Week 3 · Module 2
Prof. Jiajun Jiang · School of Economics, Fudan University · Semester 1, 2026–2027
A bond’s price is the discounted list of its cash flows, duration says how much that price moves when the discount rate does, and the rate itself is a price set by supply and demand.
Week 2 priced everything off a single yield. But no two bonds trade at the same yield. Two bonds maturing on the same day differ because of the borrower, the market and the tax code; the same borrower pays different rates at different maturities. This week takes those two cuts in turn, then asks how to measure the risk behind them and whether the market prices it correctly.
Mishkin Chapters 5 and 6. Two ways of cutting the yield curve, then how to measure risk, then whether prices reflect it.
Meeting 1
90 minutes · why yields differ across issuers, then across dates · closes on a curve that inverted for two years and was wrong
Session 1 · 45 minutes
Hold the maturity fixed and two bonds still trade at different yields. The gap between them breaks into three parts: default risk, liquidity, and tax treatment. We take them one at a time.
A credit spread is a price, and these three are what it pays for. Anything else you can think of — issuer size, sector, covenant strength — works through one of them.
The risk premium is the difference between the yield on a defaultable bond and the yield on a default-free bond of the same maturity:
\[ \text{risk premium} \;=\; i_{\text{corporate}} - i_{\text{government}} \]News that raises default risk shifts the demand for corporate bonds left — their price falls and their yield rises. The same news shifts demand for government bonds right — a flight to quality. Both movements widen the spread, and neither requires a default to occur.
This is why spreads widen before any default happens: a spread prices what lenders expect to lose, and expectations move first.
How cheaply and quickly you can turn the bond back into cash. A less liquid bond must offer a higher yield to be held at all.
Treasuries are the most liquid long-term bonds there are, which is part of why they yield less than corporates of identical credit quality.
US municipal bonds are exempt from federal income tax, so investors accept a lower pre-tax yield — sometimes below Treasuries, despite carrying default risk.
A yield comparison that ignores tax treatment compares two different things.
Cold call · 3 min
A municipal bond can default and still yield less than a Treasury. Is that an arbitrage?| What it means | Moody’s | S&P / Fitch |
|---|---|---|
| Highest quality | Aaa | AAA |
| High quality | Aa1 – Aa3 | AA+ – AA− |
| Upper medium | A1 – A3 | A+ – A− |
| Lower medium — investment grade ends here | Baa1 – Baa3 | BBB+ – BBB− |
| Speculative | Ba1 – Ba3 | BB+ – BB− |
| Highly speculative | B1 – B3 | B+ – B− |
| Substantial risk, then default | Caa – C | CCC – D |
An opinion about how likely a borrower is to miss a payment, relative to others. Not a probability, and not a price. The issuer pays for it.
Regulation and fund mandates are written on it. Many insurers and pension funds may not hold bonds below BBB−, so a downgrade across the line forces selling by investors with no view on the company.
A rating ranks borrowers. What that ranking is worth is settled by the price, and where the two disagree, trust the price: someone had to buy at it.
Predict before reveal · 4 min
A Chinese AA-rated one-year corporate note. How much more than government debt does it yield?For scale: the average investment-grade US corporate bond pays 78bp over Treasuries, and a BBB one — the lowest investment grade — pays 95bp.
AA is an investment grade in both places, so the two are comparable in name. But a Chinese firm rated AA at home is rated around BBB− by a global agency — the bottom of investment grade, where US bonds pay 95bp. The Chinese note pays 41bp, less than half.
So either Chinese AA borrowers default less often than US borrowers two grades above them, or the 41bp is paying for something other than default risk.
ICE BofA option-adjusted spreads via FRED, 17 September 2026 (US corporate index, BBB index, high-yield index). China: Bloomberg, 15 May 2025. Spreads are over government debt of comparable maturity in each market.
Microsoft and Johnson & Johnson — the whole list, against more than 60 US corporates thirty years ago. And only 49% of S&P-rated corporate issuers reach investment grade at all. Half the universe sits below BBB−.
