do it?
The project is not traded, so it has no market price. But its payoffs can still be replicated by traded bonds, and that is all the argument needs. Suppose the project costs £1,550 today and delivers £400, £600 and £800 at dates 1, 2 and 3.
| Asset | Price at date 0 | Date 1 | Date 2 | Date 3 |
|---|---|---|---|---|
| Project P | (1,550) | 400 | 600 | 800 |
| Z1 | (96.30) | 100 | — | — |
| Z2 | (91.20) | — | 100 | — |
| Z3 | (85.60) | — | — | 100 |
Replicate date by date exactly as before: 4 units of Z1, 6 of Z2, 8 of Z3.
An outside investor who wanted this exact stream of cash would have to pay £1,617.20 for it in the bond market. Your firm can manufacture it for £1,550. The difference is the net present value:
The £67.20 is not a notional accounting gain. The firm can realise it in cash today. Adopt the project, then issue 4, 6 and 8 units of the three zeros against it, raising £1,617.20 immediately. Spend £1,550 building the project. The remaining £67.20 is simply banked. At each future date the project throws off precisely the amount needed to honour the bonds the firm sold — £400, £600, £800 — so the firm is never out of pocket again.
Run the same logic with a cost of £1,700 and NPV is −£82.80. Building it would be strictly worse than spending £1,617.20 on the replicating bonds and pocketing the difference, so the firm destroys £82.80 of value by proceeding.
The two calculations were identical in structure and differ in one respect only. For the bond, the cost of acquiring the payoffs is its market price, so a gap between price and replication value is a mispricing that arbitrageurs will close. For the project, the cost is the implementation cost, and the gap is not an arbitrage at all — nobody outside the firm can exploit it, because nobody outside the firm can build the project. The gap is the return on whatever made the firm capable of it in the first place.
Everything so far has been done in payoffs and prices. No interest rate has appeared, and none was needed. But practitioners quote rates, not zero prices, so we need the translation — and the two languages carry identical information.
Buying Z1 means paying £96.30 today to receive £100 in a year. The ratio of what you get out to what you put in is the growth factor:
Subtract one and you have the growth rate: r1 = 3.842%. Equivalently, treat the £96.30 as a loan principal and the £3.70 difference as interest paid on it — 3.842% of £96.30 is £3.70. A bank offering a one-year deposit at 3.842% and the bond market offering Z1 at £96.30 are making the same offer, and arbitrage keeps them aligned.
Over two years, Z2 turns £91.20 into £100, a total growth factor of 1.09649. That is growth over the whole two years, which makes it useless for comparing against a one-year investment. So we annualise: we ask what constant rate, applied twice, produces the same result.
The same reasoning with a cube root gives r3 = 5.319%. Compounding is doing real work here: over two years the borrower pays interest on the principal and then interest on the first year's interest as well, and it is the sum of those that makes up the £8.80 gap between £91.20 and £100.
The sequence r1, r2, r3, … is the yield curve, or term structure of interest rates. Ours runs 3.842%, 4.713%, 5.319% — a rising curve, which is the most common shape. It can also be flat, falling, humped, or inverted. A curve is a snapshot of the required return for each horizon at one instant, and it moves every day.
The interactive below is not a stylised drawing. It is the Bank of England's estimated UK nominal spot curve for every trading day from June to December 2022 — a period that contains one of the most violent episodes in the history of the gilt market.
Source: Bank of England yield curve estimates, UK nominal spot curve, 2022.
Drag the slider through late September. On 22 September the 30-year rate sits at 3.72%. Over the following three trading days it climbs to 4.85%, as leveraged pension strategies were forced to sell exactly the assets everyone else was selling. Then on 28 September, when the Bank announced emergency gilt purchases, the 30-year rate fell back to 3.72% in a single day — a move of 113 basis points, and the curve inverted, with the 1-year rate at 4.03% sitting above it.
We can now write the familiar present-value formula and know exactly why it is true. The date-t discount factor is (1 + rt)−t, and the present value of a stream of payoffs is the sum of each payoff scaled by its own factor:
Apply it to bond B: 70/1.03842 + 70/1.04713² + 1070/1.05319³ = £1,047.17. The same answer replication gave, because it is the same calculation wearing different clothes. This is worth pausing on. The PV formula is usually presented as a definition to be accepted. It is not — it is a restatement of "buy the identical cash flows from the bond market and see what they cost", and it inherits all of arbitrage's force.
In practice people often want one number for an asset rather than a whole curve. So we ask a hypothetical question: if the yield curve were flat, what single rate would justify the price we actually observe? That rate is the yield to maturity.
For bond B, priced at £1,047.17, the answer is i = 5.259%. Compare that with the zero yields of 3.842%, 4.713% and 5.319%: the YTM sits inside the range, below the highest and above the lowest. That is no accident. YTM is a payoff-weighted average of the zero yields, and because most of B's money arrives at date 3, its YTM sits close to r3.
Two consequences follow. First, when the curve is not flat, two correctly priced bonds will generally have different YTMs, because they have different payoff timings and therefore different exposure to each date. A YTM is a property of an asset, not of the market. Second, YTM is nonetheless enormously useful in practice, because it can be computed from a single bond's price and payoffs without estimating a zero curve at all — and estimating a zero curve from noisy market data is genuinely hard.
Run the identical calculation on a payoff stream that is not a traded asset, and the resulting rate is the internal rate of return. For project P at a cost of £1,550, the IRR is 7.030%. Since that exceeds every rate on the yield curve, the project beats the bond market at every horizon — which is another way of saying its NPV is positive.
