---
title: "Money Markets"
book: "Markets II: Rates, FX and Credit"
subject: quant
language: en
chapter: 2
exercises: 8
source: https://one-course.com/books/quant/2/en/chapter/2-money-markets
---

# Chapter 2 — Money Markets

On Thursday 29 December 2022 [money-market funds](#def-m2-money-markets-mmf) and other eligible institutions left USD 2 308 billion overnight with the Federal Reserve through its [reverse repo facility](#def-m2-money-markets-rrp). On Friday 30 December, the last business day of the year, they left USD 2 554 billion: 245 billion more in one day, a record. On the first business day of 2023 the figure fell back to 2 188 billion. Nothing about monetary policy changed over those three days. What changed was the calendar: on the last day of the year the banks and dealers that normally borrow that cash would not take it, because the size of their balance sheet on that one night is what their regulators and investors measure. The money markets, where cash is lent for a day to a year, are among the deepest markets there are, and they are shaped by accounting dates as much as by interest rates. This chapter is about their instruments, their quoting conventions and the dates on which they misbehave.

## 2.1 Bills and their quoting conventions

**Definition 2.1 (Treasury bill).**

A *Treasury bill* is a government security with a maturity of a year or less that pays no coupon: it is sold below its face value and repays the face value at maturity. The US Treasury auctions bills of 4, 6, 8, 13, 17, 26 and 52 weeks, and cash management bills of irregular terms when its cash needs require.

A bill’s price is quoted through a rate, and the market uses three, which differ by what the interest is expressed on and by the length of the year.

**Definition 2.2 (Discount, money-market and bond-equivalent yields).**

For a bill with $t$ days to maturity and price $P$ per 100 of face value: the *discount yield* $d$ (bank discount basis) is the discount expressed on the face value over a 360-day year, $P = 100\,(1 - d\,t/360)$; the *money-market yield* $m = \frac{100 - P}{P}\frac{360}{t}$ is simple interest on the amount paid over a 360-day year, the rate of a deposit of the same term; the *bond-equivalent yield* $i$ (the Treasury’s *investment rate*) is the same on a 365-day year, $i = \frac{100 -
P}{P}\frac{y}{t}$ with $y = 365$, or 366 if the year after issue contains a 29 February, for bills of at most half a year; beyond half a year it solves

$$
P\Bigl[1 + \Bigl(t - \frac y2\Bigr)\frac{i}{y}\Bigr]\Bigl(1 + \frac i2\Bigr) = 100,
$$

a semiannual compounding that makes it comparable with the yield of a coupon bond ([Chapter 3](https://one-course.com/books/quant/2/en/chapter/3-government-bonds#ch-m2-government-bonds)).

**Proposition 2.3 (Ordering of the three yields).**

For $0 < t \le y/2$ and $d > 0$,

$$
m = \frac{d}{1 - d\,t/360} > d, \qquad i = m\,\frac{y}{360} > m .
$$

The [discount yield](#def-m2-money-markets-yields) always understates the return on the cash invested; the gap grows with the rate and with the term.

**Proof.** From $P = 100(1 - dt/360)$, $100 - P = 100\,dt/360$, so $m = \frac{dt/360}{1 -
dt/360}\frac{360}{t} = \frac{d}{1 - dt/360}$, larger than $d$ because the denominator is below one. The second identity is the definition with $y$ in place of 360. ∎

**Example 2.4 (A thirteen-week bill).**

A 91-day bill auctioned at a discount rate of 3.82% costs $100(1 - 0.0382 \times 91/360) = 99.034389$ per 100. Its [money-market yield](#def-m2-money-markets-yields) is 3.8572% and its [bond-equivalent yield](#def-m2-money-markets-yields) 3.9108%: nine basis points separate the number on the auction screen from the number to compare with a Treasury note. The formulas are the Treasury’s own, and they reproduce its published examples to the last digit ([Section 2.6](#sec-2-6)).

