What a Spot Rate Describes
A quoted rate is incomplete unless its time period is clear. In a term structure, a spot rate is attached to a particular maturity: it describes the rate for an investment made at the valuation date and held to that maturity. Rates can differ across maturities, so the rate for a two-year investment need not equal the rate for a three-year investment.
This tutorial uses annual effective rates and maturities measured in whole years. The focus is how to interpret each rate and use the rate that matches an investment’s maturity. A maturity-specific rate is not, by itself, a schedule of the rates that will apply in future years.
The subscript matters. \(s_1\) is the spot rate for a one-year investment, \(s_2\) is the rate for a two-year investment, and \(s_5\) is the rate for a five-year investment. Each is measured from the same valuation date. A table of such rates is called a spot-rate term structure or spot-rate curve.
For example, a schedule might report \(s_1=3.0\%\), \(s_2=3.8\%\), and \(s_3=4.4\%\). These are three maturity-specific rates, not three successive annual rates to apply in order. The two-year spot rate applies to an investment running from the valuation date through year two; the three-year spot rate applies to one running from the valuation date through year three.
Why the Maturity Is Part of the Rate
The accumulation formula follows directly from the meaning of an annual effective rate. If the annual rate for the whole investment is \(s_n\), then each year’s accumulated amount is multiplied by \(1+s_n\). After one year, \(C\) has grown to \(C(1+s_n)\); after two years, it has grown to \(C(1+s_n)(1+s_n)\). Repeating this multiplication for \(n\) years gives:
This is the accumulation of one investment using the spot rate for its full term. It does not say that the market will offer the same rate for a new one-year investment in every future year. A one-year investment made at a later date would have a rate observed at that later date; it is not the same contract as an investment made today and held for several years.
The rate is also not the total percentage increase over the entire term. That total increase, expressed relative to the initial amount, is \((1+s_n)^n-1\). The annual effective spot rate is the annualized rate that produces this total growth over \(n\) years under annual compounding.
Reading a Schedule of Maturity-Specific Rates
When comparing rates in a schedule, first identify the valuation date and the maturity attached to each rate. Then use the rate with the matching term. A higher rate at a longer maturity does not mean that the rate rises automatically each year for an existing investment; it means the quoted annual effective rate for that entire longer-term investment is higher.
| Quotation | Investment it describes | What it does not directly describe |
|---|---|---|
| \(s_1=3.0\%\) | An investment made today and held for one year | The rate for a future one-year investment starting at year one |
| \(s_2=3.8\%\) | An investment made today and held for two years | A two-year sequence of separately reset one-year rates |
| \(s_3=4.4\%\) | An investment made today and held for three years | The change in the rate from year two to year three |
This distinction is important whenever an investment’s term is specified. Do not select a rate merely because its number of years appears somewhere in the calculation. Match the spot rate to the length of the single investment being described.
Worked Example: Accumulating the Same Principal to Different Maturities
At one valuation date, a hypothetical institution can invest \(C=\$8{,}000\) for one, two, or three years. The annual effective spot rates are \(s_1=4.0\%\), \(s_2=4.5\%\), and \(s_3=5.0\%\). Find the maturity value for each separate investment and interpret the results.
Method. Treat each term as a separate investment from the same valuation date. Apply the annual effective spot rate quoted for that investment’s full maturity.
For one year, the matching rate is \(s_1=0.04\):
For two years, the matching rate is \(s_2=0.045\). The annual accumulation factor is \(1.045\), and the two-year factor is:
Thus the two-year maturity value is:
For three years, the matching rate is \(s_3=0.05\):
The separate investments therefore mature at \(\$8{,}320\), \(\$8{,}736.20\), and \(\$9{,}261\). These amounts are consistent with the specified annual effective rates and terms. The three-year result uses 5.0% for each annual accumulation factor of that three-year investment; it does not use 4.0%, then 4.5%, then 5.0% as three successive rates.
Common mistake avoided: applying the one-year spot rate to every maturity. For instance, using \(4.0\%\) for the three-year investment would give \(\$8{,}000(1.04)^3=\$8{,}998.912\), or \(\$8{,}998.91\) to the nearest cent. That is not the amount under the stated three-year spot rate of \(5.0\%\); the correct amount is \(\$9{,}261\).
Compare Rates Without Confusing Their Terms
A longer-term spot rate can be compared with a shorter-term spot rate as a quotation, but the rates apply to different investment horizons. Their corresponding maturity values also cover different holding periods. To make a sound comparison, state both the rate and the maturity; do not interpret the difference between two spot rates as the rate earned in a particular intervening year.
Worked Example: Comparing Two Different Maturities
A hypothetical investor compares two separate \(\$15{,}000\) investments made on the same date. The two-year annual effective spot rate is \(3.8\%\), and the four-year annual effective spot rate is \(4.2\%\). Calculate each maturity value and the total percentage growth over its own term. Then explain what the comparison does and does not show.
Method. Use each spot rate for its stated full term. Calculate total percentage growth by dividing the increase in value by the original principal.
For the two-year investment:
The total increase is \(\$16{,}161.66-\$15{,}000=\$1{,}161.66\). The percentage growth over two years is:
For the four-year investment:
Therefore:
The four-year investment increases by \(\$2{,}698.27\), which is approximately \(17.9884\%\) of \(\$15{,}000\). Its annual effective spot rate is \(4.2\%\), compared with \(3.8\%\) for the two-year investment. The larger total increase also reflects the longer term. These calculations do not show that the investor earned 4.2% in years three and four after first earning 3.8% for two years.