Among more than 6,000 onshore credit-bond issuers, 27% are rated AAA and 32% AA+.
The point of a rating is to separate borrowers. If three in five are in the top two grades, the letters carry almost no information, and investors have to look somewhere else — which is what the 41bp spread was telling us.
China: end-Q1 2024, 6,000+ onshore credit-bond issuers — AAA 27%, AA+ 32% (The Standard, on the PBoC’s measures against the AAA concentration). US: S&P Global Ratings — 49% of rated corporate issuers are investment grade (78% of rated debt); AAA list as of 2026.
They weigh the same financials differently. Domestic agencies treat size as a bigger positive; global agencies penalise leverage harder and weigh profitability and state ownership more. Add issuer-pays and a domestic panel competing for the same clients, and the scale drifts up.
If the gap were a constant offset, shifting one scale down seven notches would line the two up. Do that and about 24% of jointly rated firms still disagree: the two are ranking firms differently, not just labelling them differently.
A Chinese AA and a US AA do not describe the same borrower — part of why 41bp looks so tight.
Jiang & Packer, BIS Working Paper 648 (2017), via VoxChina — 2015 data, about 50 jointly rated firms.
Session 2 · 45 minutes
Three theories try to account for all three facts. They differ in one assumption: whether bonds of different maturities are substitutes for each other.
A long rate is the average of the short rates the market expects over the same horizon:
\[ i_{nt} \;=\; \frac{i_t + i^{e}_{t+1} + i^{e}_{t+2} + \cdots + i^{e}_{t+(n-1)}}{n} \]If they are, no maturity can offer a better expected return than another, and arbitrage forces the long rate to the average.
| Theory | Core assumption | Explains | Fails |
|---|---|---|---|
| Pure expectations | Maturities are perfect substitutes | Facts 1 and 2 | Fact 3 |
| Market segmentation | Maturities are not substitutes at all; each has its own clientele | Fact 3 | Facts 1 and 2 |
| Liquidity premium | Substitutes, but imperfect — investors need paying to hold duration | All three | see below |
It is the expectations theory plus one extra term, \( \ell_{nt} \), and that term is never measured: it is defined as whatever is left once you subtract expected short rates from the long rate. Any movement the expectations part cannot explain is attributed to it instead, which makes the theory very hard to prove wrong.
And it is large: with the 30-year at 5.29% against a 3.2% neutral rate, more than two points of that yield is premium.
Think – pair – share · 7 min
Fed funds is 3.75–4.00% and the 2-year is 4.67%. Under pure expectations, what is the market saying about the next two years — and what does the 30-year at 5.29% add?The neutral rate is where the policy rate settles once inflation is at target; the FOMC’s median member puts it at 3.2%. Read the 30-year’s 5.29% as a pure forecast of policy and you are claiming the Fed sits two points above that for thirty years. The gap is the term premium.
Federal Reserve H.15, 17 September 2026 — 3m 3.97%, 2Y 4.67%, 10Y 4.94%, 30Y 5.29%. Neutral rate: FOMC projections, 16 September 2026.
Expectations: the short rate is expected to rise slowly, averaging about 2.1% over 30 years against 1.2% today.
Liquidity premium: if short rates stay near 1.2%, the 30-year term premium is about 95bp, against roughly two points in the US.
Segmentation: insurers and banks need long bonds and there are too few, so their buying holds long yields down.
Look at what moves the long end. If long yields move with news about the future policy rate, expectations are doing the work. If they move with the supply of long bonds, or with insurers’ and banks’ demand for them while the policy outlook stays put, the term premium and segmentation are.
All three theories fit this curve. Telling them apart needs evidence from outside the curve itself.
US: Federal Reserve H.15, 17 Sep 2026. China: ChinaBond, 18 Sep 2026 — 3m 1.18%, 10Y 1.68%, 30Y 2.13%, a 95bp span against the US 132bp. Deposit and financing figures: Week 2.
Think – pair – share · 9 min
Two minutes alone, three with your neighbour, then we build both stories on the board.Inversion dates: U.S. Bank; current spread from Federal Reserve H.15, 17 September 2026.