![One illustrative bill curve (the seven auctioned maturities) in the three conventions. Discount yields fall steadily with maturity; money-market yields do not, because the gap between them widens with the term: the 52-week bill yields more than the 26-week in that convention. Only the bond-equivalent curve can be joined to the coupon curve. Levels are illustrative, not quotes. Data: the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-2/m2-money-markets/fig-700b7be0328a.svg)

***Figure 2.1.** One illustrative bill curve (the seven auctioned maturities) in the three conventions. [Discount yields](#def-m2-money-markets-yields) fall steadily with maturity; [money-market yields](#def-m2-money-markets-yields) do not, because the gap between them widens with the term: the 52-week bill yields *more* than the 26-week in that convention. Only the bond-equivalent curve can be joined to the coupon curve. Levels are illustrative, not quotes. Data: the chapter’s tutorial.*

## 2.2 Commercial paper and certificates of deposit

**Definition 2.5 (Commercial paper).**

*Commercial paper* is unsecured short-term debt issued by companies, banks and special-purpose vehicles to investors, usually at a discount. In the United States it is exempt from registration if its maturity does not exceed 270 days and it finances current transactions; most of it is issued for much shorter terms and rolled over continuously. *Asset-backed* paper is issued by a vehicle that holds receivables and repays from them.

**Definition 2.6 (Certificate of deposit).**

A *certificate of deposit* (CD) is a bank deposit for a fixed term, issued as a negotiable instrument: the bank pays principal and simple interest at maturity, and the holder can sell the certificate before then.

Paper and CDs pay a spread over bills for three reasons: credit (a company or bank can fail and a government that borrows in its own currency does not in the same way), liquidity (there is no deep secondary market in one issuer’s thirty-day paper), and the investor base (a bill can be held by any money fund, paper only by some). The spread is small in calm markets and is the first thing to move when the calm ends, because a holder who cannot sell paper must wait for it to mature, and an issuer that cannot roll paper must find the cash elsewhere, from a bank line it arranged for the purpose.

**Method 2.7 (Comparing short instruments).**

Convert every quote to the [money-market yield](#def-m2-money-markets-yields) (simple interest on the amount paid, actual/360) before comparing: a bill’s discount rate, a paper’s discount rate, a CD’s coupon, a repo rate. Then compare equal terms, and ask of each spread which of credit, liquidity and investor base it pays for.

## 2.3 Money-market funds

**Definition 2.8 (Money-market fund).**

A *money-market fund* (MMF) is a mutual fund restricted to short, high-quality instruments that offers daily liquidity and a share value that moves little or not at all. *Government* funds hold bills, other government securities and repo backed by them; *prime* funds also hold paper and bank deposits; many government and retail funds keep a stable share price of one dollar.

A fund that promises a stable dollar and invests in anything that can lose value makes a promise it can break. On 16 September 2008, the day after Lehman Brothers filed for bankruptcy, the Reserve Primary Fund, a USD 62 billion prime fund that held USD 785 million of Lehman debt, marked that debt at zero and its shares at 97 cents. Investors withdrew about USD 300 billion, 14%, from prime funds that week; the funds sold paper to pay them, the paper market stopped, and the Treasury guaranteed the dollar share price of more than USD 3 trillion of fund shares. Every reform of money funds since then, the latest adopted by the US regulator in 2023, has been about which funds may promise a stable price and what they must do when investors leave faster than their assets mature.

**As of September 2026 — US money-market funds.**

**Size** (week to 16 September 2026): USD 7.92 trillion, of which government funds 6.53 trillion, prime 1.24 trillion and tax-exempt 0.15 trillion; retail 3.11 trillion, institutional 4.81 trillion. **Rules** (reforms adopted July 2023, in force by October 2024): at least 25% of assets liquid within a day and 50% within a week (previously 10% and 30%); redemption gates removed; institutional prime and institutional tax-exempt funds must charge a liquidity fee on days when net redemptions exceed 5% of net assets; other non-government funds may charge up to 2%.

**Definition 2.9 (Reverse repo facility).**

A central bank’s *reverse repo facility* lets eligible counterparties lend it cash overnight against securities at a rate it administers; the rate puts a floor under what those counterparties will accept from anyone else.