Check. The accumulation factors exceed 1, as expected for positive rates. The four-year factor, \(1.179884462596\), produces more total growth than the two-year factor, \(1.077444\), and the computed values agree with those factors.
Common mistake avoided: comparing only dollar increases and concluding that the four-year rate is the two-year rate plus an extra rate for the remaining years. The quoted rates are annualized rates for different full terms; the four-year rate is not an incremental rate.
Using a Spot Rate to Fund a Dated Amount
A spot rate can also be used to determine the initial amount that will accumulate to a specified amount at the matching maturity. This reverses the accumulation calculation: if an initial amount \(C\) grows to a target \(L\), set \(C(1+s_n)^n=L\) and solve for \(C\). The calculation assumes the investment can be made at the stated rate for the entire term, with annual compounding and no intervening withdrawals, fees, or taxes.
Worked Example: Funding a Three-Year Liability
A hypothetical fund must have \(\$12{,}000\) available exactly three years from the valuation date. It can invest today for the full three years at an annual effective spot rate of \(5.0\%\). Find the exact initial amount required, then state the amount to invest if contributions must be in whole cents.
Method. Set the three-year accumulated value equal to the required liability and solve using the three-year spot rate. Do not round the initial amount to the nearest cent if doing so could leave the liability underfunded.
Let \(C\) be the initial investment. The three-year accumulation factor is:
The funding equation is:
The exact initial amount required is \(\$12{,}000/(1.05)^3\approx \$10{,}366.051182\). If the investment must be made in whole cents, invest \(\$10{,}366.06\); this accumulates to \(\$12{,}000.0102075\), which is sufficient.
Check. The exact unrounded amount accumulates to \(\$12{,}000\) by construction. The cent-denominated amount is slightly larger, so multiplying it by the positive factor \(1.157625\) produces a maturity value slightly above the target.
Common mistake avoided: rounding \(\$10{,}366.051182\) to the nearest cent, which would give \(\$10{,}366.05\). That amount accumulates to \(\$11{,}999.99863125\), which is less than \(\$12{,}000\). When the requirement is to ensure a minimum liability amount, round the required contribution up to the next cent if necessary.
Assumptions and Common Traps
The calculations here use annual effective spot rates for integer-year maturities. The rate for each investment is assumed to be available at the stated valuation date, and the investment is held to its specified maturity under annual compounding. If rates are quoted using a different convention or the term is not a whole number of years, the convention and time units must be addressed before applying a formula.
- Ignoring the maturity subscript. A rate without its term is ambiguous in a spot-rate schedule. Record the maturity alongside the rate.
- Reading the curve as a sequence of future rates. A two-year spot rate applies from today through year two. It does not give the rate for a one-year investment that begins one year from today.
- Using the difference between rates as an annual rate. If \(s_3=5.0\%\) and \(s_2=4.5\%\), the difference \(0.5\%\) is simply the difference between the two quoted annualized spot rates. It is not, by itself, the return in the third year.
- Confusing annualized rate with total growth. Over \(n\) years, the total growth factor is \((1+s_n)^n\), not \(1+ns_n\) under annual compounding.
- Rounding a required contribution carelessly. If a target must be met and the amount is restricted to cents, verify that the rounded contribution accumulates to at least the target.
Spot rates matter because investments and obligations have particular maturities. A maturity-specific rate lets an analyst describe the accumulation associated with a single investment horizon without pretending that all terms share one rate. In practical work, the valuation date, maturity, rate convention, and compounding assumptions must remain attached to every quoted rate.
Check Your Understanding
For each question, identify the maturity-specific rate before calculating. Unless stated otherwise, assume annual effective rates, annual compounding, and no fees or withdrawals.
- A hypothetical account invests \(\$6{,}000\) for two years at a two-year spot rate of \(4.2\%\). What is its maturity value? (Answer: \(\$6{,}514.58\))
- A \(\$10{,}000\) investment is held for three years at \(s_3=3.5\%\). Find its maturity value to the nearest cent. (Answer: \(\$11{,}087.18\))
- For a principal of \(\$4{,}000\), compare the maturity values under a two-year spot rate of \(5.0\%\) and a three-year spot rate of \(4.0\%\). Give both values to the nearest cent and state which quotation has the higher annual effective rate. (Answer: Two-year value \(\$4{,}410.00\); three-year value \(\$4{,}499.46\); the two-year quotation has the higher annual effective rate.)
- A hypothetical liability of \(\$10{,}000\) is due in two years. The two-year spot rate is \(25\%\). Find the exact amount to invest today to meet the liability, assuming annual compounding. (Answer: \(\$6{,}400\))
- A schedule gives \(s_1=2.5\%\) and \(s_3=4.0\%\). A \(\$1{,}000\) investment is made today for three years. Find its maturity value, and explain why applying \(2.5\%\) in the first year and \(4.0\%\) in each of the next two years does not follow from the schedule. (Answer: \(\$1{,}124.86\); the three-year spot rate applies to the full three-year investment, and the schedule does not specify future one-year rates.)