Finance-Theory · Unit 6 · Video 5 · Interactive Practice

Hedge, Don't Bet: Locking In a Forward Rate with Treasuries, and Quoting a Forward Loan

IKey Formulas

Formula	Name	What you need
$1 + f_t = \dfrac{(1+r_{0,t})^t}{(1+r_{0,t-1})^{t-1}}$	Forward rate for year $t$ from spot rates	The $(t-1)$ -year and $t$ -year spot rates
$1 + f_t = \dfrac{P_{0,t-1}}{P_{0,t}}$	The same forward rate from discount bond prices	Two discount bond prices, $P_{0,t} = \dfrac{1}{(1+r_{0,t})^t}$
$\text{lock in: } 1 + f_2 \quad \text{vs} \quad \text{wait: } 1 + R_2$	Year-2 value of each dollar held at year 1	$f_2$ today; the future spot rate $R_2$ is known only at year 1
$S = \dfrac{L\,P_{0,t-1}}{P_{0,t}}, \qquad \dfrac{S}{L} - 1 = f_t$	Bank's hedge of a forward loan $L$ from year $t-1$ to $t$	Buy face $L$ of $(t-1)$ -year zeros, short face $S$ of $t$ -year zeros

Key Insight: a forward rate is fixed today by the prices of two discount bonds, and opposite positions in those bonds lock it in. Locking in removes the bet on $R_2$ ; it does not remove the chance of regret, which happens exactly when $R_2 > f_2$ .

IILocked In vs Waiting

The lock-in pays $10.904M whatever happens; waiting pays $10(1+R_2)$ million, which wins only when $R_2 > f_2$ .

Future spot rate R₂ 7.0%

💡 At year 0 only $f_2 = 9.04\%$ is known and $R_2$ is not, so choosing to wait is a bet that $R_2$ ends above $f_2$ : the position of an interest-rate speculator, not of a CFO hedging a known cash need.

IIIThe Forward Rate Inside Today's Curve

The implied year-two forward rate sits on the far side of the two-year yield from the one-year yield.

Curve Your guess f₂ 4.60%

IVQuoting and Hedging a Forward Loan

Long $(t-1)$ -year zeros plus short $t$ -year zeros cost nothing today and lock in the forward rate $f_t$ .

$20M loan runs Bank's quote 8.51%

VQuiz Questions

Problem 1 · Forward Rate from Two Spot Rates

Given: one-year spot rate $r_{0,1} = 4\%$ and two-year spot rate $r_{0,2} = 6\%$ — find the forward rate $f_2$ you can lock in today for year two.

✅ Correct!

1.06^2 / 1.04 - 1 = 8.04\%

: two years at 6% equal one year at 4% followed by one year at 8.04%.

❌ Not quite. 5% is the average of the two spot rates. The two-year rate is itself (roughly) the average of

r_{0,1}

and

f_2

, so

f_2

lies above 6%, not between the two.

❌ Close, but…

2(6\%) - 4\% = 8\%

is the simple-interest shortcut. Compounding gives

1.06^2 / 1.04 - 1

, which is slightly more.

❌ Not quite.

1.06^2 - 1 = 12.36\%

is the two-year growth. Divide out the first year:

1 + f_2 = 1.06^2 / 1.04

Show solution

Growing a dollar for two years at the two-year rate must equal growing it one year at $r_{0,1}$ and then one year at $f_2$ :

(1.06)^2 = (1.04)(1 + f_2) \quad\Rightarrow\quad 1 + f_2 = \frac{1.1236}{1.04} = 1.08038

So $f_2 \approx 8.04\%$ .

Problem 2 · Regret Is Not a Mistake

Given: the CFO locked in $f_2 = 9.04\%$ on the $10M that arrives at year 1, so it will be $10.904M at year 2. At year 1 the one-year spot rate turns out to be $R_2 = 11\%$ — which statement is true?

✅ Correct!

R_2 = 11\% > f_2 = 9.04\%

, so waiting would have paid more: this is the regret case.

❌ Check the direction. Waiting gives $10M × 1.11 = $11.100M, which is more than $10.904M; the lock-in is behind, not ahead.

❌ Not quite. At year 0

R_2

was unknown; only

f_2

was available. Regret after the fact does not make the hedge wrong: the alternative was a bet on rates.

❌ Not quite.

f_2

is fixed today;

R_2

is the rate the market actually offers at year 1. Waiting earns

R_2 = 11\%

, not

f_2

Show solution

Locked in: $10M × 1.0904 = $10.904M at year 2, fixed at year 0.

Waiting: invest the $10M at year 1 at $R_2 = 11\%$ , giving $10M × 1.11 = $11.100M.

Difference: $11.100M − $10.904M = $0.196M in favour of waiting. Because $R_2 > f_2$ , this is regret; had $R_2$ come in below 9.04% it would have been celebration. Neither outcome was knowable at year 0.

