Zero Rates and Forward Rates

When you value interest rate swaps and bonds or measure market risk, you start from the rate for each maturity. From it you work out two things: the rate for converting future cash flows to present value, and the rates for future periods. How are spot rates and forward rates related?

What spot rates and forward rates mean

A spot rate is the interest rate for the period from today to a future date. Discounting a cash flow due at that date at the spot rate gives its present value¹. Formally, it is the yield, with compound interest, on a bond such as a discount bond. Such a bond has cash flows only when you invest and when you are repaid. Because there is no interest in between, it is also called the zero-coupon rate. The ideas behind discount factors and present value are covered in Present Value and Discount Factors.

A spot rate covers a period that starts today. The rate for a period that starts at a future date, derived from today's spot rates, is called a forward rate². The derivation assumes that there are no arbitrage opportunities in the market. A rate derived this way is also called an implied forward rate.

Consider a hypothetical example in which the 1-year spot rate is 2.00% and the 2-year spot rate is 2.25% (annual compounding). Comparing two ways to invest 100 for two years gives the following table.

If you invest at the 2-year spot rate, nothing is paid in between, and after two years you have 100 × 1.0225 × 1.0225, or about 104.55. The rate that turns 102.00 after one year into 104.55 after two years is 104.55 ÷ 102.00 − 1, or about 2.50%³. This 2.50% is the forward rate from year 1 to year 2, and it is set so that both ways of investing give the same amount after two years.

How to calculate a forward rate

The forward rate for $m$ years starting $n$ years from now is the rate that makes two investments grow to the same amount. The first invests at the $(n+m)$-year spot rate. The second invests at the $n$-year spot rate and then reinvests at the forward rate from year $n$. If $r_n$ is the $n$-year spot rate and $F_{n,m}$ is the forward rate for $m$ years starting $n$ years from now, then with annual compounding the following relationship holds⁴.

$$(1 + r_{n+m})^{n+m} = (1 + r_n)^{n}(1 + F_{n,m})^{m}$$

Solving this for $F_{n,m}$ gives the formula for the forward rate.

$$F_{n,m} = \left( \frac{(1 + r_{n+m})^{n+m}}{(1 + r_n)^{n}} \right)^{1/m} - 1$$

Using the discount factor $DF(t) = 1 / (1 + r_t)^t$, you can write the same formula as a ratio of discount factors⁵. For one-year forward rates ($m = 1$), the forward rate from year $t - 1$ to year $t$ is as follows.

$$F_{t-1,1} = \frac{DF(t-1)}{DF(t)} - 1$$

When a yield curve (the spot rates for each maturity) is stored as discount factors, you can calculate forward rates in this form. Multiplying $1 + F_{s-1,1}$ for $s = 1, 2, \dots, t$ cancels the discount factors in between, and the product is $1 / DF(t) = (1 + r_t)^t$. So discounting one year at a time at the forward rates gives the same value as discounting at the $t$-year spot rate.

Deep diveWhether the forward rate is above or below the spot rate

Substituting $DF(t) = 1 / (1 + r_t)^t$ into the one-year forward rate formula gives the following.

$$1 + F_{t-1,1} = \frac{(1 + r_t)^t}{(1 + r_{t-1})^{t-1}}$$

If $r_t > r_{t-1}$ (where spot rates rise with maturity), the denominator $(1 + r_{t-1})^{t-1}$ is smaller than $(1 + r_t)^{t-1}$. So the right-hand side is larger than $1 + r_t$, and $F_{t-1,1} > r_t$. If $r_t < r_{t-1}$, the inequality is reversed and $F_{t-1,1} < r_t$. If the spot rate is the same for all maturities, the forward rate equals the spot rate.

Worked example: five years of an upward-sloping curve

Consider a hypothetical example in which the 1-year spot rate is 2.00% and rises by 0.25 percentage points each year to 3.00% at 5 years (annual compounding). The table shows the discount factor for each maturity and the one-year forward rates⁶.

Figure description

The chart plots the spot rates and forward rates from the table above as lines, with maturity on the horizontal axis and the rate (%) on the vertical axis. Each forward rate point is the rate for the one-year period that starts one year before its maturity, plotted at that maturity. The spot rate line rises by 0.25 percentage points per year, while the forward rate line rises by about 0.50 percentage points per year, roughly twice the slope. The two lines coincide in year 1. From year 2 on, the forward rate line lies above the spot rate line, and the gap widens as maturity gets longer.