[Figure 2.2](#fig-m2-money-markets-plumbing) is where the money goes. A government fund’s cash lands in three places: bills, repo with dealers, and, when dealers will not take it, the Federal Reserve’s [reverse repo facility](#def-m2-money-markets-rrp) ([Remark 1.8](https://one-course.com/books/quant/2/en/chapter/1-central-banks-and-the-short-rate#rem-m2-central-banks-and-the-short-rate-below)). Which of the three wins each day is decided by their rates to the basis point, and the fund’s choice moves the Treasury’s financing cost and the dealers’ funding cost.

![Where a money-market fund’s cash goes each day. A government fund chooses among bills, repo with dealers and the central bank’s reverse repo facility, at their rates of the day; a prime fund can also buy paper and bank CDs. At quarter-ends and at the year-end the arrow to the dealers shrinks and the one to the Federal Reserve grows ().](https://one-course.com/images/onecourse/chapters/quant-2/m2-money-markets/fig-b1e71b81d583.svg)

***Figure 2.2.** Where a [money-market fund](#def-m2-money-markets-mmf)’s cash goes each day. A government fund chooses among bills, repo with dealers and the central bank’s [reverse repo facility](#def-m2-money-markets-rrp), at their rates of the day; a prime fund can also buy paper and bank CDs. At quarter-ends and at the year-end the arrow to the dealers shrinks and the one to the Federal Reserve grows ([Figure 2.3](#fig-m2-money-markets-rrp)).*

## 2.4 Turns: quarter-end and year-end

**Definition 2.10 (Turn).**

A *turn* is a date across which borrowing costs more, or lending earns less, than on the neighbouring days, because a balance sheet is measured on it: a quarter-end, and above all the year-end. The *turn premium* is the excess of the rate for that night over the rate for ordinary nights.

Banks and dealers are assessed, for capital ratios, for supervisory scores and by their shareholders, partly on the size of their balance sheet at reporting dates. Repo and deposits are large and cheap to shrink, so they shrink on those dates: a dealer that normally borrows a fund’s cash overnight and lends it on declines it for one night, or takes it only at a price. The cash goes to the central bank’s facility if there is one, which is why the facility’s take-up jumps on those days ([Figure 2.3](#fig-m2-money-markets-rrp)), and the rate that remains in the market jumps with it.

![Daily take-up of the Federal Reserve’s overnight reverse repo facility, July 2022 to March 2023. The spikes are the quarter-ends (and, smaller, some month-ends): on the reporting dates banks and dealers take less of the funds’ cash and the facility takes the rest. The year-end spike, 245 billion in one day, is the largest. Data: Federal Reserve Bank of New York, via FRED series RRPONTSYD.](https://one-course.com/images/onecourse/chapters/quant-2/m2-money-markets/fig-04b05895ff30.svg)

***Figure 2.3.** Daily take-up of the Federal Reserve’s overnight [reverse repo facility](#def-m2-money-markets-rrp), July 2022 to March 2023. The spikes are the quarter-ends (and, smaller, some month-ends): on the reporting dates banks and dealers take less of the funds’ cash and the facility takes the rest. The year-end spike, 245 billion in one day, is the largest. Data: Federal Reserve Bank of New York, via FRED series RRPONTSYD.*

**Proposition 2.11 (A turn priced in a term rate).**

A deposit for $N$ days at the simple rate $R$ spans $N_0$ ordinary days at the overnight rate $r_0$, compounded, and a turn of $N_T$ days at the rate $r_T$. If the two ways of placing the cash are equivalent,

$$
r_T \;=\; \Bigl[\frac{1 + R\,N/B}{(1 + r_0/B)^{N_0}} - 1\Bigr]\frac{B}{N_T} .
$$

**Proof.** Rolling overnight grows 1 to $(1 + r_0/B)^{N_0}(1 + r_T N_T/B)$; the deposit grows it to $1 + RN/B$. Equate and solve for $r_T$. ∎

A term rate therefore contains every turn it spans, diluted by its length: a turn of eight basis points for one night adds a little over one basis point to a one-week rate and a quarter of a basis point to a one-month rate ([Figure 2.4](#fig-m2-money-markets-oneweek)). The dilution is also why a turn is priced long before it arrives: the one-month rate quoted in early December already contains the year-end.

