The forward rate f(t1,t2) of a bond, is the implicit interest rate in a future period between time t1 and t2. For example, assuming continuous time returns, if the discount rate from period 0 to t1 is: exp(-r t1), and from period 0 to t2 (greater than t1) is: exp(-r t2), then, the forward rate f from t1 to t2 maintains the following no arbitrage relationship: exp(-r t1) exp(-f (t2-t1)) = exp(-r t2). Suppose we observe the prices of a 6-year zero-coupon bond (with a face value of $93.7), where P(t1,t2) means the price of the bond between t1 and t2, and a year 2-to-6 forward rate as follows:P(0,6) = $93.1003200000 and f(2,6) = 0.1201716274%. Calculate the price of a 2-year zero-coupon, with face value $96.37:

We are given the following information:

- The price of a 6-year zero-coupon bond, $ P(0, 6) = 93.1003200000 $
- The forward rate from year 2 to 6, $ f(2, 6) = 0.1201716274\% = 0.001201716274 $ (in decimal form)
- The face value of the 2-year bond, $ \text{Face value of 2-year bond} = 96.37 $

We are asked to find the price of a 2-year zero-coupon bond, $ P(0, 2) $, which represents the price today (at time 0) of a bond that matures in 2 years with a face value of $96.37.

### Step 1: Using the forward rate formula
We know from the forward rate no-arbitrage relationship:

\[
P(0, t_2) = P(0, t_1) \times e^{-f(t_1, t_2)(t_2 - t_1)}
\]

Where:

\[
P(0, 6) = P(0, 2) \times e^{-f(2, 6) \times (6 - 2)}
\]

Rearranging this to solve for $ P(0, 2) $:

\[
P(0, 2) = \frac{P(0, 6)}{e^{-f(2, 6) \times (6 - 2)}}
\]

### Step 2: Plug in the values
Now we substitute the given values into the equation:

\[
P(0, 2) = \frac{93.1003200000}{e^{-0.001201716274 \times (6 - 2)}}
\]

Let me calculate that.The price of the 2-year zero-coupon bond, $ P(0, 2) $, is approximately \$93.55.

Would you like further details or have any questions?

Here are five related questions to expand on this topic:
1. How does the continuous compounding assumption affect bond pricing?
2. What would happen to the bond prices if the forward rate increased?
3. Can you explain the difference between forward rates and spot rates?
4. How is the yield curve related to forward rates in bond markets?
5. How does the no-arbitrage condition ensure consistency in bond pricing?

**Tip:** Forward rates are often used to anticipate market expectations for future interest rates, making them useful for investment strategies and risk management.

The forward rate f(t1,t2) of a bond is the implicit interest rate in a future period between time t1 and t2. For example, assuming continuous time returns, if the discount rate from period 0 to t1 is: exp(-r t1), and from period 0 to t2 (greater than t1) is: exp(-r t2), then, the forward rate f from t1 to t2 maintains the following no arbitrage relationship: exp(-r t1) exp(-f (t2-t1)) = exp(-r t2). Suppose we observe the prices of a 6-year zero-coupon bond (with a face value of $93.7), where P(t1,t2) means the price of the bond between t1 and t2, and a year 2-to-6 forward rate as follows: P(0,6) = $93.1003200000 and f(2,6) = 0.1201716274%. Calculate the price of a 2-year zero-coupon bond, with face value $96.37.

Learn how to calculate the price of a 2-year zero-coupon bond with a face value of $96.37 using forward rates and no-arbitrage principles. This finance problem covers forward rate calculation, continuous compounding, and bond pricing with detailed step-by-step solutions. Suitable for undergraduate students studying finance or economics.

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