Continuous Spot Rates and Discount Factors

Introduction

A spot rate describes the return associated with a particular maturity. It can be quoted as an annual effective rate or as a continuously compounded rate. Although these quotations use different compounding conventions, each can describe the same accumulation over a specified term. The key is to match the rate to its maturity and use a consistent time unit.

This tutorial derives the continuous discount factor and shows how to convert between continuous and annual effective spot-rate quotations. The formulas apply to a stated maturity and do not imply that spot rates are the same at every maturity.

Definition: For a maturity of \(t\) years, let \(\delta_t\) be the continuously compounded spot rate, expressed per year. Under continuous compounding over that term, the accumulation factor is \(e^{\delta_t t}\), and the corresponding discount factor is \(e^{-\delta_t t}\). The annual effective spot rate \(i_t\) is the annual effective rate that produces the same accumulation over the same \(t\)-year term.

From Continuous Accumulation to a Discount Factor

Under a constant force of interest \(\delta\) over an interval of length \(t\), an amount of 1 accumulates to \(e^{\delta t}\). A discount factor reverses that accumulation: it is the present value of 1 due at the end of the interval. Therefore, the discount factor is the reciprocal of the accumulation factor.

a(t)=e^{\delta_t t}, \qquad v(t)=\frac{1}{a(t)}=e^{-\delta_t t}.

Here the subscript on \(\delta_t\) identifies the maturity associated with the spot quotation. For a term structure, the continuous spot rate can differ from one maturity to another. The expression uses the rate for the particular maturity \(t\); it does not say that a single rate applies to all terms.

Important condition: The exponent \(\delta_t t\) must be dimensionless. If \(t\) is measured in years, \(\delta_t\) must be an annual rate. A rate stated per month cannot be multiplied directly by a time stated in years without first converting the units.

Conversion to an Annual Effective Spot Rate

The annual effective rate \(i_t\) is defined so that 1 accumulates to \((1+i_t)^t\) over \(t\) years. If this accumulation is to equal the continuous-compounding accumulation for the same maturity, then the two accumulation factors must be equal:

(1+i_t)^t=e^{\delta_t t}.

For \(t>0\), taking the \(t\)-th root gives \(1+i_t=e^{\delta_t}\). Thus the annual effective rate corresponding to a continuous spot rate is:

i_t=e^{\delta_t}-1.

The corresponding discount factor can also be written using the annual effective quotation:

v(t)=e^{-\delta_t t}=(1+i_t)^{-t}.

The conversion works in the reverse direction as well. Starting from an annual effective spot rate, equate its accumulation to the continuous accumulation and take natural logarithms:

\delta_t=\frac{\ln\left((1+i_t)^t\right)}{t} =\ln(1+i_t).

Both conversions preserve the accumulation and discount factor for the specified term. The annual effective rate is not obtained by multiplying the continuous rate by the maturity: it is obtained by exponentiating the continuous rate, or, in reverse, by taking the natural logarithm of one plus the annual effective rate.

Worked Applications

Worked Example: Converting a Two-Year Continuous Spot Rate

A hypothetical investment has a two-year continuous spot rate of \(4\%\) per year. Find its two-year discount factor and the annual effective spot rate equivalent over the same term. Report the discount factor to six decimal places and the annual effective rate to four decimal places as a percentage.

Method. Use \(v(t)=e^{-\delta_t t}\) for the discount factor and \(i_t=e^{\delta_t}-1\) for the annual effective equivalent. The rate is \(0.04\) per year and the maturity is \(2\) years.

Discount factor.

\delta_2=0.04,\qquad t=2, \qquad v(2)=e^{-(0.04)(2)}=e^{-0.08}=0.9231163464\ldots

Rounding only at the final step gives \(v(2)=0.923116\).

Annual effective equivalent.

i_2=e^{0.04}-1 =1.0408107742\ldots-1 =0.0408107742\ldots

As a percentage, this is \(4.08107742\ldots\%\), or \(4.0811\%\) to four decimal places.

Check. The annual effective accumulation over two years is \((1.0408107742\ldots)^2=e^{0.08}\). Its reciprocal is \(e^{-0.08}=0.9231163464\ldots\), which matches the discount factor. The common mistake avoided here is using \(e^{-0.04}\) as the two-year discount factor, which omits one year of accumulation.

Worked Example: Converting an Annual Effective Rate to a Continuous Quotation

A hypothetical four-year spot rate is quoted as \(5\%\) effective annually. Find the equivalent continuous spot rate and the discount factor for a payment due in four years. Report the rate to four significant figures and the discount factor to six decimal places.

Method. Use \(\delta_t=\ln(1+i_t)\), then calculate the discount factor from the continuous quotation. Here \(i_4=0.05\) and \(t=4\) years.

Continuous spot rate.

\delta_4=\ln(1+0.05)=\ln(1.05) =0.04879016417\ldots

Thus the continuous quotation is \(0.04879\), or approximately \(4.879\%\) per year to four significant figures.