![The rate of a one-week deposit by its start date, when ordinary nights earn 3.87% and the quarter-end night of 30 September earns 3.95%. Every deposit that spans the quarter-end costs 1.1 basis points more; the turn is priced for a week before it happens. Rates are illustrative. Data: the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-2/m2-money-markets/fig-7e0369cc731d.svg)

***Figure 2.4.** The rate of a one-week deposit by its start date, when ordinary nights earn 3.87% and the quarter-end night of 30 September earns 3.95%. Every deposit that spans the quarter-end costs 1.1 basis points more; the turn is priced for a week before it happens. Rates are illustrative. Data: the chapter’s tutorial.*

**Remark 2.12 (Floors and ceilings on the turn).**

A [floor system](https://one-course.com/books/quant/2/en/chapter/1-central-banks-and-the-short-rate#def-m2-central-banks-and-the-short-rate-corridor) bounds the turn from both sides for those who can reach the facilities: a fund can always earn the reverse repo rate, and a bank can borrow against Treasuries at the standing repo rate. Neither bound is absolute. Institutions without access are outside them, and the standing repo facility lends only against collateral and adds to the borrower’s balance sheet on exactly the night it wants to show a small one.

## 2.5 Tutorial: one bill, three rates

**Goal.** Convert the bill curve of [Figure 2.1](#fig-m2-money-markets-bills) between conventions, check the conversions against the Treasury’s own worked examples, and price a turn. **End state:** [Figure 2.1](#fig-m2-money-markets-bills) and the numbers of [Example 2.4](#ex-m2-money-markets-13w).

1. **Price from discount, and back.** The Treasury rounds prices to six decimals. `def price_from_discount (d: float , days: int ) -> float : """P = 100 (1 - d r / 360), rounded to six decimals as the Treasury does.""" return round (100.0 * (1.0 - d * days / 360.0 ), 6 ) def discount_from_price (p: float , days: int ) -> float : return (100.0 - p) / 100.0 * 360.0 / days` **Listing 2.1.** A bill’s price from its discount rate, and the inverse. code/firm/mmyield/firm_mmyield.py Check: 7.610% for 90 days gives 98.097500, as in the regulation.
2. **The investment rate**, with its leap-year rule and its quadratic beyond half a year. `def investment_rate (p: float , days: int , y: int = 365 ) -> float : """Coupon-equivalent (bond-equivalent) yield of a bill. Up to half a year: simple interest on price over a y-day year. Beyond: solve P [1 + (r - y/2) i / y] (1 + i / 2) = 100.""" if days <= y / 2 : return (100.0 - p) / p * y / days a = days / (2.0 * y) - 0.25 b = days / y c = (p - 100.0 ) / p return (-b + math.sqrt(b * b - 4.0 * a * c)) / (2.0 * a)` **Listing 2.2.** The bond-equivalent yield, short and long bills. code/firm/mmyield/firm_mmyield.py Check: a 52-week bill at 92.265000 gives 8.237%.
3. **The curve.** `money_market_demo.bill_curve()` converts the seven maturities; `fig_money_market.py` writes [Figure 2.1](#fig-m2-money-markets-bills) . The 13-week row reads 99.034389, 3.8572%, 3.9108%.
4. **A turn.** `def implied_turn (term_rate: float , term_days: int , normal_rate: float , normal_days: int , turn_days: int , basis: int = 360 ) -> float : """Overnight rate over the turn implied by a term rate spanning it, when the other days of the term are assumed to earn normal_rate each day, compounded daily.""" total = 1.0 + term_rate * term_days / basis normal = (1.0 + normal_rate / basis) ** normal_days return (total / normal - 1.0 ) * basis / turn_days` **Listing 2.3.** The overnight rate over a turn implied by a term rate. code/firm/mmyield/firm_mmyield.py

**What to change next.** Replace the discount rates by a flat 3.80% and read the bond-equivalent curve ([Exercise 2.7](#exo-m2-money-markets-7)); then move the issue date to 1 March 2027 and see which bills change.

