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VIPSolutions · Accepted Answer

## Problem's Objective

**a.** Compute the expected price of the four-year zero-coupon bond at the end of each year 1-4, assuming the expectations hypothesis is valid.  
**b.** Find the rate of return for the bond in years 1, 2, 3, and 4. Show that the expected return equals the forward rate for each year.

---

## Given Data

| Beginning of Year | Price of Bond ($) |
|:----------------:|:----------------:|
|        1         |      930.00      |
|        2         |      902.97      |
|        3         |      829.62      |
|        4         |      772.99      |

---

# Required A: Compute the Expected Price at the End of Each Year

### Formula (Zero-Coupon Bond Price):

$$
P_t = \frac{F}{(1+y_t)^n}
$$

But here we will use the prices as given and calculate the expected price at the end of each year by assuming the bond "rolls down the yield curve", i.e., its price at the end of the year equals the market price for a bond with one less year to maturity.

#### Step 1: End of Year 1
- **Expected Price:** Price at Beginning of Year 2 = **\$902.97**

#### Step 2: End of Year 2
- **Expected Price:** Price at Beginning of Year 3 = **\$829.62**

#### Step 3: End of Year 3
- **Expected Price:** Price at Beginning of Year 4 = **\$772.99**

#### Step 4: End of Year 4
- **Expected Price:** Face Value (Assume \$1,000 if not stated, but since only prices are asked, use the maturity price which is generally \$1,000 for a zero-coupon bond.)

---

### **Summary Table**

| Year-End | Expected Price (\$) |
|:--------:|:------------------:|
|    1     |      902.97        |
|    2     |      829.62        |
|    3     |      772.99        |
|    4     |     1,000.00*      |

\*Assumed as standard face value, unless otherwise specified.

---

# Required B: Calculate the Rate of Return for Each Year

## Formula for Rate of Return:

$$
\text{Rate of Return} = \frac{\text{Ending Price} - \text{Beginning Price}}{\text{Beginning Price}}
$$

### **Year 1:**
$$
\text{Rate of Return}_1 = \frac{902.97 - 930.00}{930.00}
$$
$$
= \frac{-27.03}{930.00}
$$
$$
= -.02907 \text{ or } -2.91\%
$$

---

### **Year 2:**
$$
\text{Rate of Return}_2 = \frac{829.62 - 902.97}{902.97}
$$
$$
= \frac{-73.35}{902.97}
$$
$$
= -.08124 \text{ or } -8.12\%
$$

---

### **Year 3:**
$$
\text{Rate of Return}_3 = \frac{772.99 - 829.62}{829.62}
$$
$$
= \frac{-56.63}{829.62}
$$
$$
= -.06828 \text{ or } -6.83\%
$$

---

### **Year 4:**
$$
\text{Rate of Return}_4 = \frac{100.00 - 772.99}{772.99}
$$
$$
= \frac{227.01}{772.99}
$$
$$
= .2938 \text{ or } 29.38\%
$$

---

## **Conclusion: Expected Return and Forward Rate**

According to the expectations hypothesis, the expected return on the bond each year equals the forward rate for each year. Our calculations above reflect the annual returns if the bond is held and "rolled down" the yield curve.

---

# Final Answer

- **Expected bond prices at the end of each year**:  
  - End of Year 1: \$902.97  
  - End of Year 2: \$829.62  
  - End of Year 3: \$772.99  
  - End of Year 4: \$1,000.00 (assumed face value)
- **Annual rates of return**:  
  - Year 1: -2.91%  
  - Year 2: -8.12%  
  - Year 3: -6.83%  
  - Year 4: 29.38%

**Crux:** The expected return for each year matches the forward rate implied by the sequence of bond prices under the expectations hypothesis.

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