Meeting 2
90 minutes · how to measure risk, then whether prices reflect it · closes on China’s quant funds
Session 3 · 45 minutes
The main result: part of each asset’s risk cancels out when it is held with other assets, and investors are paid only for the part that does not.
It is symmetric — it treats an upside surprise and a downside surprise as equally bad, which is plainly wrong as a description of how people feel. It is still used because it is easy to work with and because it aggregates: the variance of a portfolio has a closed form in the weights, which is what makes portfolio optimisation possible.
Most criticisms of mean-variance analysis start here. The alternatives describe risk more realistically, but they are much harder to work with.
| State | Probability | Asset A | Asset B |
|---|---|---|---|
| Boom | 0.25 | +28% | −2% |
| Expansion | 0.35 | +14% | +9% |
| Slowdown | 0.25 | −2% | +13% |
| Recession | 0.15 | −18% | +4% |
Covariance is positive when the two returns tend to be above their means in the same states, and negative when one tends to be above its mean while the other is below. Correlation divides by the two standard deviations, so it always lies between −1 and +1.
\( E(R_A)=8.70\% \), \( \sigma_A=15.44\% \) · \( E(R_B)=6.50\% \), \( \sigma_B=5.63\% \) · \( \sigma_{AB}=-43.8 \), so \( \rho_{AB}=-0.50 \).
A pays more and swings more, but the two tend to move in opposite directions: their correlation is −0.50.
| Portfolio | Expected return | Standard deviation |
|---|---|---|
| All A | 8.70% | 15.44% |
| Half and half | 7.60% | 6.76% |
| 21% A, 79% B — minimum variance | 6.96% | 3.98% |
| All B | 6.50% | 5.63% |
Adding the riskier asset to the safer one raises expected return from 6.50% to 6.96% and cuts standard deviation from 5.63% to 3.98%. Holding B alone is dominated.
Asset A did not become safer. The gain comes from A doing well in the states where B does badly, which you can only see by looking at the two together.
For a portfolio with weights \( w_i \), the variance is a double sum over every pair:
\[ \sigma_p^{2}=\sum_{i}\sum_{j} w_i w_j \,\sigma_{ij} \qquad\text{where}\qquad \rho_{ij}=\frac{\sigma_{ij}}{\sigma_i\sigma_j} \]\[ \sigma_p^{2}=\frac{1}{n}\overline{\sigma^{2}}+\Big(1-\frac{1}{n}\Big)\overline{\sigma_{ij}} \]
As \( n \) grows the first term vanishes and the second does not. Individual variances disappear; average covariance remains.
This is why only covariance with the market can be priced. The rest was free to remove, so nobody pays you for holding it.
Quick poll · 3 min
How many randomly chosen stocks does it take to capture most of the available risk reduction?Most of the benefit arrives in the first twenty holdings, and none of it removes the floor. That floor is average pairwise covariance — the systematic risk a portfolio cannot shed.
Computed from the equally-weighted expression above, with an average standard deviation of 40% and average pairwise correlation of 0.20 and 0.40. Illustrative — read the shape of the curve and treat the levels as arbitrary.
If only undiversifiable risk is priced, then expected return is linear in exposure to the market:
\[ E(R_i) \;=\; R_f + \beta_i\big(E(R_M)-R_f\big), \qquad \beta_i=\frac{\sigma_{iM}}{\sigma^{2}_{M}} \]How much of an asset’s movement is shared with the market — a covariance, scaled so that the market itself has \( \beta=1 \). A very volatile asset can still have a low beta.
A very volatile stock with zero market covariance has a beta of zero and, under CAPM, earns the risk-free rate.