## 2.6 Build: the money-market quote converter

**Purpose.** Every short rate the miniature firm reads arrives in its own convention; the curve builder ([Chapter 9](https://one-course.com/books/quant/2/en/chapter/9-interest-rate-swaps#ch-m2-interest-rate-swaps)), the repo book ([Chapter 5](https://one-course.com/books/quant/2/en/chapter/5-repo-and-specials#ch-m2-repo-and-specials)) and the cash desk need them on one basis. This component converts.

**Interface.** `price_from_discount(d, days)`; `discount_from_price(p, days)`; `money_market_yield(p, days, basis)`; `year_days(issue)`; `investment_rate(p, days, y)`; `term_proceeds(amount, rate, days, basis)`; `implied_turn(term_rate, term_days, normal_rate, normal_days, turn_days)`.

**Rules.** Prices per 100, rounded to six decimals when computed from a discount rate; rates in decimals; the 365/366 rule of the regulation; the long-bill formula beyond $y/2$ days.

**Acceptance tests.** `code/firm/mmyield/tests/`: the four worked examples of 31 CFR 356, Appendix B, to the printed digit; the leap-year rule on its two examples; discount $<$ money-market $<$ bond-equivalent; the long formula continuous at half a year; a flat rate implies a flat turn.

**Stretch.** Commercial-paper and CD conventions by currency (sterling is actual/365); a turn calendar that knows every quarter-end and holiday for USD, EUR, GBP and JPY.

Sources and further reading

- US Treasury, *Treasury Bills* (TreasuryDirect); 31 CFR Part 356, Appendix B, *Formulas and Tables* , section VI.
- Investment Company Institute, *Money Market Fund Assets* , weekly release of 17 September 2026. US Securities and Exchange Commission, press release 2023-129, *SEC Adopts Money Market Fund Reforms* , 12 July 2023.
- M. L. Schapiro, testimony on money market funds before the Senate Committee on Banking, Housing and Urban Affairs, 21 June 2012.
- T. K. Hahn, “Commercial paper”, Federal Reserve Bank of Richmond *Economic Quarterly* 79(2), 1993.
- Federal Reserve Bank of New York, reverse repo operation results (FRED series RRPONTSYD).

## 2.7 Exercises

**Exercise 2.1 ★.**

Give the price per 100 of a 26-week (182-day) bill auctioned at a discount rate of 3.75%.

**Solution of Exercise 2.1.**

$100\,(1 - 0.0375 \times 182/360) = 98.104167$ per 100.

**Exercise 2.2 ★.**

A 91-day bill trades at 99.040000. Give its [discount yield](#def-m2-money-markets-yields), [money-market yield](#def-m2-money-markets-yields) and [bond-equivalent yield](#def-m2-money-markets-yields) (year of 365 days).

**Solution of Exercise 2.2.**

[Discount yield](#def-m2-money-markets-yields) $(100 - 99.04)/100 \times 360/91 = 3.7978\%$; [money-market yield](#def-m2-money-markets-yields) $0.96/99.04 \times 360/91 = 3.8346\%$; [bond-equivalent yield](#def-m2-money-markets-yields) $0.96/99.04
\times 365/91 = 3.8879\%$.

**Exercise 2.3 ★.**

A company buys a USD 10 million 90-day CD paying 4.00% (actual/360). What does it receive at maturity? Which convention of [Definition 2.2](#def-m2-money-markets-yields) is the CD’s rate already in?

**Solution of Exercise 2.3.**

$10\,000\,000 \times (1 + 0.04 \times 90/360) = \text{USD}~10\,100\,000$. The CD’s rate is simple interest on the amount invested, actual/360: it is already a [money-market yield](#def-m2-money-markets-yields), directly comparable with the $m$ of a bill.

**Exercise 2.4 ★★.**

Show that for a bill of at most half a year the [bond-equivalent yield](#def-m2-money-markets-yields) always exceeds the [discount yield](#def-m2-money-markets-yields). Is the gap larger for a 4-week or a 26-week bill at the same discount rate? Why?

**Solution of Exercise 2.4.**

By [Proposition 2.3](#prop-m2-money-markets-order), $i = \frac{y}{360}\,\frac{d}{1 -
dt/360}$, a product of two factors above one times $d$. The gap $m - d =
\frac{d^2 t/360}{1 - dt/360}$ grows with $t$: at the same discount rate the 26-week bill’s yields exceed its discount rate by more than the 4-week’s, because the discount is taken on face value and the longer bill’s price is further below it.

