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Lesson 2: Forward Contracts I Part 1

Forward Contracts on Underlying with No Income, No Storage Cost (continued)

Present Value of an existing forward contract
Example 2.5. (Value of an Existing Forward Contract):

Consider the MPC's forward contract mentioned in the Overview of this lesson – MPC is long on a crude oil forward contract with contract priced at F(1/9/2012,9/1/2012) = $100/bbl since 1/9/2012. What would be the value of the MPC's long forward position on March 1, 2012 if the spot price on March 1st is $105 and the risk-free rate is 4%?

Click on Example 2.5. Solution to view the solution.

Example 2.5. Solution:

The market value of MPC's long position as of 3/1/2012 can be realized by taking a short position on 3/1/2012 leaving the net underlying = zero at the delivery time.

The forward price on March 1st for 9/1/2012 delivery is the FV of S3/1/2012, $105. Since it is 0.5 year between 3/1/2012 and 9/1/2012, we get:

  • F(3/1/2012,9/1/2012) = 105 · EXP(4% · 0.5) = 107.12
  • By selling on 3/1/2012 the forward contract (delivery 9/1/2012) at 107.12, on Sept 1. 2012 MPC will receive 107.12 from closing the short position and pay 100 from closing the long position. There will be no net delivery of the underlying as the long and short positions cancel out each other. Thus the market value of the MPC's long position, V3/1/2012(1/9/2012,9/1/2012), is the PV of $7.12, the difference between the forward price on March 1 and the forward price of the existing contract:
  • V3/1/2012(1/9/2012,9/1/2012) = PV[F(3/1/2012,9/1/2012) - F(3/1/2012,9/1/2012)]
    = (107.12 - 100) · EXP(-4% · 0.5) = $6.98
  • The market value of the forward contract made on 1/9/2012 at $100 is worth $6.98 on 3/1/2012.

Note: If you need further assistance in solving this example, view the Value of an Existing Forward Contract video by clicking on the Instructional Videos link in the left menu.

By formalizing this, the market value at t = t1 of a long forward position created earlier can be found through one of two approaches:

Click on Approach 1 Solution to view the solution.

Approach 1 Solution: Take an offseting short forward

Approach 1: Short forward at t = t1 as in the example above
  1. Short forward at t = t1: F(t1,T) = St1er(T-t1)
  2. CFs at maturity (T)
    • Close the existing long forward: pay F(0,T) & receive the underlying
    • Close the short forward: deliver the underlying & receive F(t1,T)
    • The PV of the net proceeds:
    • Vt1(0,T) = PV{F(t1,T) - F(0,T)}
                    = {(St1er(T-t1)) - F(0,T)}e-r(T-t1)
                    = St1 - F(0,T)e-r(T-t1)
      Vt1(0,T) = SpotPrice(t1) - PV (the price of the existing forward) for an underlying asset with no income and no storage cost

Click on Approach 2 Solution to view the solution.

Approach 2 Solution: Short sell the underlying

Approach 2: Short sell the underlying at t = t1
  1. Borrow the underlying
    • Sell it at the spot price of St1
  2. CFs at maturity (T)
    • Close the long forward: pay F(0,T) & receive the underlying
    • Close the short selling: return the underlying
  3. The two CFs are at two different points in time:
    • +St1 at t=t1
    • -F(0,T) at t=T
    • Thus the PV of the two CFs at t = t1 is:
      Vt1(0,T) = St1 - PV{F(0,T)} = St1 - F(0,T)e-r(T-t1)
      Vt1(0,T) = SpotPrice(t1) - PV (the price of the existing forward)
Value of an existing long forward contract on underlying with zero income:

Vt1(0,T) = PV{F(t1,T) – F(0,T)}
              = {(St1er(T-t1) ) – F(0,T)}e-r(T-t1)
              = St1 – F(0,T)e-r(T-t1)

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