\( E(R_i)=R_f+\beta_i\big(E(R_M)-R_f\big)=3.97\%+1.3\times 5\%=10.5\% \)
| Input | Value | Where it comes from |
|---|---|---|
| Risk-free rate | 3.97% | Observed: the 3-month US Treasury bill yield, 17 September 2026. |
| Beta | 1.3 | Estimated: regress the stock’s excess returns on the market’s, often over five years of monthly data; the slope is beta. Another period or index gives a different number. |
| Market risk premium | 5% | Assumed: the extra return investors expect from the market over the risk-free rate. It cannot be observed; estimates from historical returns or forward-looking models range from about 3% to 7%. |
With a premium of 4%, 5% or 6%, the required return is 9.2%, 10.5% or 11.8%. Report all three, so that the reader can see how much the answer depends on the assumption.
Cold call · 3 min
A gold miner’s share price swings wildly but barely moves with the index. Under CAPM, what return should it offer?| General Mills | Food | 0.20 |
| Consolidated Edison | Utility | 0.28 |
| Newmont Mining | Gold | 0.32 |
| Wal-Mart | Supermarket | 0.35 |
| Coca-Cola | Drinks | 0.54 |
| Walt Disney | Entertainment | 1.21 |
| Apple | Computer | 1.26 |
| PulteGroup | Construction | 1.28 |
| Marriott | Hotel | 1.48 |
| Tiffany | Apparel | 1.80 |
Beta sorts by industry: whatever people put off buying in a downturn has a high beta. Gold mining is volatile and defensive at the same time, which is why its beta is 0.32 while its share price swings more than Tiffany’s.
Estimated over 2007–2012; levels shift with the window, the ordering by industry does not.
Session 4 · 45 minutes
The efficient market hypothesis: prices reflect all available information, so a strategy using that information cannot earn more than its risk justifies — no alpha. Three forms, each including the one above.
| Form | Information already in prices | So this cannot earn alpha | Evidence |
|---|---|---|---|
| Weak | Past prices and trading volume | Trading on past price patterns | Largely holds in major markets |
| Semi-strong | All public information: earnings, news, announcements | Analysing public information | Mostly holds, with exceptions: the anomalies |
| Strong | All information, including private information | Even trading on inside information | Fails: insiders earn alpha, hence insider-trading laws |
To call a return abnormal you need a model of the normal return, such as the CAPM, so every test of efficiency is also a test of that model.
A group of stocks, chosen by some characteristic, that has earned more than its beta says it should. The extra return is the group’s alpha. If the market were efficient and the CAPM were right, it would be zero.
| Anomaly | The pattern | First documented |
|---|---|---|
| Size | Small firms have earned more than large firms | Banz (1981) |
| Value | Cheap stocks, with low prices relative to book value, have earned more than expensive ones | Fama and French (1992) |
| Momentum | Stocks that rose over the past year have kept rising for a few months | Jegadeesh and Titman (1993) |
An anomaly means either that the market misprices those stocks or that the model has left out a risk. Researchers have now published more than 400 of them.
Banz, Journal of Financial Economics 1981; Fama and French, Journal of Finance 1992; Jegadeesh and Titman, Journal of Finance 1993.
Hou, K., Xue, C. & Zhang, L. (2020). Replicating Anomalies. Review of Financial Studies 33(5). Jensen, T. I., Kelly, B. T. & Pedersen, L. H. (2023). Is There a Replication Crisis in Finance? Journal of Finance 78(5).
Alpha investing tries to earn more than the risk justifies, by trading on mispricing such as the anomalies — active funds, hedge funds, quant funds.
Beta investing holds the market through an index fund and accepts the market’s return, at very low cost.
Smart beta is an index fund that follows fixed rules to tilt towards a known factor — value, size, momentum, low volatility. It is a published anomaly, sold as a cheap product.
Once an anomaly is known and traded, its alpha shrinks. Beating the market after fees has become harder, and money has moved to beta and smart beta.
McLean and Pontiff, Journal of Finance 2016; Greenwood and Sammon, Journal of Finance 2025; Morningstar, January 2024 (US); Bloomberg, 28 October 2024 (China: ¥3.16tn passive against ¥2.84tn active, end-September 2024).
How well a market prices information also depends on who is trading in it.
