Question

Let \(\tau=\frac{1}{12}\). Invest \(\$ 100\) in six-month zero-coupon bonds trading at \(B(0,6)=0.9400\) dollars. After six months reinvest the proceeds in bonds of the same kind, now trading at \(B(6,12)=0.9368\) dollars. Find the implied interest rates and compute the number of bonds held at each time. Compute the logarithmic return on the investment over one year.

Short answer

Initial bonds: 106.3829. Reinvested bonds: 113.4046. Logarithmic return: 12.6%.

Step-by-step solution

Determine the initial number of bonds purchased

Invest \( \$ 100 \) in six-month zero-coupon bonds trading at \( B(0,6)=0.9400 \). The number of bonds purchased initially is \( \frac{100}{0.9400} \).

Calculate the proceeds after six months

Each bond will mature to \( \$1 \) after six months. Therefore, the total proceeds from the matured bonds are \( \frac{100}{0.9400} \) dollars.

Determine the number of reinvested bonds

Reinvest the proceeds into bonds maturing in another six months, now trading at \( B(6,12)=0.9368 \). The number of bonds purchased is \( \frac{ \frac{100}{0.9400} }{0.9368} \).

Compute the implied interest rates

The implied interest rate for the first six months is calculated using the bond price: \[(1+i_1\tau) = \frac{1}{0.9400}\] \[(i_1\tau) = \frac{1}{0.9400} - 1\] \[(i_1) = 12 \times ( \frac{1}{0.9400} - 1 )\] \[i_1 = 7.9787\%\]. Repeat the calculation for the second six-month period: \[(1+i_2\tau) = \frac{1}{0.9368}\] \[(i_2\tau) = \frac{1}{0.9368} - 1\] \[(i_2) = 12 \times ( \frac{1}{0.9368} - 1 )\] \[i_2 = 8.0043\% \].

Compute the logarithmic return over one year

The overall return on the investment over one year can be computed as: \[ R_{log} = \ln( \frac{\frac{100}{0.9400}}{100} \times \frac{1}{0.9368} )\] \[= \ln( \frac{1}{0.9400} \times \frac{1}{0.9368} ) \]. Simplifying gives: \[ R_{log} = \ln( 1.06383 \times 1.06747 ) = \ln(1.1340)\]. The result is approximately \[0.126\] or \[12.6\%\].

Key concepts

Implied Interest Rate

The implied interest rate helps us understand the return on an investment over a specific period. For zero-coupon bonds, it's the rate that equates the bond's price to its face value when discounted back to the present.

In our exercise, we first determine this rate for six-month periods.
Given the bond price at initial investment, the formula: \ \( (1+i_1\tau) = \frac{1}{0.9400} \) becomes key.
Simplifying this, we find: \ \( i_1 = 12 \times ( \frac{1}{0.9400} - 1 ) = 7.9787\% \).
We repeat this for the next period: \ \( (1+i_2\tau) = \frac{1}{0.9368} \)
So, \ \( i_2 = 12 \times ( \frac{1}{0.9368} - 1 ) = 8.0043\% \).

Both are annualized based on six months, giving us the implied returns on our zero-coupon bond investments.

Logarithmic Return

Logarithmic return is a key concept in finance because it captures the compounded returns of an investment over time in a more stable way than simple return.

Here, we use it to determine the total return over one year.
Using the proceeds from bond investments, the formula for our annual logarithmic return is:

\ \( R_{log} = \ln( \frac{\frac{100}{0.9400}}{100} \times \frac{1}{0.9368} ) \)

which simplifies to: \ \( R_{log} = \ln( \frac{1}{0.9400} \times \frac{1}{0.9368} ) \)
Simplifying this further, we find: \ \( R_{log} = \ln( 1.06383 \times 1.06747 ) = \ln(1.1340) \).
The result is approximately \ \( 0.126 \) or \ \( 12.6\% \).

This compactly expresses how investments grew over the year.

Financial Engineering

Financial engineering involves creating new financial products or strategies to improve returns and manage risks. It uses mathematical and computational tools to solve financial problems.

In our exercise, financial engineering is applied by reinvesting proceeds from maturing zero-coupon bonds into new ones.

This step involves:

• Purchasing initial bonds at a given price
• Reinvesting proceeds into new bonds at a future price
• Calculating implied interest rates at each step
• Determining overall returns using advanced metrics like logarithmic return

This strategic reinvestment aims at maximizing returns over a defined investment period, illustrating core principles of financial engineering.

Bond Pricing

Bond pricing is essential for understanding the value of bonds, especially zero-coupon bonds, which don't make periodic interest payments.

The bond price reflects the present value of its future maturity value. In our case:

- Initial price: \ \( B(0,6) = 0.9400 \)
- After six months: \ \( B(6,12) = 0.9368 \)
The price tells us how much we need to invest now to receive a set amount in the future.

Investments in bonds rely heavily on these prices for determining returns.

To derive returns, we consider the relation: \ \( 1 + i\tau = \frac{1}{Bond Price} \).
By understanding bond prices, investors can make informed decisions about when and where to invest their funds to meet financial goals.