Problem 3 · Hedging a Forward Loan

Given: discount bond prices $P_{0,2} = 0.8900$ and $P_{0,3} = 0.8278$ . A customer wants to borrow $10,000,000 at year 2 for one year. The bank buys $10,000,000 of face value in two-year zeros and pays for them by short-selling three-year zeros — find the face value of the short sale and the rate the bank should quote.

Face value of the three-year short sale?

Rate to quote the customer?

✅ Correct! The short liability of $10,751,389 at year 3 is 7.51% above the $10,000,000 lent, so the customer's repayment at 7.51% covers it.

❌ Not quite. The short sale must raise the $8,900,000 cost, so its face value is

8{,}900{,}000 / 0.8278

; multiplying by the price goes the wrong way.

❌ Not quite. Three-year zeros sell for only 0.8278 per dollar of face, so raising $8,900,000 takes more than $10,000,000 of face value.

❌ Close, but…

10{,}000{,}000 / 0.8900

divides by the two-year price. The short is in three-year zeros: divide the $8,900,000 cost by 0.8278.

❌ Check the short sale. Cost today

= 10{,}000{,}000 \times 0.8900

; the short must raise exactly that at 0.8278 per dollar of face.

❌ Wrong year. 7.01% is

f_2

, for a loan from year 1 to year 2. This loan runs from year 2 to year 3, so it needs

f_3 = P_{0,2}/P_{0,3} - 1

❌ Not quite. 6.5% is the three-year spot rate, an average over years 1 to 3. The loan covers only year 3, priced by the forward rate

f_3

❌ Wrong year. 8.51% is

f_4

, for a loan from year 3 to year 4, the video's example. This loan starts at year 2.

❌ Check the quote. The quote is the short liability divided by the loan, minus 1.

Show solution

Long: $10,000,000 of face in two-year zeros costs

10{,}000{,}000 \times 0.8900 = 8{,}900{,}000

dollars today and pays $10,000,000 at year 2, which goes to the customer as the loan.

Short: selling three-year zeros at 0.8278 per dollar of face to raise $8,900,000 takes

\frac{8{,}900{,}000}{0.8278} = 10{,}751{,}389

dollars of face value, owed at year 3. Net cost today: 0.

Quote:

\frac{10{,}751{,}389}{10{,}000{,}000} - 1 = 7.51\% = \frac{P_{0,2}}{P_{0,3}} - 1 = f_3

From spot rates, $f_3 = 1.065^3 / 1.06^2 - 1 \approx 7.51\%$ as well.

Problem 4 · Lending Forward, One Year Later

Given: a firm will receive $5M at year 2 and must hold it until a payment at year 3. Spot rates are $r_{0,2} = 4\%$ and $r_{0,3} = 5\%$ — find the Treasury positions that lock in a rate for year 3 at zero net cost today, and that rate.

Which positions lock it in?

Which rate is locked in for year 3?

✅ Correct! Borrowing for 2 and lending for 3 lends forward over year 3, at

1.05^3 / 1.04^2 - 1 = 7.03\%

❌ Reversed. Lending short and borrowing long borrows forward, the bank's hedge. The firm has cash arriving and needs to lend forward.

❌ Not quite. Waiting leaves the year-3 rate unknown until year 2: a bet on rates, not a lock-in.

❌ Wrong window. Borrow 1 / lend 2 locks in a rate for year 2. The cash arrives at year 2 and is needed at year 3, so the horizons are 2 and 3.

❌ Check the positions. The $5M at year 2 must repay the short leg, and the long leg must pay out at year 3.

❌ Not quite. 5% is the three-year spot rate, an average over three years; the rate for year 3 alone is the forward rate.

❌ Close, but…

3(5\%) - 2(4\%) = 7\%

is the simple-interest shortcut; with compounding

1 + f_3 = 1.05^3 / 1.04^2

❌ Not quite.

1.05^3 / 1.04 - 1

divides out only one year at 4%; the two-year spot rate compounds for two years, so divide by

1.04^2

❌ Check the rate. Use

1 + f_3 = (1 + r_{0,3})^3 / (1 + r_{0,2})^2

Show solution

Positions: borrow today for 2 years (short two-year zeros with face $5M), so the $5M arriving at year 2 repays the loan exactly. Lend the proceeds for 3 years (buy three-year zeros). Nothing is paid today, $5M goes out at year 2 and more comes back at year 3.

Rate:

1 + f_3 = \frac{(1.05)^3}{(1.04)^2} = \frac{1.157625}{1.0816} = 1.07029

So $f_3 \approx 7.03\%$ : the firm lends forward at 7.03% over year 3, exactly as the CFO lent forward at 9.04% over year 2.

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