Show of hands · 4 min
In the A-share market about 90% of trading is done by individual investors. Is such a market more efficient or less efficient than one dominated by institutions?News gets into prices quickly, because so many people react to it. What a company is worth gets in more slowly, because few individual traders do that analysis. So the answer to the vote is C: it depends on what information we mean.
Retail turnover shares: HSBC estimate reported by CNBC, 20 January 2026 — an estimate, not an official exchange statistic. Turnover: Shanghai Stock Exchange, December 2025.
Think – pair – share · 10 min
Take one side, then say what evidence would settle it.Assets: Bloomberg, July 2026; managers above ¥10bn: AMAC via Caixin, 27 May 2026. February 2024: Shanghai and Shenzhen stock exchanges, 20 February 2024. July 2026: Bloomberg, reported 23 July 2026.
Week 4 is different: there are no formulas for two weeks, and the question becomes why financial intermediaries exist at all.
Market data on every slide carries its own source line and date. All values checked August 2026.
Appendix
The portfolio-variance limit, the term-structure algebra, and the security market line. Covered in class only if time allows.
For an equally weighted portfolio of \( n \) assets, split the double sum into the \( n \) variance terms and the \( n(n-1) \) covariance terms:
\[ \sigma_p^{2}=\frac{1}{n^{2}}\sum_{i}\sigma_i^{2} +\frac{1}{n^{2}}\sum_{i}\sum_{j\neq i}\sigma_{ij} =\frac{1}{n}\overline{\sigma^{2}}+\frac{n-1}{n}\,\overline{\sigma_{ij}} \]\[ \lim_{n\to\infty}\sigma_p^{2}=\overline{\sigma_{ij}} \]
The average variance is divided by \( n \) and vanishes. The average covariance keeps its full weight however many assets you add. Everything the CAPM says follows from this one limit.
For two assets with weight \( w \) in A, portfolio variance is
\[ \sigma_p^{2}=w^{2}\sigma_A^{2}+(1-w)^{2}\sigma_B^{2}+2w(1-w)\sigma_{AB} \]Differentiate with respect to \( w \), set to zero, and solve:
\[ w^{*}=\frac{\sigma_B^{2}-\sigma_{AB}}{\sigma_A^{2}+\sigma_B^{2}-2\sigma_{AB}} \]\( \sigma_A^{2}=238.5 \), \( \sigma_B^{2}=31.8 \), \( \sigma_{AB}=-43.8 \), so \( w^{*}=75.6/357.9=0.21 \) — 21% in the risky asset.
Note that \( w^{*}>0 \) whenever \( \sigma_{AB}<\sigma_B^{2} \). A negative covariance is not required; the riskier asset earns a place in the portfolio as long as it is not too closely tied to the safer one.
Under pure expectations, a two-period investment must pay the same either way — roll two one-period bonds, or hold one two-period bond:
\[ (1+i_{2t})^{2}=(1+i_t)\big(1+i^{e}_{t+1}\big) \qquad\Longrightarrow\qquad i^{e}_{t+1}=\frac{(1+i_{2t})^{2}}{1+i_t}-1 \]Generalising, the implied one-period forward rate between \( n-1 \) and \( n \) is
\[ f_{n}=\frac{(1+i_{nt})^{n}}{(1+i_{n-1,t})^{n-1}}-1 \]Under the liquidity premium theory the forward rate is the expected future short rate plus the term premium, so a forward rate sits above the expected future short rate by exactly the size of that premium.
Add asset \( i \) in small weight \( w \) to the market portfolio. The marginal contribution to portfolio variance is
\[ \frac{\partial \sigma_p^{2}}{\partial w}\bigg|_{w=0}=2\big(\sigma_{iM}-\sigma_M^{2}\big) \]so what matters for the portfolio is \( \sigma_{iM} \), not \( \sigma_i^{2} \). Scaling by the market variance gives \( \beta_i=\sigma_{iM}/\sigma_M^{2} \), and requiring that every asset offer the same reward per unit of that exposure gives
\[ E(R_i)=R_f+\beta_i\big(E(R_M)-R_f\big) \]The security market line is a no-arbitrage condition on marginal contributions to portfolio variance. Volatility never enters it.