**Exercise 2.5 ★★.**

Reproduce the regulation’s 52-week example: a discount rate of 7.65% for 364 days gives a price of 92.265000 and an investment rate of 8.237%. Why does the formula beyond half a year compound semiannually?

**Solution of Exercise 2.5.**

$P = 100\,(1 - 0.0765 \times 364/360) = 92.265000$. With $y = 365$: $a =
364/730 - 0.25 = 0.248630$, $b = 364/365 = 0.997260$, $c = (92.265 -
100)/92.265 = -0.083835$, and $i = (-b + \sqrt{b^2 - 4ac})/(2a) = 8.237\%$. A coupon security of the same maturity would pay a coupon after six months, which its holder could reinvest; the bill pays nothing until maturity. The formula credits the bill with that reinvestment so that its yield compares with a semiannual bond yield.

**Exercise 2.6 ★★.**

An institutional prime fund of USD 20 billion receives redemption requests of USD 1.1 billion in one day. What does the 2023 rule require? What would a government fund have to do?

**Solution of Exercise 2.6.**

Net redemptions are $1.1/20 = 5.5\%$ of net assets, above the 5% threshold: an institutional prime fund must charge the mandatory liquidity fee to the redeeming investors. It pays the redemptions from its daily liquid assets, at least 25% of the fund. A government fund has no mandatory fee; it pays from its liquid assets, which for a fund of bills and repo are most of the fund.

**Exercise 2.7 ★★★.**

*Coding.* With `firm_mmyield`, compute the [bond-equivalent yield](#def-m2-money-markets-yields) of all seven bill maturities at a flat discount rate of 3.80% (issue in September 2026). Report the 4-week and the 52-week values and explain the slope.

**Solution of Exercise 2.7.**

4 weeks 3.8642%, 6 weeks 3.8699%, 8 weeks 3.8757%, 13 weeks 3.8901%, 17 weeks 3.9018%, 26 weeks 3.9282%, 52 weeks 3.9675%. A flat discount curve is an *upward-sloping* yield curve: the conversion adds more at longer terms ([Exercise 2.4](#exo-m2-money-markets-4)), and beyond half a year the semiannual formula takes over. Ten basis points of slope come from the convention alone.

**Exercise 2.8 ★★★.**

*Find the flaw.* “The 13-week bill was auctioned at 3.80% and the 52-week at 3.79%: the bill curve is inverted, and the market expects a cut.” Convert and correct.

**Solution of Exercise 2.8.**

In bond-equivalent terms the 13-week bill yields 3.8901% and the 52-week 3.9567%: the curve slopes *up* by 6.7 basis points. Discount rates of different maturities are not comparable yields. To read the market’s expectations one must also allow for term premia and for the supply of each maturity, but first one must compare like with like.

## 2.8 Problem: Parking Cash over Year-End

**Problem 2.1.**

Weekend problem — the implied year-end turn

A corporate treasurer holds USD 500 million from Monday 28 December 2026 to Monday 4 January 2027. Thursday 31 December is the last business day of the year; Friday 1 January is a holiday. A bank quotes a one-week deposit at 3.95% (actual/360). Ordinary overnight placements are at 3.88%, and the Federal Reserve’s [reverse repo facility](#def-m2-money-markets-rrp) pays 3.75% to the funds that can use it.

**Part I — The quotes.**

1. How many days does the deposit run, and how many days does the overnight placement of 31 December cover?
2. What interest does the one-week deposit pay?
3. Express its rate as a [bond-equivalent yield](#def-m2-money-markets-yields) .
4. A money fund places the same amount at the [reverse repo facility](#def-m2-money-markets-rrp) every night of the week, reinvesting the interest. What does it earn?
5. Which of these placements carries credit risk, and to whom?

**Part II — The turn.**

6. What would the three ordinary nights (28, 29 and 30 December) earn at 3.88%, compounded?
7. Give the overnight rate on 31 December implied by the deposit quote ( [Proposition 2.11](#prop-m2-money-markets-turn) ).
8. Give the [turn premium](#def-m2-money-markets-turn) in basis points.
9. What does the turn cost a bank that must borrow USD 500 million over it, relative to ordinary nights?
10. Why would a bank pay it rather than simply not borrow?

