1. Forward pricing Assets with no cash flows
Assets with known discrete cash flows
Assets with continuous cash flows (index)
Finance Spring 2004
Associate Professor Steven C. Mann
The Neeley School of Business at TCU
S. Mann, 2004

2. Forward price of a stock
S(0) = $ Stock pays no dividends
is(6 month) = 7.12% ( T = 1/2 )
if you borrow $25 today you repay $25[ (1/2)] = $25.89
f = six month forward price of stock
Consider strategy:
now T=6 months later
borrow $ $25.89
buy stock S(T)
sell 6-month forward [ S(T) - f ]
total f
= f - S(0)[ 1 + is T]
Arbitrage-free forward pricing:
f (0,T) = S(0)(1 + isT) = $25.89

3. Cash and Carry forward pricing
Forward contract with delivery date T; spot asset with no cash flows
“Cash and carry” strategy:
now at date T
buy asset at cost S(0) - S(0) S(T)
borrow asset cost + S(0) -S(0)(1+isT)
sell forward at f(0,T) [ S(T) -f(0,T)]
Total f(0,T) - S(0)(1+isT)
f(0,T) = S(0)( 1 + isT)
f(0,T) [ 1/(1+isT)] = S(0)
S(0) = f(0,T)B(0,T)
S(t) = f(t,T) B(t,T)
Forward price is “future value”
of spot price
Spot price is “present value”
of the forward price

4. Forward valuation (post-initiation and “off-market”)
Value = V[forward price, time]
at initiation, value is zero: V[ f(0,T), 0 ]  0.
At maturity: V[ f(0,T), T ] = S(T) - f(0,T)
at some time t (post initiation): V[ f(0,T), t ] = ?
1) Valuation by offset:
At time t: long f(0,T). Sell f(t,T)
value at T: S(T) - f(0,T) - [ S(T) - f(t,T) ]
total value at T: f(t,T) - f(0,T)
value at t: B(t,T)[ f(t,T) - f(0,T) ]
2) Valuation by algebra:
V[f(0,T),t] = PV( V[f(0,T),T]) = PV[ S(T) - f(0,T)]
note S(t) = PV[S(T)]; and S(t) = f(t,T)B(t,T) (prior page)
so V(f(0,T),t] = f(t,T)B(t,T) - f(0,T)B(t,T)
=B(t,T) [ f(t,T) - f(0,T)] = PV(price difference)

5. Example: forward pricing and valuation
Non-dividend paying asset; S(0) = $65
contract maturity is 90 days
simple interest rate is 4.50% ; daycount is actual/365.
a) find forward price
f(0,90/365) = S(0)(1+is(90/365)) = $65( (90/365) ) = $65.72
value of contract is zero.
b) You are asked to value a 90-day forward on this asset with
delivery price = $60. This is an “off-market’ forward: value is nonzero.
Value of long forward with off-market delivery price:
value = PV( difference in forward prices)
= B(0,T)[ market forward price - contract forward price]
= B(0,T) [ ]
= ( (90/365)) -1 [$ 5.72]
=( )($5.72) = $5.66

6. Example: forward pricing and valuation
S(0) = $ Non-dividend paying asset
contract maturity is 100 days
simple interest rate is 4.75% ; daycount is actual/365.
a) find current forward price f(0,100/365)
f(0,100/365) = $45.00( (100/365)) = $45.59
b) You are long 100 day forward to buy asset at $50.25.
If you sell a 100 day forward at current price, what is payoff at T?
At maturity: long: S(T) - $50.25
short -[ S(T) - $45.59]
net: $ $50.25 = $4.66.
c) what is the present value of your net position?
PV = B(t,T)[ f(t,T) - f(0,T)]
= [1/( (100/365)](-4.66) = (0.9872)(-4.66) = -$4.60

