Question

Explain how a deposit of \(\$ 50,000\) for six months can be arranged to start in six months and find the rate if \(y(0,6)=6 \%\) and \(y(0,12)=7 \%\), where \(\tau=\frac{1}{12}\) In general, the initial forward rate \(f(0, M, N)\) is an interest rate such that $$ B(0, N)=B(0, M) \mathrm{e}^{-(N-M) \tau f(0, M, N)} $$ so $$ f(0, M, N)=-\frac{1}{\tau(N-M)} \ln \frac{B(0, N)}{B(0, M)}=-\frac{\ln B(0, N)-\ln B(0, M)}{\tau(N-M)} $$ Note that this rate is deterministic, since it is worked out using the present bond prices. It can be conveniently expressed in terms of the initial term structure. Insert into the above expression the bond prices as determined by the yields, \(B(0, N)=\mathrm{e}^{-\tau N y(0, N)}\) and \(B(0, M)=\mathrm{e}^{-\tau M y(0, M)}\), to get $$ f(0, M, N)=\frac{N y(0, N)-M y(0, M)}{N-M} $$

Short answer

The interest rate for the deposit starting in six months for six months is 8%.

Step-by-step solution

Identify the given variables

Given: - Deposit amount: $50,000 - Time period: 6 months - Rate for 6 months: 6% - Rate for 12 months: 7% - \( \tau = \frac{1}{12} \)

Understand the formula for the forward rate

The formula for the forward rate \( f(0, M, N) \) is: \[ f(0, M, N) = \frac{N y(0, N) - M y(0, M)}{N - M} \] where \- \( N \) and \( M \) are the time periods in years,- \( y(0, N) \) and \( y(0, M) \) are the yields for those periods.

Insert the given values into the formula

Here, M = 6 months = 0.5 years, N = 12 months = 1 year, \( y(0, 6) = 6\% = 0.06 \), and \( y(0, 12) = 7\% = 0.07 \). Substitute these values into the forward rate formula: \[ f(0, 0.5, 1) = \frac{1 \times 0.07 - 0.5 \times 0.06}{1 - 0.5} = \frac{0.07 - 0.03}{0.5} = \frac{0.04}{0.5} = 0.08 \]

Convert the forward rate to a percentage

The forward rate is 0.08, which means the interest rate is 8%.

Key concepts

Forward Rate Calculation

Forward rate calculation is a critical concept in financial engineering. It helps in determining the future interest rates for investments or loans. In simpler terms, it's the interest rate agreed upon today for a loan that will occur in the future. To explain this, consider a deposit of $50,000 for six months starting in six months from now. If the annual yields are given for the 6-month timeframe (6%) and the 12-month timeframe (7%), we use the forward rate formula to find the 6-month rate starting six months from today.
Using the formula:
\[ f(0, M, N) = \frac{N y(0, N) - M y(0, M)}{N - M} \]
where:

• M = the initial period (in years)
• N = the final period (in years)
• y(0, M) = yield for the initial period
• y(0, N) = yield for the final period

We input:

• M = 0.5 years (6 months)
• N = 1 year (12 months)
• y(0, M) = 0.06 (6%)
• y(0, N) = 0.07 (7%)

So, we calculate:
\[ f(0, 0.5, 1) = \frac{(1 \times 0.07) - (0.5 \times 0.06)}{1 - 0.5} = \frac{0.07 - 0.03}{0.5} = \frac{0.04}{0.5} = 0.08\]
This reveals an 8% forward rate.

Term Structure of Interest Rates

The term structure of interest rates, also known as the yield curve, is a vital concept that shows the relationship between the yield of bonds and their maturities. Essentially, it helps investors understand how the interest rates vary with different terms to maturity. Usually, the yield curve is upward sloping, which implies longer-term bonds have higher yields compared to short-term ones. Here's why this happens:

• Longer-term bonds require higher yields to compensate for the increased risk of holding the bond over a more extended period.
• Economic expectations can alter the shape of the yield curve, reflecting higher or lower future interest rates.

In our exercise, the yields for the 6-month and 12-month periods (represented as 6% and 7%, respectively) are used to understand the term structure. These yields tell us that the market expects a higher rate of return for the additional risk of holding a bond for one year compared to six months.

Bond Pricing

Bond pricing refers to determining the fair price of a bond based on its future cash flows, including interest and principal payments. The price of a bond is influenced by its yield, which can be understood using the term structure of interest rates. Here’s how it works:

• Bonds with similar maturities but different interest rates are generally priced differently based on their yield.
• The yield is the discount rate that equates the bond's cash flow with its current price.

For calculating forward rates, we use the present bond prices defined by these yields. The relationship is given by:
\[ B(0, N) = e^{-\tau N y(0, N)} \] and \[B(0, M) = e^{-\tau M y(0, M)}\]
where:

• B(0, N) and B(0, M) are the bond prices for periods N and M, respectively
• y(0, N) and y(0, M) are the yields for these periods

Understanding bond pricing helps clarify how differences in yields are used to derive forward rates, as shown in the example.

Yield to Maturity

Yield to maturity (YTM) is a vital concept in bond investments. It represents the total return expected on a bond if held until it matures. It’s a useful measure because it incorporates all future coupon payments and the difference between its current price and its par value. For investors, YTM helps in comparing bonds with different prices and maturities. Here’s how it works:

• YTM assumes that all coupon payments are reinvested at the same rate.
• It gives an annualized rate of return, making it easier to compare bonds.

In our exercise, the yields, 6% for 6 months and 7% for 12 months, could be seen as YTMs for respective periods. Calculating YTM ensures investors understand the anticipated return from holding a bond till maturity. Using these, the forward rate formula gives us an insight into interest rates for the period between two maturities, simplifying investment decisions.