Finance-Theory · Unit 6 · Video 3 · Interactive Practice

Three Basis Points: A Flight to Quality, and the Rates the Fed Cannot Set

IKey Formulas

Formula	Name	What it says
$P_{0,T} = \dfrac{F}{(1+r_{0,T})^T}$	Price of a Treasury bill	A bill pays no coupon, so its price is the discounted face value. For a three-month bill $T = \tfrac14$ : with $F = 100$ , a yield of 0.84% gives a price of about 99.79, and a yield of 0.03% about 99.9925
$r_{0,T} = \left(\dfrac{F}{P_{0,T}}\right)^{1/T} - 1$	Yield from price	The same relation solved for the rate. With $T = \tfrac14$ the exponent is 4, so a price above $99.99 per $100 gives a yield of about 4 basis points or less
$r_{0,T} = \big[(1+R_1)(1+R_2)\cdots(1+R_T)\big]^{1/T} - 1$	Spot rate as a geometric average	For a whole number of years $T$ , the spot rate averages the one-year rates $R_1, \dots, R_T$ . Plotted against $T$ , the spot rates are the yield curve
$r_{0,T+1} > r_{0,T} \iff R_{T+1} > r_{0,T}$	Slope of the yield curve	The curve rises exactly when the added year's rate is above the average of the years before it; read as a forecast, a rising curve says one-year rates are expected to go up

Key Insight: The Fed sets a target for one rate, the overnight federal funds rate (2% in September 2008). The market sets the bill and bond yields that make up the curve: on 17 September 2008 the three-month bill yielded 0.03% while overnight federal funds averaged 2.80%.

IIThree Basis Points Means a Price Near $100

For a three-month bill the yield is $(100/P)^4 - 1$ : only a price within a cent of $100 yields about 4 basis points or less.

Price of the $100 bill 99.79

IIIWhere the Flight to Quality Landed

Between 16 and 17 September 2008 the near end of the curve collapsed and the long end did not move.

Maturity 3m Axis: categories → years categories

💡 Beyond the first step the curve slopes upward: the market expected rates to rise again once the panic passed, from inflation or from the economic consequences of the crisis. The steeper that slope, the faster and further the expected rise.

IVThe Rate the Fed Targets and the Rates the Market Sets

The Fed targets the overnight federal funds rate; that week even the overnight rate strayed, and the bill yield went its own way.

Day 17 Sep

💡 None of these lines is the discount rate: that is a separate rate, what the Fed itself charges banks that borrow at its discount window.

VQuiz Questions

Problem 1 · Price a Three-Month Bill

Given: as of 17 September 2026 the three-month Treasury bill yields about 4.12%. Find the price of a bill paying $100 in three months, using $P_{0,T} = F/(1+r_{0,T})^T$ with annual compounding.

✅ Correct! With

T = \tfrac14

(1.0412)^{1/4} \approx 1.010144

and

100/1.010144 \approx 98.996

, so the bill costs about $99.00.

❌ That discounts for a full year.

100/1.0412 \approx 96.04

is the price of a one-year bill. The three-month bill is discounted for a quarter of a year: the exponent is

T = \tfrac14

❌ The exponent is upside down.

100/(1.0412)^4 \approx 85.09

discounts for four years. Three months is

T = \tfrac14

of a year, so the denominator is

(1.0412)^{1/4}

❌ That splits the rate into quarters.

100/(1 + 0.0412/4) = 100/1.0103 \approx 98.98

treats 4.12% as quarterly simple interest. With annual compounding the quarter-year growth factor is

(1.0412)^{1/4} \approx 1.010144

, slightly less than 1.0103.

Show solution

A bill pays no coupons, so its price is the face value discounted for the fraction of a year until it pays: $F = 100$ , $T = \tfrac14$ , $r_{0,T} = 0.0412$ .

P_{0,T} = \frac{100}{(1.0412)^{1/4}} \approx \frac{100}{1.010144} \approx 98.996

The price is about $99.00. The buyer earns about $1.00 on the $100 over three months, which compounds to 4.12% over a year.

Problem 2 · Read the Yield Off the Price

Given: a three-month bill paying $100 trades at $99.95. Find its annual yield $r_{0,T}$ .

✅ Correct!

r_{0,T} = (100/99.95)^4 - 1 \approx (1.00050025)^4 - 1 \approx 0.0020025

, about 0.20% a year, or 20 basis points.

❌ That is the three-month return, not the annual yield.

100/99.95 - 1 \approx 0.05\%

is what the buyer earns over a quarter of a year. The yield is annual, so that growth factor is raised to the power

1/T = 4

❌ The exponent is inverted.