7. Assets with known cash flows
Example:
12 month T-note
par = $1000.
Coupon=10% semi-annual
Zero-coupon yield curve
(simple interest)
month T is(T) B(0,T)
6 1/2 7.18%
9 9/ % %
$ $1050
Spot bond price Bc(0,12) = $50.00 B(0,6) + $ B(0,12)
= $50.00(.9653) + $ (.9268)
= $ $
= $
Forward contract does not receive coupon at T=6 months

8. Forward pricing: assets with known cash flows
Strategy 1: cost now t1 =6 months at T=9 months
a) buy bond Bc(9,12)
b) borrow PV(coupon) B(0,6)
total Bc(9,12)
[S(0) - d(t1)B(0,t1)]
Strategy 2: cost now t1= 6 months at T=9 months
a) enter long forward Bc(9,12) - f (0,9)
b) lend PV( f (0,9)) f (0,9) B(0,9) f (0,9)
(buy bill )
total f (0,9) B(0,9) Bc(9,12)
Each strategy has same payoff: must have same cost to avoid arbitrage
f (0,9) B(0,9) = $ = $973.12
f (0,9) = $ / = $
in general: f (0,T)B(0,T) = S(0) - d(t1)B(0,t1)

9. General forward pricing for assets with known cash flows
Strategy 1: cost now at t1 at T
a) buy asset S(0) + d(t1) S(T)
b) borrow PV(d(t1)) - d(t1) B(0,t1) - d(t1)
total S(0) - d(t1)B(0,t1) S(T)
Strategy 2: cost now at t at T
a) enter long forward S(T) - f (0,T)
b) lend PV( f (0,T)) f (0,T) B(0,T) f (0,T)
(buy bill )
total f (0,T) B(0,T) S(T)
Each strategy has same payoff: must have same cost to avoid arbitrage
in general: f (0,T)B(0,T) = S(0) - d(t1)B(0,t1)
for N known flows:
f (0,T)B(0,T) = S(0) - S d(ti)B(0,ti) N
i=1

10. Example: forward pricing - asset pays dividends
S(0) = $ Bill prices:
stock pays dividends: $1.50 in 1 month 1 month:
$2.00 in 7 months 7 month :
12 month:
Find price of one-year forward contract written on stock.
Use: f (0,T) B(0,T) = S(0) - PV(dividends)
f (0,12) B(0,12) = S(0) - d(1)B(0,1) - d(7)B(0,7)
f (0,12) (0.9512) = $ $1.50(0.9967) - $2.00(0.9741)
f (0,12) = $59.93/(0.9512)
f (0,12) = $63.01

11. Assets with continuous payouts (index, currency)
for N known flows:
f (0,T)B(0,T) = S(0) - S d(ti)B(0,ti)
= S(0) - PV(cash payout to time T)
if asset pays continuous yield
then PV(dividends to time T) = S(0)[ 1 - exp(-dyT)]
so that f (0,T)B(0,T) = S(0) - [ S(0) [ 1 - exp(-dyT)]]
f (0,T)B(0,T) = S(0) exp(-dyT)
write B(0,T) as continuous discount factor: B(0,T) = exp(-rT)
then f (0,T) = S(0) exp ( (r-dy)T) N
i=1

12. Example: Index forward pricing
S&P500 Index =
div yield = 2.50% continuous (365 day year)
a) Given 95-day discount rate =5.75% (360 day year), find f (0,95 days)
B(0,95) = (95/360) =
exp(-dyT) = exp(-0.025(95/365)) =
f (0,95)B(0,95) = S(0)exp(-dyT)
f (0,95) = ( ) / ( ) =
b) One day later index is at day discount is 5.75%.
What is the value of the contract in part (a)?
B(0,94) = (94/360) =
exp(-dyT) = exp(-0.025(94/365)) =
f (0,94) = ( )/( ) =
value of prior contract = B(0,94) ( )
= (-2.17) = -2.04

