Finance-Theory · Unit 6 · Video 4 · Interactive Practice

Forward Rates: Locking In Next Year's Interest Rate From Today's Spot Rates

IKey Formulas

Formula	Name	What you need
$1 + f_t = \dfrac{P_{0,t-1}}{P_{0,t}} = \dfrac{(1+r_{0,t})^t}{(1+r_{0,t-1})^{t-1}}$	One-year forward rate, dates $t-1$ to $t$	Two discount bond prices, or the two spot rates for maturities $t-1$ and $t$
$(1+r_{0,2})^2 = (1+r_{0,1})(1+f_2)$	Two routes to year 2	The case $t = 2$ : two years at the two-year spot rate, or one year at the spot rate then one at the forward rate
$f_1 = r_{0,1}$	The first forward rate	Nothing more: its period starts today
$(1+f_{3\to5})^2 = \dfrac{(1+r_{0,5})^5}{(1+r_{0,3})^3}$	Two-year forward rate $f_{3\to5}$ , years 3 to 5, quoted per year	The five-year and three-year spot rates; $(1+f_{3\to5})^2 = (1+f_4)(1+f_5)$

Key Insight: Keep three rates apart. A spot rate $r_{0,t}$ starts today and is known today; a future spot rate $R_t$ starts later and is unknown today; a forward rate $f_t$ is set today for a period that starts later. When year 1 arrives, $R_2$ need not equal the $f_2$ quoted today.

IIReading Forwards off the Spot Curve

Each one-year forward rate $f_t$ comes from comparing two spot rates, $r_{0,t}$ and $r_{0,t-1}$ .

Forward rate

IIIThe CFO Locks In Year 2

Borrowing for one year and lending for two, both today, turns the unknown $R_2$ into a rate fixed now.

Step trades r₀,₁ 5.0% r₀,₂ 7.0%

Position ($M)	Year 0	Year 1	Year 2
1-Yr Borrowing
2-Yr Lending
Repatriation
Net

IVCompounding, Not Averaging

If two years together earn $r_{0,2}$ and year one earns $r_{0,1}$ , what must year two earn?

Spot rates Guess for year 2 7.00%

VQuiz Questions

Problem 1 · Forward Rate from Two Spot Rates

Given: $r_{0,1} = 4\%$ and $r_{0,2} = 6\%$ — find the one-year forward rate $f_2$ for year 1 to year 2.

✅ Correct!

1 + f_2 = (1.06)^2 / 1.04 = 1.1236 / 1.04 \approx 1.0804

: year 2 has to earn more than 6% because year 1 earns only 4%.

❌ Not quite.

5\%

averages the two spot rates. But

r_{0,2} = 6\%

already covers both years;

f_2

is the rate for the second year alone, which must make up for year 1 earning only 4%.

❌ Close, but that is averaging, not compounding.

2 \times 6 - 4 = 8

treats

6\%

as the simple average of

4\%

and

f_2

. Compounding decides it:

1.04 \times 1.08 = 1.1232

falls just short of

(1.06)^2 = 1.1236

, so

f_2

is slightly above

8\%

❌ Not quite.

6\% - 4\% = 2\%

is the gap between the two spot rates, not a rate for any period. Compare the growth factors:

(1.06)^2

over

1.04

Show solution

Two years at the two-year spot rate must grow a dollar as much as one year at the one-year spot rate followed by one year at the forward rate:

(1+r_{0,2})^2 = (1+r_{0,1})(1+f_2)

1 + f_2 = \frac{(1.06)^2}{1.04} = \frac{1.1236}{1.04} \approx 1.0804

So $f_2 \approx 8.04\%$ . Check: $1.04 \times 1.0804 = 1.123616 \approx 1.1236 = (1.06)^2$ .

Problem 2 · Forward Rate vs Future Spot Rate

Given: Today $r_{0,1} = 5\%$ and $r_{0,2} = 7\%$ , so $f_2 \approx 9.04\%$ , and the CFO borrows $9.524 million for one year at 5% and lends it for two years at 7%. When year 1 arrives, the one-year rate on offer is $R_2 = 7.5\%$ . Which statement is correct?

✅ Correct! Both trades were agreed today, so the year-2 payout of $10.904 million was fixed today; the $10 million repatriated in year 1 still earns

10.904 / 10 - 1 = 9.04\%

, whatever

R_2

turns out to be.

❌ Not quite. Nothing resets. The two-year loan at 7% pays

9.524 \times (1.07)^2 \approx 10.904

million in year 2 no matter what rate is quoted in year 1; that is the whole point of locking in.

❌ Not quite.

f_2

is set by today's bond prices;

R_2

is set only when year 1 arrives and cannot be observed today. The two usually differ, and a gap between them is no evidence that today's prices were wrong.

❌ Not quite.

