The problem in the image is asking to compute the expected one-year rate (forward rate) using the yields on Treasury bonds.

Given:

• One-year bond yield (YTM for 1 year) = 3.8%
• Two-year bond yield (YTM for 2 years) = 7.8%

We need to calculate the one-year rate expected to be in effect one year from now.

This can be done using the following formula for the forward rate:

$(1 + YTM_2)^2 = (1 + YTM_1) \times (1 + \text{{Expected One-Year Rate}})$

Where:

• $YTM_1$ is the yield for the one-year bond.
• $YTM_2$ is the yield for the two-year bond.
• The "Expected One-Year Rate" is what we are solving for.

Steps:

1. Convert the given percentages into decimal form: $YTM_1 = 0.038, \quad YTM_2 = 0.078$
2. Apply the formula to find the forward rate: $(1 + 0.078)^2 = (1 + 0.038) \times (1 + f)$ Simplifying: $(1.078)^2 = (1.038) \times (1 + f)$ $1.162084 = 1.038 \times (1 + f)$
3. Solve for $f$ (the expected one-year rate): $1 + f = \frac{1.162084}{1.038}$ $1 + f = 1.1196$ $f = 1.1196 - 1 = 0.1196$

So, the expected one-year rate is approximately 0.1196, or 11.96%.

If you need further clarification or additional details, feel free to ask!

Here are some related questions you might find interesting:

1. How does the yield curve shape affect the forward rate calculation?
2. Can this approach be extended to bonds of different maturities, such as 5-year and 10-year bonds?
3. What factors influence the changes in bond yields over time?
4. How do expectations of future interest rates impact current bond prices?
5. Can forward rates be used to predict economic conditions accurately?

Tip: Forward rates are not direct predictions of future interest rates, but rather reflect the market's expectations based on current bond yields.