IRR rarely solves in closed form, so in practice it is found numerically. The standard hand method is linear interpolation: compute NPV at two trial rates that bracket zero, then draw a straight line between them and read off where it crosses. For cash flows of £2,600, £1,800 and £1,200 against a cost of £5,000, NPV is +£62.37 at 6% and −£96.78 at 8%, giving an interpolated IRR of 6.784% against a true value of 6.770%.
This section goes beyond what the core valuation argument requires. It is here because "long" and "short" are words that are easy to nod along to and surprisingly easy to get wrong when the cash actually moves — and because seeing a short position built, maintained and closed on a balance sheet is what makes the replication arguments above feel concrete rather than algebraic.
Follow one bond through its life. Three parties hold positions that change as it is issued, traded on, borrowed and finally repaid. At each step, watch three things for each party: cash held, long positions, and short positions. Every transaction moves at least two of them, and the total across all parties always balances.
Every replication argument in this week assumed we could hold negative quantities of an asset — that "short 10.7 units of Z3" is a thing a trader can actually do. This section is where that assumption is cashed out. The arbitrage argument earlier, in Arbitrage Is What Enforces It, is not a thought experiment: it is a sequence of ordinary, collateralised transactions, and the profit at date 0 is real cash in a real account.
Discounting, Present Value and Yield to Maturity used linear interpolation as a recipe. This section takes it apart — because the recipe hides a simple geometric idea, and because almost every way of getting an IRR question wrong is a misreading of what the number means rather than an arithmetic slip.
Take the two trial points from before. Point A sits at 6% with an NPV of +£62.37; point B sits at 8% with −£96.78. Join them with a straight line and call its zero crossing X. Everything follows from one property of straight lines: the same fraction of the total horizontal movement produces the same fraction of the total vertical movement.
Drop a horizontal line from A. The vertical through X meets it at D; the vertical through B meets it at E. That gives a small triangle ADX and a large one AEB. Both have a right angle, and the angle at A belongs to both, so one is simply a scaled copy of the other — every side of the small one is the same fraction of its partner in the large one.
Put the numbers in. The total fall from A to B is 62.37 − (−96.78) = 159.15, and the fall from A down to zero is just 62.37. So the small triangle covers 62.37/159.15 = 39.19% of the large one's height — and therefore 39.19% of its base too. The base is the full two percentage points from 6% to 8%, so the step right is 0.3919 × 2 = 0.784 percentage points, giving 6% + 0.784% = 6.784%.
rL is the lower rate, whichever you happened to try first. Try 8% then 6% and rL is still 6%. Write each NPV down next to the rate that produced it — mixing the two up is the commonest way this formula goes wrong.
rL does not mean "the positive one". On a falling NPV profile the lower rate usually gives the higher NPV, but that is a consequence, not a definition.
Work in decimals throughout, or percentages throughout. Mixing 0.06 with 8 is the kind of error that produces an answer like 0.68% and survives unnoticed.
Nothing in the derivation required the line to cross zero between the trial points. If it crosses outside them, the same formula extends the line until it does. That is extrapolation, and the fraction tells you it has happened.
| Trials | NPVs | Fraction | Estimate | Error |
|---|---|---|---|---|
| 6% and 8% | +62.37, −96.78 | 0.392 | 6.784% | +0.014 |
| 8% and 12% | −96.78, −389.49 | −0.331 | 6.677% | −0.093 |
| 2% and 4% | +409.91, +231.00 | 2.291 | 6.582% | −0.188 |
True IRR 6.770%. Errors in percentage points. The middle row extrapolates downwards, the last row upwards.
Read the fraction as a position within the interval. Between 0 and 1 means the crossing lies inside the bracket. Negative means it lies below both rates — which is what two negative NPVs give you. Greater than one means it lies above both, which is what two positive NPVs give. Both extrapolations here are worse than the bracketed estimate, and they get rapidly worse the further the line is extended. Finding one NPV of each sign remains the dependable starting point.
Exam papers frequently supply discount factors rounded to three decimal places. Using them changes the answer, and it is worth being clear that this is a separate error from the one interpolation itself introduces.
Ask what 6.770% is actually the return on. Not on £5,000 for three years: by the end of year 1 you have had £2,600 back, so only part of the original stake is still at work. The IRR is the one rate that, applied each year to whatever is still tied up, runs the balance down to exactly zero on the final date — and that balance shrinks with every receipt. Four things it does not mean:
It is not the mean of three annual percentages. The receipts total £5,600, a cumulative undiscounted gain of 12% — and dividing that by three gives 4%, nowhere near the IRR.
It does not mean 6.77% of the original £5,000 arrives each year. Most of the £2,600 at date 1 is capital coming back, not return on capital.
The 6% was a trial rate, nothing more. What the project offers is separate from what an investor requires, which comes from what else their money could earn at the same level of risk.
Computing the IRR assumes nothing about what happens to early receipts. But earning 6.77% compound on the whole £5,000 through to date 3 would require reinvesting them at 6.77%.
The relationship to the required return is the one that decides anything: NPV is positive exactly when the required return is below the IRR, and negative when it is above. That is the sense in which IRR and NPV give the same answer — for a project shaped like this one.
Everything above relies on the NPV profile falling steadily as the rate rises, which it does when an outflow is followed only by inflows. Change the sign more than once and that guarantee disappears.
Consider a project that costs £8,400 today, returns £20,150 at date 1, and then requires £12,000 of decommissioning at date 2 — a mine or an oil field, where the clean-up bill arrives after the profits.
The profile rises, peaks and falls, crossing zero twice: at 9.94% and again at 29.94%. Both satisfy the definition — both make NPV exactly zero — and there is no principled reason to prefer one. "The IRR is 9.94%" and "the IRR is 29.94%" are equally true and equally useless.
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