7\%

is the rate for the two years together. The first year earns

5\%

, so the second-year piece is

f_2 \approx 9.04\%

1.05 \times 1.0904 = 1.14492 \approx (1.07)^2

Show solution

Follow the CFO's cash in $ millions. Year 0: borrow $10/1.05 \approx 9.524$ , lend the same $9.524$ , net $0$ . Year 1: the loan's repayment $\tfrac{10}{1.05} \times 1.05 = 10$ is covered by the $10$ repatriated, net $0$ . Year 2: the two-year loan pays $\tfrac{10}{1.05} \times (1.07)^2 \approx 10.904$ .

Every one of these amounts was fixed by the two trades made today, so the year-2 rate is $10.904 / 10 - 1 = 9.04\%$ whatever $R_2$ turns out to be. A forward rate is observed today for a future period; the future spot rate $R_2$ is unknown today and will rarely equal $f_2$ .

Problem 3 · Forward Rates from STRIPS Prices

Given: STRIPS prices per $1 of face value: $P_{0,1} = 0.95$ , $P_{0,2} = 0.88$ , $P_{0,3} = 0.80$ — find the one-year forward rates $f_2$ and $f_3$ (to two decimal places).

What is f₂ (year 1 to year 2)?

What is f₃ (year 2 to year 3)?

✅ Correct! Each forward rate is the ratio of two adjacent prices:

0.95/0.88 \approx 1.0795

and

0.88/0.80 = 1.10

❌ Not quite.

1/0.95 - 1 \approx 5.26\%

is the one-year spot rate

r_{0,1}

, which is

f_1

. For

f_2

compare the bonds maturing at 1 and 2:

P_{0,1}/P_{0,2}

❌ Not quite.

(1/0.88)^{1/2} - 1 \approx 6.60\%

is the two-year spot rate

r_{0,2}

, spread over both years.

f_2

covers year 1 to year 2 only:

P_{0,1}/P_{0,2}

❌ Check the ratio's direction.

0.88/0.95 - 1 \approx -7.37\%

puts the later price on top. The earlier bond is worth more, so

1 + f_2 = P_{0,1}/P_{0,2}

❌ Not quite.

0.95/0.80 - 1 = 18.75\%

compares maturities 1 and 3, so it spans two years.

f_3

uses the adjacent pair:

P_{0,2}/P_{0,3}

❌ Check the ratio's direction.

0.80/0.88 - 1 \approx -9.09\%

puts the later price on top;

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

❌ Not quite.

(1/0.80)^{1/3} - 1 \approx 7.72\%

is the three-year spot rate

r_{0,3}

, spread over all three years.

f_3

is the year-3 piece alone:

P_{0,2}/P_{0,3}

❌ Check f₂. Use

1 + f_2 = P_{0,1}/P_{0,2}

❌ Check f₃. Use

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

Show solution

The one-year forward rate between $t-1$ and $t$ is the ratio of the two discount bond prices; the face values cancel:

1 + f_t = \frac{P_{0,t-1}}{P_{0,t}}

1 + f_2 = \frac{0.95}{0.88} \approx 1.0795 \quad\Rightarrow\quad f_2 \approx 7.95\%

1 + f_3 = \frac{0.88}{0.80} = 1.10 \quad\Rightarrow\quad f_3 = 10.00\%

The spot rates are different numbers: $r_{0,1} \approx 5.26\%$ , $r_{0,2} \approx 6.60\%$ , $r_{0,3} \approx 7.72\%$ . Each covers a period that starts today; each forward covers one year that starts later.

Problem 4 · A Two-Year Forward Rate

Given: A CFO will receive $5 million at year 3 and pay it out at year 5, and wants to lock in today the rate it earns in between. Today $r_{0,3} = 5\%$ and $r_{0,5} = 6\%$ . Find the annual two-year forward rate $f_{3\to5}$ for years 3 to 5.

✅ Correct! The ratio of growth factors is the two-year growth,

\approx 1.156010

; its square root,

\approx 1.0752

, is the annual rate.

❌ Not quite.

15.60\%

is the total growth over the two years,

(1 + f_{3\to5})^2 - 1

. The annual rate needs a square root.

❌ Close, but that is averaging, not compounding.

(5 \times 6 - 3 \times 5)/2 = 7.5

treats spot rates as simple averages. Compounding:

(1.06)^5 / (1.05)^3 \approx 1.156010

, and

\sqrt{1.156010} \approx 1.0752

❌ Not quite. Halving the two-year growth,

15.60\% / 2

, ignores compounding within the two years. Take the square root of

1.156010

instead.

Show solution

Draw the timeline: the loan is agreed today, the money is received at year 3 and repaid at year 5. Compare the five-year bond with the three-year bond:

(1+f_{3\to5})^2 = \frac{(1+r_{0,5})^5}{(1+r_{0,3})^3} = \frac{(1.06)^5}{(1.05)^3} = \frac{1.3382255776}{1.157625} \approx 1.156010

1 + f_{3\to5} = \sqrt{1.156010} \approx 1.0752 \quad\Rightarrow\quad f_{3\to5} \approx 7.52\%

The $5 million received at year 3 grows to about $5.780 million by year 5 ( $5 \times 1.156010 = 5.78005$ ).

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