Spot Rates and Forward Rates

A forward rate is the interest rate on a loan beginning at some time in the future.  A spot rate, on the other hand, is the interest rate on a loan beginning immediately. Therefore, the forward market rate is for future delivery after the usual settlement time in the cash market.

Forward Rates

Forward rates on bonds or money market instruments are traded in forward markets. For instance, let’s assume that in a cash market, a 4-year zero-coupon bond is priced at 85 on a par value of 100. On a semi-annual bond basis, the yield-to-maturity is 4.105%.

$$85=\frac { 100 }{ (1+r)^{ 8 } } ;\quad r=2.052%×2=4.105\%$$

The most common market practice is to name forward rates by, for instance, “2y5y”, which means “2-year into 5-year rate”. The first number refers to the length of the forward period from today while the second number refers to the tenor or time-to-maturity of the underlying bond.

Implied Forward Rates

Implied forward rates (forward yields) are calculated from spot rates. The general formula for the relationship between the two spot rates and the implied forward rate is:

$$ (1+Z_A)^A×(1+IFR_{A,B-A} )^{B-A}=(1+Z_B )^B $$

Where IFR_A,B-A is the implied forward rate between time A and time B.

Example: Computing an Implied Forward Rate

Suppose the yields-to-maturity on a 3-year and 4-year zero coupon bonds are 3.5% and 4% on a semi-annual basis. The “3y1y” implies that the forward rate could be calculated as follows:

A = 6 periods

B = 8 periods

B − A = 2 periods

z₆ = 0.035/2 =  0.0175

z₈ = 0.04/2 = 0.02

$$ (1+0.0175)^6×(1+IFR_{6,2} )^2=(1+0.02)^8$$

=> IFR_6,2 = 0.0275

The “3y1y” implies that the forward rate or forward yield is 5.50% (0.0275% × 2).

Question

Suppose the current forward curve for 1-year rates is 0y1y=2%, 1y1y=3%, and 2y1y=3.75%. The 2-year and 3-year implied spot rates are:

1. 1% and 0.75%, respectively.
2. 2.75% and 2%, respectively.
3. 2.5% and 2.91%, respectively.

Solution

The correct answer is C.

The 2-year and 3-year implied spot rates are 2.5% and 2.91%, respectively.

(1.02 × 1.03) = (1+z₂)²; z₂= 0.0250

(1.02 × 1.03 × 1.0375) = (1+z₃)³; z₃= 0.0291