For example, the forward rate for year 2 comes from the ratio of the 1-year and 2-year discount factors: 0.980392 ÷ 0.956474 − 1, or about 2.50%. The forward rate from year 4 to year 5 is 0.897166 ÷ 0.862609 − 1, or about 4.01%, roughly 1 percentage point above the 5-year spot rate of 3.00%. Discounting one year at a time from year 1 to year 5 at the forward rates in the table gives a 5-year discount factor of 0.862609, the same as discounting at the 5-year spot rate.

The curve shape and forward rates

Whether a forward rate is above or below the spot rate depends not on the level of spot rates but on how they change as the maturity gets longer. Where spot rates rise with maturity, as in the example above, forward rates are higher than spot rates. Where spot rates fall with maturity, forward rates are lower, and where spot rates are the same for all maturities, the two are equal. In a downward-sloping example with 3.00% at 1 year and 2.00% at 5 years, the forward rate from year 4 to year 5 is 1.01%.

Changing the compounding frequency changes the discount factors and forward rates slightly, even with the same 2.00% and 3.00%. With 2.00% at 1 year and 3.00% at 5 years under continuous compounding, the 5-year discount factor is 0.8607 and the forward rate from year 4 to year 5 is 4.00%.

Use in practice

You cannot read spot rates directly from the rates traded in the market. They are backed out from market rates such as interest rate swap rates, one maturity at a time, starting with the shortest: 1 year, 2 years, 3 years, and so on. For maturities with no market rate, a rate is first filled in from the market rates for the maturities on either side⁷. The procedure is covered in detail in Bootstrapping a Yield Curve.

Spot rates are the basis for calculating the present value (theoretical value) of financial instruments. Discount factors and forward rates are derived from spot rates and used to measure and understand market risk⁸.

Assumptions and limitations

A forward rate is a value derived from today's spot rates on the assumption that the market allows no arbitrage. The Bank of Japan's material says that forward rates can be taken as future rates that reflect today's market expectations. Market expectations change from day to day, and today's expectations may not be realized in the future market¹⁰. So the rate that actually applies in the future may differ from the forward rate calculated today.

Except for the last continuous-compounding example in the section "The curve shape and forward rates," the formulas and worked examples in the text use whole years and annual compounding.

Notes

1. Bank of Japan Financial System and Bank Examination Department, Center for Advanced Financial Technology, "スポットレートとその応用" (Spot rates and their applications, reference material 5, in Japanese), "２．スポットレート" (2. Spot rates), page 6. The same center's "Ⅱ．市場リスクの把握と管理" (Understanding and managing market risk, June 2012, in Japanese) gives the same definition in "（参考）割引率（スポットレート）の定義" (Reference: definition of the discount rate (spot rate)), page 10. ↩

2. "スポットレートとその応用" (Spot rates and their applications), "４．インプライド・フォワード・レート" (4. Implied forward rates), page 12. The section heading spells it "インプライド・フォワード・レート", and the heading on page 13 spells it "インプライド・フォワードレート". ↩

3. Amounts are rounded to two decimal places. Before rounding, the amount after two years is 104.550625 and the forward rate is about 2.5006%. ↩

4. "スポットレートとその応用" (Spot rates and their applications), "インプライド・フォワードレートの計算式" (Formula for the implied forward rate), page 13. This page denotes the material's symbol ｎＦｒｍ (the m-year rate n years from now) as $F_{n,m}$. ↩

5. "スポットレートとその応用" (Spot rates and their applications), "３．ディスカウント・ファクター" (3. Discount factors), page 11. ↩

6. Spot rates and discount factors are calculated from unrounded values and then rounded to the digits shown. Calculating the forward rates from the ratios of the discount factors in the table (to 6 decimal places) gives the same second decimal place as the table. ↩

7. "スポットレートとその応用" (Spot rates and their applications), "スポットレートの計算式" (Formula for spot rates), page 8, and "（例）スポットレートの計算式" (Example: formula for spot rates), page 9. The treatment of maturities with no market rate is in the note on page 8. ↩

8. "スポットレートとその応用" (Spot rates and their applications), "スポットレートの重要性" (Why spot rates matter), page 10. ↩

9. "Ⅱ．市場リスクの把握と管理" (Understanding and managing market risk, June 2012), "（２）ＧＰＳ（グリッド・ポイント・センシティビティ）" ((2) GPS (grid point sensitivity)), page 14. The BPV is on page 13 of the same document. ↩

10. "スポットレートとその応用" (Spot rates and their applications), "留意点" (Points to note), page 16. ↩