FIN513:

Forward - Pricing and Valuation

Forward Contract on Investment Assets with Known Dollar Income

Note: If you need further assistance in understanding this concept, view the Forward Contract on Asset with One-time Known Income video by clicking on the Instructional Videos link in the left menu.

The Issue: If the underlying asset of a forward contract has income before the maturity of the forward contract, a synthetic forward method requires adjustments with the income. The table below compares cash flows between the forward (top half) and the synthetic forward (bottom half).

The equality of the cash flows requires:

Equation 2.1. - F(0,9) = - FV₉(S₀) + FV₉(D_3mo)

Since

FV₉(D_3mo) = PV₀(D_3mo) · (1+r_9mo)^9/12

and

FV₉(S₀) = S₀ · (1 + r_9mo)^9/12

⇒ Equation 2.1. yields:

F(0,9) = FV₉(S₀) - FV₉(D_3mo) = {S₀ - PV₀(D_3mo)} · (1 + r_9mo)^9/12

Example 2.6. (Forward Price on Assets with Known Income):

On January 15th, ABC stock is priced at $62.50 and will pay a $0.75 dividend in April 15th. The 3-month interest rate is 4%, the 9-month interest rate 6%, and the 12-month rate 7%.

1. What would be an equilibrium price of a forward contract on the stock expiring on October 15th?
2. What would be the value of a long forward contract above on April 15th, ex dividend? The stock price is $65.00, and the 3-month interest rate 3%, the 6-month rate 4%, and 9-month rate 5%.
3. On October 15th, the expiration date, the stock price is $61.50. What is the value of the forward contract?

Click on Example 2.6. Solution to view the solution.

Example 2.6. Solution:

Forward Contract on Investment Assets with Known Dollar Income (continued)

Formalize: Forward Contracts on Investment Assets with Known Dollar Income

Forward Price:

Assume I = PV₀(the income during life of forward contract)

            ⇒       F(0,T) = (S₀ – I )e^rT                          [continuous]

Proof: Consider two portfolios:

1. Long forward at forward price F(0,T):
   t = 0: no CF
   t = T: pay F(0,T) & receive the asset
2. Synthetic Forward: Buy the asset spot with borrowed money
   t = 0: CF₀ = + S₀ (loan) – S₀ (pay for stock) = 0
                     + Asset (assume no storage cost)
   t = T: CF_T = - S₀ · e^rT (←loan pmt)
                    + I · e^rT  (←FV at t=T of income)
3. Since the two portfolios should be equivalent, we get:
   - F(0,T) = - S₀ · e^rT + I · e^rT  ⇒   F(0,T) = (S₀ – I )e^rT

Valuation of an Existing Forward Position:

Present Value of a Long Forward at t

Note: The following refers to the equation below found in section i. Discrete Compunding
The term, D/(1+r) (T-t), is PV of the dividend.
The term, F(0,T)/(1+r) (T-t), is PV of F(0,T).

1. Discrete compounding:
   At t = t:   Short sell at S_t
   At t = T:  Close the existing long forward: Pay F(0,T) and receive one share
               Close the short sell: Return the share & pay div (–D)
   → V_t (0,T)   = S_t – PV{D + F(0,T)}
                      = S_t – D/(1+r) ^(T-t) – F(0,T)/(1+r) ^(T-t)
2. Continuous compounding:
        V_t (0,T)  = S_t - De^-r(T-t) – F(0,T)e ^–r(T-t)
        

Copyright 2002, CFA Institute. Reproduced and republished from Analysis of Derivatives for the CFA Program by Don M. Chance, with permission from CFA Institute. All rights reserved.

Forward Contracts on Investment Assets with Known Yield

Example 2.7. (Assets with Known Yield):

The spot price of ABC stock is $100 today. It will pay 4% dividend in two months. What should be the equilibrium forward price, F₀, maturing in 8 months?

S₀= $100 [Stock price at t = 0]
T = 8 months: Time to maturity of the forward
Dividend Dates        Dividend Rates
t = 2 mo                     q = 4%
Risk-free rate = 6%

Keep in mind that when you close the forward position in 8 months, you will have one share of stock, but you will have missed the dividend to be paid in 2 months. Furthermore, the amount of dividend is not known at t = 0. All that is known is that it will be 4% of the market price in 2 months, S_2month, whatever it will be.

Analysis:

1. Create a synthetic forward:
   
   t = 0:       Borrow $100 and buy one share
   t = 8 mo: Pay off the loan, $100 · (1 + 6%)^8/12
                  Receive one share (S_8mo + FV_8mo(Div_2mo))
   
   Difficulty: This will not help because the amount of dividend is not known - it is 4% of the stock price at t = 2 mo, which is unknown at t = 0

2. Create an alternative synthetic forward:
   
   t = 0:       Borrow $100 · (1 – 4%) and buy 96% of one share
   t = 2 mo: Buy additional stock with the dividend:
                  Dividend received: 4% on 0.96 shares = 4% · (0.96 · S_2mo)
                  Ex div stock price = S_2mo – 4% div = 0.96 · S_2mo
                  Buy additional stock with the dividend:
                  => Additional share = [4% · (0.96*S_2mo) /[0.96 · S_2mo] = 4% of a share

   Now you have (96% + 4%) =100% of a share

   T = 8 mo: Pay off the loan, $100(1 - 4%) · (1 + 6%)^8/12
                   Have one share worth S_8mo
   → F(0,8mo) = $100(1 - 4%) · (1 + 6%)^8/12 = 99.80 (= FV{S₀(1 - q)})

Note: If you need further assistance in solving this example, view the Forward Contract on Assets With One-time Known Yield video by clicking on the Instructional Videos link in the left menu.

