A spot rate, r(T), is the interest rate today for a specified period of time T.

Consider two zero coupon bonds. Bond A is a 1-year bond and bond B is a 2-year bond. Both have face value of $1,000. Assume annual compounding. The 1-year interest rate, r(1), is 8%. The 2-year interest rate, r(2) is 10%.

The present value for bond A and B:

• PV_A=$1,000/1.08¹ = $925.93
• PV_B=$1,000/1.10² = $826.45

P(T) = 1/(1+r(T) )^T is called the discount factor. In our example, P(1) = 0.92593, and P(2) = 0.82645

A forward rate, f(T*, T), is the interest rate beginning some time in the future, T*, for a period of time T.

Let's say an investor wants to invest $1,000 for 2 years. He is faced with 2 choices:

• Directly invest in the 2-year bond. The value of his investment after 2 years will be $1,000 x (1 + 0.1)² = $1,210
• Invest in the 1-year bond, and again invest the proceeds after 1 year in a 1-year bond.

The returns from the 2 choices should be the same. The value of the second choice after 2 years should be 1,000 x (1 + 0.08) x (1 + F(1,1)) = 1,210 so F(1,1) = 12.04%

F(1,1) is the 1-year forward rate starting 1 year from now.

Given spot rates for maturities of T* and T* + T, we can compute the forward rate starting at T* for T:

[1+ r(T*+T)]^(T*+T)=[1+r(T*)]^T* [1+f(T*,T)]^T This is called the forward rate model.

Example

Ed is considering a bond purchase but cannot decide if he wants to hold the bond for 10 years or 12 years. The spot rate is 5% for 10 years, and 6% for 12 years. What is the forward rate starting 10 years from now, for 2 years?

(1 + r(10 + 2)]^{10 + 2} = [1 + r(10)]¹⁰ [1 + f(10, 2)]²
f(10,2) = 11.14%

Similarly, we can calculate spot rates from forward rates. The spot rate is the geometric mean of the forward rates.

[1+r(T)]^T = [1+r(1)] [1+f(1,1)] [1+f(2,1)]...[1+f(T-1,1)] You can think of the spot rate for a given maturity as a sort of average of the forward rates for all maturities up to and including the spot rate's maturity. If the spot rate is increasing, so spot rate n is greater than spot rate n - 1, then forward rate must be above spot curve. If the spot curve is downward sloping, the forward curve will lie below the spot curve.

The yield-to-maturity measures an investor's return from the bond correctly only if these assumptions are true:

• The bond is held to maturity. The issuer does not default: All coupon and principal payments are made in full when due.
• The coupon reinvestment rate is the same as the YTM.
• A flat yield curve.

For example, consider a 10%, 2-year bond selling for $1,036.30 (selling at premium). The cash flows for this bond are (1) 4 payments every six months of $50, and (2) a payment 2 years from now of $1,000.

$1036.30 = $50/(1 + YTM/2)¹ + $50/(1 + YTM/2)² + $50/(1 + YTM/2)³ + $1050/(1 + YTM/2)⁴. By trial and error or using a financial calculator, YTM is found to be 8%.

YTM can be viewed as a weighted average of the spot rates applying to the bond's cash flows.

YTM measures an investor's return from the bond correctly only if these assumptions are true. However, in reality the YTM of a bond may not be realized because:

• Future interest rates may be less than the YTM. This is known as reinvestment risk.
• The bond cannot be held to maturity, and may have to be sold for less than the purchase price because the interest rate required by the market is higher than the YTM. That is, there is interest rate risk.

The yield computed at the time of purchase (i.e., YTM) considers all three sources of return, including, coupon payments, capital gain/loss, and reinvestment income. The amount of actual reinvestment income affects the return the investor can realize from the bond. It depends on the level of reinvestment rate. If the reinvestment rate is less than YTM, the realized return will be less than the expected level of return from the bond.

The realized return is the actual return on the bond during the time an investor holds the bond.