**Part III — The balance sheet.**

11. Explain the jump of [Figure 2.3](#fig-m2-money-markets-rrp) on 30 December 2022 with your answer to question 10.
12. A money fund that finds no bank on 31 December places at the [reverse repo facility](#def-m2-money-markets-rrp) instead. What does it give up over the four days, against the implied turn rate?
13. Give the rate of a two-week deposit from 28 December spanning the same turn, if the ten other nights are ordinary.
14. Why is the turn smaller in the two-week rate?
15. The implied turn is almost exactly the Fed’s standing repo rate of 4.00%. Coincidence? What would you expect if it were far above?

**Part IV — Judgement.**

16. Why is the year-end turn already in one-month rates quoted on 1 December?
17. Name two things that would make turns disappear.
18. The treasurer could buy a bill maturing 5 January instead. What does she gain and what does she risk?
19. State the *named result* : the implied year-end turn and its premium.
20. In one sentence: why is one night different from the others?

**Solution of Problem 2.1.**

**1.** Seven days; the 31 December placement runs from Thursday to Monday, four days. **2.** $500 \times 10^6 \times 0.0395 \times 7/360 = \text{USD}~384\,028$. **3.** $3.95\% \times 365/360 = 4.0049\%$. **4.** $500 \times 10^6 \times [(1 + 0.0375/360)^3(1 + 0.0375 \times 4/360) -
1] = \text{USD}~364\,665$. **5.** The deposit: the treasurer lends unsecured to the bank. The facility: none, but the treasurer cannot use it; only eligible funds and dealers can. **6.** $500 \times 10^6 \times [(1 + 0.0388/360)^3 - 1] = \text{USD}~161\,684$. **7.** $r_T = [(1 + 0.0395 \times 7/360)/(1 + 0.0388/360)^3 - 1] \times
360/4 = 4.0009\%$. **8.** $4.0009 - 3.88 = 12.09$ basis points. **9.** $500 \times 10^6 \times 0.001209 \times 4/360 = \text{USD}~6\,716$. **10.** Its positions (securities it holds, loans it has made) must be funded on that night whatever happens; the lenders who can still take the risk and the balance sheet charge for them. What it cannot do cheaply on one night is sell its assets. **11.** On the reporting date dealers and banks took less of the funds’ cash; the funds placed it with the Fed, 245 billion more in one day, and took it back on 3 January. **12.** $500 \times 10^6 \times (0.040009 - 0.0375) \times 4/360 =
\text{USD}~13\,938$. **13.** $[(1 + 0.0388/360)^{10}(1 + 0.040009 \times 4/360) - 1] \times
360/14 = 3.9171\%$. **14.** The same four expensive days are averaged over fourteen instead of seven: the turn adds 3.4 basis points to the two-week rate against 6.9 to the one-week. **15.** Not a coincidence in kind: eligible banks can borrow against Treasuries from the Fed at 4.00%, so a secured rate much above it would be arbitraged by those with collateral and balance sheet to spare. An unsecured deposit can exceed it, because the facility needs collateral and, like any borrowing, enlarges the borrower’s year-end balance sheet. A turn far above would signal stress, or reluctance to be seen using the facility. **16.** A one-month deposit from 1 December spans the year-end: its rate contains the turn, diluted over 31 days, from the day it is quoted. **17.** Measuring balance sheets as averages over the period rather than on its last day; reserves so abundant that no bank needs to borrow over the turn; central-bank facilities that take the cash without cost to anyone’s balance sheet (which the [reverse repo facility](#def-m2-money-markets-rrp) already does for funds). **18.** Gains: no bank credit risk, a liquid asset. Risks and costs: bills maturing just after the year-end are in demand and usually yield *less* than the deposit; she must buy and settle the bill, and she loses the [turn premium](#def-m2-money-markets-turn) that the bank pays. **19.** Named result: *the implied year-end turn* is 4.0009%, a premium of 12.1 basis points over ordinary nights, for the four days from 31 December. **20.** Because it is the night on which the size of balance sheets is measured, and so the night on which balance sheet is scarce.