Formalize: Forward Contracts on Investment Assets with Known Yield
(One Dividend)

How many shares of a stock do you need to buy at t = 0 such that you will end up having exactly one share of the stock at t = T, but without accumulating dividends? The stock will pay a known dividend rate of q at t  = t1 < T.

The answer to the question is buy (1 - q) shares at t = 0.

Note:

1. The dollar amount of the dividend is: D_t1 = q · S_t1.
2. The cost of obtaining a share of the stock that pays S_T at t = T is: S₀ (1 - q)
3. Therefore, F₀ = S₀ (1 - q)(1+r)^T - - - (discrete compounding) or

                            F₀ = S₀ (1 - q)e^rT - - - (continuously compounding)

Proof: If you buy one share at t = 0, you will have (S_T + div), not S_T, at the maturity, T. It is important to differentiate three different points in time: t = 0, t = t1, t = T

1. At t = 0, buy (1 - q) shares
2. At t = t1,

   ⇒
   On ex-dividend, the total number of shares =
   (1 - q) + q = 1 share. This will remain until t = T.

   • receive the dividend, qS_t1 per share on (1 - q) shares: (1 - q) D_t1 = (1 - q) q S_t1
   • Invest dividends by buying q units of the stock
     • Ex-dividend stock price = (1 - q)S_t1 [b/c Spot price declines from S_t1 after dividend by D_t1 = q S_t1]
     • No. of the stock: (1 - q) q S_t1 / [(1 - q) S_t1 ] = q shares

Formalize: Forward Contracts on Investment Assets with Known Yield
(Two Dividend Dates)

Note: If you need further assistance in understanding this concept, view the Forward on Assets With Known Yields at Two Discrete Times video by clicking on the Instructional Videos link in the left menu.

How many shares of a stock do you need to buy at t = 0 such that you will end up having exactly one share of the stock at t = T, but without accumulating dividends? The stock will pay dividend twice at known dividend rates of q₁ at t = t1 < t2, q₂ at t = t2 <T.

The answer to this question is buy (1 - q₁) (1 - q₂) shares at t = 0.

Note:

1. The dollar amounts of dividend are: D_t1 = q₁ · S_t1 and D_t2 = q₂ · S_t2 at t = t1 and t2, respectively.
2. The cost of obtaining a share of the stock that pays S_T at t = T is: S₀ (1 - q₁) (1 - q₂)
3. Therefore,  F₀ = S₀ (1 - q₁) (1 - q₂)(1+r)^T or F₀ = S₀ (1 - q₁) (1 - q₂)e^rT

Proof:

1. At t = 0, buy  (1 - q₁) · (1 - q₂) units
2. At  t = t1, receive the dividend: (1- q₁)(1- q₂) q₁S_t1
   Invest dividends: buy additional shares at the ex-div price of S_t1 (1 - q₁)/share:
   ⇒ No of shares: (1 - q₁)(1 - q₂) q₁ S_t1 / {S_t1 (1 - q₁)} = (1 -  q₂) q₁ share
   ⇒ On ex-div date, t1, the total number of shares is
       (1 - q₁) (1 - q₂) + (1 - q₂) q₁ units
3. At t = t2, receive the dividend & buy additional shares:
   ⇒ {(1 - q₁)(1 - q₂) + (1 - q₂) q₁} q₂ S_t2 / {S_t2 (1 - q₂)} =  q₂ shares
   ⇒ Then the total number of shares at t2 is:
   (1 - q₁) (1 - q₂) + (1 - q₂) q₁ + q₂ = 1 share ...Q.E.D.

Formalize: Forward Contracts on Investment Assets with Known Yield
(n Dividend Dates)

How many shares of a stock do you need to buy at t = 0 such that you will end up having exactly one share of the stock at t = T, but without accumulating dividends? The stock will pay dividend n times at known dividend rates of q₁, q₂, ...q_n at t₁ < t₂ < ...< t_n <T.

The answer to the question is buy (1 - q₁) (1 - q₂) . . . (1 - q_n) shares at t = 0.

Note:

1. The cost of obtaining a share of the stock that pays S_T at t = T is:
   S₀ (1 - q₁) (1 - q₂) . . . (1 - q_n)
2. Therefore, F₀ = S₀ (1 - q₁) (1 - q₂) . . . (1 - q_n)(1+r)^T  - - - (discrete compounding)

                           F₀ = S₀ (1 - q₁) (1 - q₂) . . . (1 - q_n)e^rT  - - - (continuously compounding)

Note: If you need further assistance in understanding this concept, view the Forward Contract on Assets with Continuous Known Yield video by clicking on the Instructional Videos link in the left menu.