(100/99.95)^{1/4} - 1 \approx 0.0125\%

shrinks a quarter's return further. Solving

P = F/(1+r)^{1/4}

for

r

gives

1 + r = (F/P)^4

❌ Price and face value are swapped.

(99.95/100)^4 - 1 \approx -0.20\%

would say the buyer loses money. The buyer pays $99.95 and receives $100, so the growth factor is

F/P = 100/99.95 > 1

Show solution

Solve the bill's price relation for the rate. From $P_{0,T} = F/(1+r_{0,T})^{1/4}$ :

1 + r_{0,T} = \left(\frac{F}{P_{0,T}}\right)^4 = \left(\frac{100}{99.95}\right)^4 \approx (1.00050025)^4 \approx 1.0020025

so $r_{0,T} \approx 0.20\%$ , about 20 basis points. Over the three months the buyer earns only 5 cents on the $100; compounding that quarter's return four times gives the annual yield. On 17 September 2008 the price was even closer to $100, about $99.9925, and the yield was 3 basis points.

Problem 3 · What Makes the Curve Rise

Given: spot rates $r_{0,2} = 2.00\%$ and $r_{0,3} = 2.50\%$ . Find the one-year rate $R_3$ for the third year that these spot rates imply.

✅ Correct!

1 + R_3 = (1.025)^3/(1.02)^2 = 1.076890625/1.0404 \approx 1.035074

. The added year's rate, about 3.51%, is above the 2.00% average of the first two years, which is exactly why

r_{0,3} > r_{0,2}

❌ That is the change in the spot rate, not a rate. The spot rate went up by 0.50 points, but

r_{0,3}

averages three years, so the third year's rate must pull a two-year average of 2.00% up to 2.50%. That takes a rate well above 2.50%.

❌ That averages the two spot rates. The third year's rate is not between them: to lift a 2.00% average to 2.50%, the added year must be above both.

❌ The added year's rate is not the new spot rate. If

R_3

were 2.50%, the three-year spot rate would be

\big[(1.02)^2(1.025)\big]^{1/3} - 1 = (1.066410)^{1/3} - 1 \approx 2.17\%

, not 2.50%: one year at 2.50% lifts a three-year average only a third of the way.

Show solution

A spot rate is the geometric average of the one-year rates, so $(1 + r_{0,3})^3 = (1+R_1)(1+R_2)(1+R_3) = (1 + r_{0,2})^2 (1 + R_3)$ . Solving for $R_3$ :

1 + R_3 = \frac{(1.025)^3}{(1.02)^2} = \frac{1.076890625}{1.0404} \approx 1.035074

so $R_3 \approx$ 3.51%. This is the slope condition at work: the curve rises from $T = 2$ to $T = 3$ because $R_3 \approx 3.51\% > r_{0,2} = 2.00\%$ . Read as a forecast, the rising curve says the one-year rate is expected to go up.

Problem 4 · Which Rate Did the Fed Set?

Given: on 17 September 2008 the federal funds target was 2%, the effective federal funds rate averaged 2.80%, and the three-month Treasury bill yielded 0.03%. Which statement is correct?

✅ Correct! The federal funds rate is what banks charge each other to borrow reserves overnight; the Fed sets a target for it and steers the actual rate by buying and selling securities. The bill yield is a market rate, read off bill prices, which the Fed cannot set directly.

❌ The bill yield is a market rate. It collapsed because cash fled into bills in a flight to quality, bidding the price to within a cent of $100. The Fed held its target at 2% on 16 September, partly because a cut would have encouraged inflation.

❌ That is a different rate. The 2.80% was the effective federal funds rate, what banks charged each other overnight. The discount rate, what the Fed itself charges banks, is separate.

❌ The target is a target, not a fixed price. The Fed steers the actual overnight rate toward it, but that week the rate strayed: 1.98% on the 16th, 2.80% on the 17th and 1.48% on the 19th.

Show solution

Three different rates are in play:

The federal funds target, 2%. The one number the Fed sets: its target for the rate banks charge each other to borrow reserves overnight.
The effective federal funds rate, 2.80%. The rate banks actually charged. The Fed steers it toward the target by buying and selling securities, but on 17 September it ran 80 basis points above.
The three-month bill yield, 0.03%. A market rate, set by trading in bills. The flight to quality pushed the bill's price to about $99.9925 per $100, and that price is the yield.

So the answer is (a). The discount rate, what the Fed charges banks at its discount window, is a fourth rate and none of these.

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