 # Spot Rates and Forward Rates

This reading will establish how interest rates and prices of bonds for different maturities are related.

## Spot Rates

#### Solution

$$P\left(T\right)=\frac{1}{\left(1+S\left(T\right)\right)^T}$$

The 10% interest rate is an example of a spot rate.

\begin{align*} P\left(2\right) &=\frac{100}{\left(1+0.10\right)^2} \\ P\left(2\right) &=82.64 \end{align*}

## Forward Rates

The forward rate $$f(0, t, T)$$ is the annualized interest rate payable on a loan, which is agreed upon today, starting at time $$t$$, to be repaid at maturity $$T$$.

In this case:

• $$0$$ is the time at which the forward rate agreement is entered.
• $$t$$ is the start time of the forward rate.
• $$T$$ is the maturity of the agreement.

This can be shown in the following timeline. $$F(0,t,T)$$ is the forward price (discount factor) of a zero-coupon bond, agreed upon at time 0, filled at time $$t$$, and maturing at time $$T$$. It is expressed mathematically as:

$$F\left(0, t, T\right)=\frac{1}{\left[1+f\left(0,t,T\right)\right]^{T-t}}$$

This can also be written as:

$$F\left(t,T-t\right)=\frac{1}{\left[1+f\left(t,T-t\right)\right]^{T-t}}$$

A forward curve is a graph showing the relationship between the forward rates and the related terms to maturity. The following figure illustrates both the forward curve and the spot curve. Notice that the forward curve lies above the spot curve for an upward sloping spot curve. Conversely, the forward curve will lie below the spot curve for a downward sloping spot curve. This will be discussed later in the forward rate model.

## Yield to Maturity

The yield to maturity (YTM) is the discount rate that equates the present value of future bond payments (includes coupons and the par value) to the bond’s market price.

In other words, YTM is the expected rate of return on a bond if:

1. The bond is held to maturity.
2. The bold does not default.
3. Reinvestment of the bond and all coupons is executed at the original yield to maturity.

If the spot rate curve of a coupon-paying bond is flat, the YTM will be the same as the spot rate. Also, note that the YTM of a zero-coupon bond is equal to the spot rate.

1. 8%.
2. 9%.
3. 47%.

#### Solution

\begin{align*} P\left(T\right)&=\frac{1}{\left(1+S\left(T\right)\right)^T} \\ \\ 34,029 &=\frac{50,000}{\left(1+S_5\right)^5} \\ \\ S_5 &=8\% \end{align*}

N/B: The YTM of a zero-coupon bond is equal to the spot rate.

Reading 28: The Term Structure and Interest Rate Dynamics

LOS 28 (a) Describe relationships among spot rates, forward rates, yield to maturity, expected and realized returns on bonds, and the shape of the yield curve.

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