A risk manager at an investment bank is tasked with valuing an existing long position in an equity index forward contract for FRM Part I. The bank initially entered into 500 contracts, each with an index multiplier of $100 per index point, with a strike price of $3,200 and a maturity of two years. One year has passed since inception.
Currently, the spot equity index stands at $3,300. The continuously compounded risk-free rate for the remaining maturity is 2.5% per annum, and the continuously compounded dividend yield for the index is 1.5% per annum. The initial 1-day 99% VaR for this position was $500,000, and the counterparty's credit spread is 75 basis points.
What is the current mark-to-market value of this long forward position for the investment bank?
A.$6,493,884
B.$6,658,278
C.$8,950,414
D.$6,526,435
Rationale:
The current mark-to-market value of an existing long forward contract is determined by the difference between the current forward price for a contract with the remaining maturity and the original contract's strike price, discounted back to the present. The relevant formula for the value of a long forward contract at time t is V_t = (F_t(T) - K) * e^(-r_t * (T-t)) * N * M, where F_t(T) is the current forward price, K is the original strike price, r_t is the current risk-free rate, (T-t) is the remaining time to maturity, N is the number of contracts, and M is the multiplier. The VaR and credit spread information provided are distractors for this specific valuation task.
First, calculate the current forward price, F_t(T), using the current spot price, risk-free rate, and dividend yield for the remaining maturity:
F_t(T) = S_t * e^((r_t - q_t) * (T-t))
F_t(T) = $3,300 * e^((0.025 - 0.015) * 1)
F_t(T) = $3,300 * e^(0.01)
F_t(T) = $3,300 * 1.010050167 ≈ $3,333.16555
Next, calculate the value of the forward position:
V_t = ($3,333.16555 - $3,200) * e^(-0.025 * 1) * 500 * $100
V_t = $133.16555 * 0.9753099 * 50,000
V_t = $6,493,884.11, which rounds to $6,493,884.
The option suggesting $6,658,278 incorrectly calculates the difference between the current and original forward prices but fails to discount this difference back to the present, thereby overstating the current mark-to-market value of the position: $133.16555 * 50,000 = $6,658,278.
The option suggesting $8,950,414 makes an error by not incorporating the dividend yield in the calculation of the current forward price (i.e., using F_t(T) = S_t * e^(r_t * (T-t)) instead of including q_t), which significantly inflates the forward price and thus the contract's value: F_t(T) = $3,300 * e^(0.025) ≈ $3,383.54, giving V_t ≈ $8,950,414.
The option suggesting $6,526,435 uses an incorrect risk-free rate for discounting (e.g., if an original interest rate of 2.0% was mistakenly used for discounting, leading to a discount factor of e^(-0.02) = 0.98019867), resulting in a value of $133.16555 * 0.98019867 * 50,000 ≈ $6,526,435, failing to use the current market rate for the remaining maturity.
