### Spot Rates and Forward Rates

A forward rate indicates the interest rate on a loan beginning at some time in the future, whereas a spot rate is the interest rate on a loan beginning immediately. Thus, 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 semiannual 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 and 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 IFRA,B-A is the implied forward rate between time A and time B.

### Example of Computing an Implied Forward Rate

Suppose that 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

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

=> IFR6,2 = 0.0275

The “3y1y” implies 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, respectively:

A. 2.5%; 2.91%

B. 1%; 0.75%

C. 2.75%; 2%

Solution

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

Define forward rates and calculate spot rates from forward rates, forward rates from spot rates, and the price of a bond using forward rates

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