The Three Year Zero Rate Is 7 Per Annum And The Four Year Ze

The Three Year Zero Rate Is 7 Per Annum And The Four Year Zero Rat

The three-year zero rate is 7% per annum, and the four-year zero rate is 7.5% per annum, both continuously compounded. We are asked to determine the one-year forward rate starting in three years’ time. This calculation involves the relationship between zero rates and forward rates, which can be expressed as:

f_t,T = (1/(T - t)) * [ln(P_t) - ln(P_T)]

where P_t and P_T are the present values of zero-coupon bonds maturing at times t and T, respectively. Given continuous compounding, these bond prices can be represented as:

P(t) = e^{-z(t) * t}

Applying this, we first compute the prices of the zero-coupon bonds for 3 and 4 years:

P(3) = e^{-0.07 * 3} ≈ e^-0.21 ≈ 0.8106

P(4) = e^{-0.075 * 4} ≈ e^-0.30 ≈ 0.7408

The one-year forward price starting in 3 years is then:

f_3,4 = - (1 / (4 - 3)) ) * [ln(P(3)) - ln(P(4))]

= - [ln(0.8106) - ln(0.7408)]

= - [ -0.2107 - ( -0.3002)]

= - (0.0895) ≈ -0.0895

Converting back to the forward rate:

f_3,4 = e^-0.0895 - 1 ≈ 0.9143 - 1 ≈ -0.0857 or about 8.57% per annum (continuously compounded). Since rates are positive, the forward rate is approximately 8.57% starting in 3 years.

Paper For Above instruction

The calculation of forward rates from zero rates is a fundamental aspect of fixed income and derivatives markets, providing insight into future interest rate expectations. In this context, the given zero rates for three and four years allow us to determine the forward rate starting in three years' time, which is crucial for investors and risk managers to hedge interest rate exposure or speculate on future movements.

Using the continuous compounding assumption, the bond prices associated with these zero rates are computed, which then facilitate the calculation of the forward rate through the relationship:

f_t,T = (1 / (T - t)) * [ln(P_t) - ln(P_T)]

This approach aligns prices of zero-coupon bonds with their respective zero rates, enabling the derivation of implied forward rates consistent with the no-arbitrage principle.

Practically, understanding this relationship helps fixed income investors to compare expected future interest rates with current market prices, informing decisions regarding bond investments, issuance, and hedging strategies. For instance, if forward rates are higher than expected future spot rates, it might indicate market expectations of rising interest rates, influencing investment flows and pricing in the bond market.

Moreover, the technique reflects the market's consensus on future rates, integrating current zero rates into forecasts. This predictive aspect of forward rate calculations underscores their importance in financial markets, where expectations form the basis of many trading strategies.

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