13. Commodity Forwards: Storage cost
Define: G = cost of storing asset for (0,T) (per unit); paid time 0.
“Cost of carry” strategy:
cost now value at date T
buy asset at cost S(0), pay storage S(0) + G S(T)
borrow asset cost and storage cost [S(0) + G] -[S(0)+G](1+isT)
sell forward at f(0,T) [ S(T) -f(0,T)]
Total f(0,T) - [S(0) +G](1+isT)
f (0,T)B(0,T) = S(0) + G
Example: 180-day Gold forward (100 troy oz.) Spot Gold S(0) = $368 / oz.
cost of storage for 180 days = 2.25 / oz., paid at time 0.
180 days simple interest rate = 3.875% annualized (actual/actual).
f (0,180) = ( )( (180/365)) = ( ) = $ / oz.
If storage cost G is defined to be paid at T, then f(0,T)B(0,T) = S(0) + G B(0,T)
so f (0,180) = 368( (180/365)) = $ / oz.

14. Commodity Forwards: Convenience yield
Define: Y(0,T) = present value (time 0) of benefits provided by holding asset.
“Cost of carry” strategy: cost now value at date T
buy asset at cost S(0), pay storage S(0) + G S(T)
borrow asset cost and storage cost [S(0) + G] -[S(0)+G](1+isT)
receive convenience yield Y(0,T)(1+isT)
sell forward at f(0,T) [ S(T) -f(0,T)]
Total f(0,T) + [S(0) + G -Y(0,T)](1+isT)
f (0,T)B(0,T) = S(0) + G - Y(0,T).
Define yn = net convenience yield = Y-G ; where Y and G are continuously compounded rates
f (0,T) = S(0) exp[(r-yn)T]
Example: 180-day Gold forward (100 troy oz.) Spot Gold S(0) = $368 / oz.
cost of storage is 0.25% as continuous annual rate.
Gold lease rate (convenience yield) is 1.50% annual continuously compounded.
Yn = = ( 125 basis points)
180 day continuously compounded rate is 4.0%
f (0,180) = 368 exp [ ( (0.0125))(180/365)] = $368 exp[ ] = 368( ) = $373.02

15. Implied Repo rates Define: iI as implied repo rate (simple interest).
iI is defined by:
f (0,T) = S(0)(1+ iI T) ; f (0,T) and S(0) are current market prices.
Example: asset with no dividend, storage cost, or convenience yield (example: T-bill)
Given simple interest rate is, the theoretical forward price (model price) is:
f (0,T) = S(0)(1+isT).
If iI > is market price > model price.
iI < is market price < model price.
If iI > is : arbitrage strategy :
cost now value at date T
buy asset at cost S(0) S(0) S(T)
borrow asset cost -S(0) -S(0)(1+isT)
sell forward at f(0,T) 0 -[S(T) -f(0,T)] =
Total f(0,T) - S(0)(1+isT) = S(0)(1+iIT) - S(0) )(1+isT)
= S(0)(iI - is)T > 0

16. Forwards compared to futures
is(1) is(1,2)
Forwards compared to futures
h1 h12
is(2) h2
Forward price f(0,2) f(1,2) S(2)
forward cash flow S(2) - f(0,2)
forward contract: total time 2 cash flow = S(2) - f(0,2)
futures price F(0,2) F(1,2) S(2)
futures cash flow F(1,2) - F(0,2) S(2) - F(1,2)
futures contract: total time 2 cash flow:
= S(2) - F(1,2) + [F(1,2)-F(0,2)](1+is(1,2)h12)
= S(2) - F(1,2) + F(1,2) - F(0,2) (F(1,2) -F(0,2))(1+is(1,2)h12)
= S(2) - F(0,2) [ F(1,2)-F(0,2)](1+is(1,2)h12)
Position:
buy futures, sell forward cost today = 0: total time 2 cash flow:
= f(0,2) - F(0,2) + [F(1,2) - F(0,2)](1+is(1,2)h12)
if f(0,2) = F(0,2) then expected margin account earnings are zero.
S. Mann, 2004