Discount factor. Using the annual effective quotation directly,

v(4)=(1.05)^{-4} =\frac{1}{1.05^4} =\frac{1}{1.21550625} =0.8227024748\ldots

The discount factor is \(0.822702\) to six decimal places.

Check. Since \(4\delta_4=4\ln(1.05)=\ln(1.05^4)\), it follows that \(e^{-4\delta_4}=1/1.05^4\), the same discount factor. The common mistake avoided is reporting \(\ln(1.05)\) as a percentage without converting the decimal rate: \(0.04879\) is about \(4.879\%\), not \(0.04879\%\).

Worked Example: Applying Different Continuous Rates at Different Maturities

A hypothetical term structure has a one-year continuous spot rate of \(3.0\%\) and a two-year continuous spot rate of \(3.6\%\), both per year. For each maturity, calculate the discount factor and the equivalent annual effective spot rate. Give discount factors to six decimal places and rates to four decimal places as percentages.

Method. Apply the conversion separately at each maturity: \(v(t)=e^{-\delta_t t}\) and \(i_t=e^{\delta_t}-1\). Do not use the one-year rate for the two-year maturity.

One-year maturity. Here \(\delta_1=0.030\) and \(t=1\):

v(1)=e^{-(0.030)(1)}=e^{-0.030} =0.9704455335\ldots

i_1=e^{0.030}-1 =1.0304545340\ldots-1 =0.0304545340\ldots

Thus \(v(1)=0.970446\) and \(i_1=3.0455\%\).

Two-year maturity. Here \(\delta_2=0.036\) and \(t=2\):

v(2)=e^{-(0.036)(2)}=e^{-0.072} =0.9305308958\ldots

i_2=e^{0.036}-1 =1.0366558465\ldots-1 =0.0366558465\ldots

Thus \(v(2)=0.930531\) and \(i_2=3.6656\%\).

Check. For each term, \((1+i_t)^t=e^{\delta_t t}\), so \((1+i_t)^{-t}=v(t)\). The calculations use the two-year rate of \(3.6\%\) for the two-year maturity, not the one-year rate. A maturity-specific spot rate must not be silently replaced by a rate from another maturity.

Common Traps and Scope of the Formula

The conversion formulas are simple, but several details determine whether they are being used correctly:

Include the maturity in the discount-factor exponent. The continuous discount factor is \(e^{-\delta_t t}\), not generally \(e^{-\delta_t}\). The latter is appropriate only when \(t=1\) in the stated time unit.
Do not multiply the rate by the maturity when converting to an annual effective rate. The correct conversion is \(i_t=e^{\delta_t}-1\). The maturity appears in the accumulation comparison and in the discount-factor exponent.
Keep the quotation tied to its maturity. A two-year spot rate and a one-year spot rate may differ. Conversion changes the compounding convention; it does not make rates at different maturities equal.
Use consistent time units. If maturity is stated in years, use a continuous rate per year. A different time unit requires a matching rate unit before calculating the exponent.

These relationships are conversion tools for a specified maturity and compounding convention. They do not, by themselves, value a collection of cash flows, determine a spot rate from a security price, or establish how a rate behaves between quoted maturities.

Key takeaway: For a maturity \(t\) in years, a continuous spot rate \(\delta_t\) gives the discount factor \(e^{-\delta_t t}\). Its annual effective equivalent is \(i_t=e^{\delta_t}-1\), and the reverse conversion is \(\delta_t=\ln(1+i_t)\). Each formula must use the rate and time unit corresponding to the same maturity.

Check Your Understanding

For each problem, use the stated maturity and quotation convention. Round only where requested.

A hypothetical three-year investment has continuous spot rate \(2.0\%\) per year. Find its discount factor to six decimal places and its annual effective equivalent to four decimal places as a percentage. (Answer: \(v(3)=0.941765\); \(i_3=2.0201\%\).)
A two-year spot rate is \(4\%\) effective annually. Find its equivalent continuous spot rate to four significant figures and its discount factor to six decimal places. (Answer: \(\delta_2=\ln(1.04)=0.03922\), or \(3.922\%\) per year; \(v(2)=(1.04)^{-2}=0.924556\).)
A hypothetical five-year discount factor is \(0.80\). Find the continuous spot rate and the equivalent annual effective spot rate, each to four decimal places as a percentage. (Answer: \(\delta_5=-\ln(0.80)/5=4.4629\%\) per year; \(i_5=e^{\delta_5}-1=4.5640\%\).)
A term structure has continuous spot rates of \(2.5\%\) at one year and \(3.0\%\) at three years. Calculate each annual effective equivalent and the three-year discount factor. (Answer: \(i_1=2.5315\%\), \(i_3=3.0455\%\), and \(v(3)=e^{-0.09}=0.913931\).)
A hypothetical four-year continuous spot rate is \(1.5\%\) per year. A candidate uses \(e^{-0.015}\) for the four-year discount factor. State the correct discount factor and explain the error. (Answer: \(e^{-(0.015)(4)}=e^{-0.06}=0.941765\); the candidate omitted the four-year maturity from the exponent.)

What You'll Learn