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Interpret how spot, forward and par rates relate to one another.

FRM® Part I · Valuation and Risk Models · Interest Rates · FRM1.VR.10.e

The idea

Spot, forward and par rates are three views of the same curve and are linked through discount factors. An n-year spot rate is a geometric average of the one-period forward rates up to n, which in continuous compounding is an exact arithmetic average. The par rate is a discount-factor weighted average of the one-period forward rates up to n, so it lies close to but not exactly at the spot rate. When the curve slopes up, the forward rate for the final period is above the n-year spot rate, which in turn is above the n-year par rate, because the forward is the marginal rate and spot and par are averages. When the curve is inverted the order reverses. On a flat curve all three coincide. Exam questions ask you to name the ordering or to compute one rate from another.

Formulas

Spot rate as a geometric average of forward rates

(1 + sn)ⁿ = (1 + f1) × (1 + f2) × … × (1 + fn)

sn
the n-year spot rate
n
the number of years
f1
the one-year rate for year one
f2
the one-year forward rate for year two
fn
the one-year forward rate for year n

Annual compounding: take the nth root to find sn. Averaging the forwards arithmetically is exact only under continuous compounding.

A practice question

From the question bank, with the reason each wrong option is wrong.

On an upward-sloping curve, which ordering of rates for the same final maturity is correct?

  1. Par rate > spot rate > forward rate for the final period.Why not: Reverses the order: the par rate is a weighted average of shorter, lower spot rates, so it is the lowest.
  2. Forward rate for the final period > spot rate > par rate. - correct
  3. Spot rate > par rate > forward rate for the final period.Why not: Places the forward rate lowest, but it is the marginal rate and is above the average spot rate.
  4. All three are equal, because they come from the same curve.Why not: Equality holds only on a flat curve.

On an upward curve the final-period forward rate is highest because it is the marginal rate. The spot rate is a geometric average of the forwards and the par rate is a discount-factor weighted average of the forwards, which gives more weight to the earlier, lower forwards, so spot > par.

A harder one

The 1-year spot rate is 2%, the 1-year rate one year forward is 8% and the 1-year rate two years forward is 20%, all annually compounded. The 3-year spot rate is closest to:

  1. 9.75% - correct
  2. 10.00%Why not: Takes the arithmetic mean of the forward rates, which overstates the geometric average.
  3. 20.00%Why not: Uses the last forward rate as the 3-year spot rate.
  4. 32.19%Why not: Stops at the total 3-year growth and does not take the third root.

(1 + s3)³ = 1.02 x 1.08 x 1.20 = 1.3219, so s3 = 1.3219^(1/3) - 1 = 9.75%.

The hardest tier

Continuously compounded spot rates are 3.0% for 2 years and 4.0% for 3 years. A bank offers you a forward loan for year 3 (from year 2 to year 3) at a fixed 5.5% continuously compounded. Which action exploits this offer?

  1. Take the forward loan at 5.5% and lend synthetically for year 3 by buying the 3-year zero and shorting the 2-year zero, earning an implied 6.0%. - correct
  2. Decline the loan and borrow synthetically for year 3 by shorting the 3-year zero and buying the 2-year zero.Why not: Reverses the replication: this borrows forward at 6.0%, which is more expensive than the 5.5% offered.
  3. Take the forward loan at 5.5% and buy both the 2-year zero and the 3-year zero.Why not: Buying both zeros does not create a year-3 forward deposit, so it leaves the rate risk unhedged.
  4. No arbitrage exists, because 5.5% is above the 3-year spot rate of 4.0%.Why not: Compares the offer with the spot rate; the right benchmark is the implied forward rate for year 3.

The implied year-3 forward rate is (4.0% x 3 - 3.0% x 2) / 1 = 6.0%. Borrowing forward at 5.5% while a long 3-year and short 2-year zero position earns 6.0% for the same period gives a risk-free margin of 0.5%.

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