## 2.9 Interview questions

**Interview question 2.1 ★ trader, bank.**

Why is a bill’s discount rate lower than its yield? Which one would you use to compare it with a two-year note?

**Solution of Interview question 2.1.**

The discount rate expresses the discount on the face value; the investor pays less than face, so the return on what she actually pays is higher, and the Treasury’s [bond-equivalent yield](#def-m2-money-markets-yields) also uses a 365-day year instead of 360. Compare with a two-year note on the [bond-equivalent yield](#def-m2-money-markets-yields) (for bills beyond half a year, the semiannual formula), since note yields compound semiannually on a 365-day basis.

*What the interviewer is looking for: face value against price paid, and 360 against 365.*

**Interview question 2.2 ★ trader, researcher.**

What is a quarter-end turn, and why does it exist?

**Solution of Interview question 2.2.**

The rate for a night on which balance sheets are measured (quarter-ends, especially the year-end) jumps, because banks and dealers shrink repo and deposits for that night; lenders who still offer balance sheet charge for it, and cash that finds no borrower goes to central-bank facilities. The premium appears in every term rate that spans the date, diluted by its length.

*What the interviewer is looking for: the balance-sheet measurement as cause, and the term-rate dilution.*

**Interview question 2.3 ★★ trader, researcher, bank.**

Why did one [money-market fund](#def-m2-money-markets-mmf)’s loss in September 2008 stop the [commercial paper](#def-m2-money-markets-cp) market?

**Solution of Interview question 2.3.**

Prime funds promised a stable dollar share and held paper. When one fund’s shares fell below a dollar because of one issuer’s default, investors in every prime fund had a reason to redeem first, before losses were shared. Funds met redemptions by not rolling paper and by selling it; issuers who relied on rolling paper could not, and the market in new paper stopped. It took a Treasury guarantee of fund shares and central-bank facilities to restart it.

*What the interviewer is looking for: the first-mover incentive of a stable-value claim on risky assets, and the rollover dependence of issuers.*

**Interview question 2.4 ★★ trader.**

A government money fund can lend in repo at 3.85% or to the Fed at 3.75%. Why might it still choose the Fed?

**Solution of Interview question 2.4.**

The Fed is a riskless counterparty with unlimited capacity at a fixed rate (up to the counterparty limit); dealers may not take all the cash, or only at a lower rate on reporting dates; a fund may have counterparty limits with each dealer, or want to keep the cash available late in the day when the private market has closed. The 10 basis points are the price of those differences.

*What the interviewer is looking for: capacity, counterparty limits and timing, not only the rate.*

**Interview question 2.5 ★★ developer, trader.**

Given a one-week rate spanning the year-end and the rate for ordinary nights, compute the implied year-end rate. What assumptions did you make?

**Solution of Interview question 2.5.**

$r_T = [(1 + R N/B)/(1 + r_0/B)^{N_0} - 1] B/N_T$ with $N$ the term’s days, $N_0$ the ordinary days, $N_T$ the days the turn fixing covers (four if the year-end falls on a Thursday before a holiday). Assumptions: ordinary nights earn $r_0$ each, compounded daily; no other turn in the period; the term and overnight quotes carry the same credit; the basis of 360 days.

*What the interviewer is looking for: the weights of the turn fixing, and naming the assumptions.*

**Interview question 2.6 ★★★ researcher, bank.**

How would you build a dollar money-market curve for the first year from bills, repo and overnight-index swaps? What would you do about turns?

**Solution of Interview question 2.6.**

Start from the policy rate path: overnight-index swaps and overnight-rate futures give compounded overnight rates to each date. Bills add the government-supply and safety component; repo adds the collateral component; build a curve per instrument class and model their spreads. Put turns in explicitly: a jump in the forward overnight rate for each quarter-end and year-end night, sized from quotes that straddle them, rather than letting the interpolation smear them over neighbouring dates. Check by repricing every input and by the forward rate’s shape around each turn.

*What the interviewer is looking for: an explicit turn model, and separate curves for instruments with different credit and collateral.*