Forward Contracts on Investment Assets with Known Yield
(Continuous Payment)

How many shares of a stock do you need to buy at t = 0 such that you will end up having exactly one share of the stock at t = T, but without accumulating dividends? The stock will pay dividend continuously at a rate of d per year.

The answer to this question is buy e^-δT shares at t = 0.

Note:

• The cost of obtaining a share of the stock that pays S_T at t = T is: S₀ e^-δT
• Therefore, F₀ = S₀ e^-δT e^rT = S₀ e^(r-δ)T

Proof:

Assume dividends are paid equal amount m times a year at an annual rate of δ % at an equal interval:

q₁ = q₂ = ... q_m = δ/m
Then the number of shares to purchase at t = 0 is:
(1 - q₁) (1 - q₂) ... (1 - q_m) = (1 - δ/m)^mT shares

If m becomes infinitely large, it is equivalent to the instantaneous (continuous) dividend payment:
⇒ lim_{m→ ∞} (1 - δ/m)^mT = e^-δT (remember lim_{x→ ∞} (1 + 1/x)^x = e)

δ = the instantaneous (or average) yield during the life of the contract
Thus, the cost of obtaining a share of the stock that pays S_T at t = T is:

S₀ e^-δT at t = 0

The forward price: F₀ = (S₀ e^-δT) e^rT = S₀ e^(r-δ)T at t = 0

Formalize: Forward Contracts on Investment Assets with Known Yield
(Continuous Payment - continued)

Value of a forward contract on assets with known yield

Consider an existing forward contract on an asset with known yield, δ:
T = a maturity date
F₀ = delivery price of the existing position, = S₀e^{(r-δ)(T- 0)}
What is the value of a long forward contract at t = t1? Short forward contract?

The answer to this question is:

F_t1 = forward price of the contract prevailing at t₁ > 0
      =S_t1e^{(r-δ)(T- t1)}

Long: V_t1  = (F_t1 – F₀ )e^–r(T-t1) = S_t1e^{- δ (T-t1)} – F₀ e^–r(T-t1)
Short: V_t1 = (F₀ – F_t1 )e^–r(T-t1) = F₀ e^–r(T-t1) – S_t1e ^{- δ (T-t1)}

Summary: Forward Pricing & Valuation: Investment Assets with Known Dollar Income

Table 2.3. Forward Pricing & Valuation: Investment Assets with Known Dollar Income

Notations:
T = Forward contract maturity I0 = PV0(the income during the period (0, T)) , discounted back to time 0 It = PVt(the income during the period (t, T)), discounted back to time t S0 = spot price of the underlying at t=0 St = spot price of the underlying at t > 0, but before maturity F(t,T) = price at t (any time before maturity) of a forward contract with maturity T Vt2(t1,T) = The present value as of t2 of F(t1,T), t1 ≤ t2 ≤ T
Pricing of a forward at  t = T:
F(t,T) = (St – It) · EXP(rt,T  · (T-t))       continuous compounding F(t,T) = (St – It) · (1+ rt,T)(T-t)             discrete compounding Note:  When t=0, the two equations yield: F(0,T) = (S0 – I0) · EXP(r0,T · T) F(0,T) = (S0 – I0) · (1+r0,T)T
Value of an existing long forward, F(0,T), discounted back to time t (0 < t < T):
Vt (0,T)= St - It – F(0,T)/EXP(rt,T · (T-t))     continuous compounding Vt (0,T)= St - It – F(0,T)/(1+rt,T)(T-t)           discrete compounding

Summary: Forward Pricing & Valuation: Investment Assets with Known Yields

Table 2.4. Forward Pricing & Valuation: Investment Assets with Known Yields

Investment Assets with Known Yields, discrete number of times
Dividends paid n times at known dividend rates of q1, q2, … qn  at t1 < t2 < . . . < tn < T Pricing of a forward price at t = 0: F0 = S0 (1 - q1) (1 - q2) . . . (1 - qn)erT Value at t = t1 of an existing forward long position, F0, discounted back to time t (0 < t < T): Vt1  = (Ft1 – F0 )e–r(T-t1) where Ft1 = St1Πi(1 - qi) where  Πi  (Xi) = Xi · Xi+1 · Xi+2 · . . . · XT · for all i such that t1 < i < T
Investment Assets with a Known Yield, continuous time
Dividends paid continuously at an annual rate of δ (delta) Pricing of a forward price at t = 0: F0 = S0 e(r – δ)T Value at t = t1 of an existing forward long position, F0, discounted back to time t (0 < t < T): Vt1  = (Ft1 – F0 )e–r(T-t1) = St1e- δ (T-t1) – F0 e–r(T-t1)

OTC Derivative Statistics

View the following over-the-counter (OTC) derivatives from the Bank for International Settlements. Notice the relative size of notional amount and gross market value for each OTC derivative category.

Source: Bank for International Settlements ("BIS"). (2011). Semiannual OTC derivatives statistics at end-June 2011. Retrieved from http://www.bis.org/statistics/derstats.htm