Reading 70
MODULE 70.1: FORWARD CONTRACT VALUATION
Explain how the value and price of a forward contract are determined at initiation, during the life of the contract, and at expiration.
Consider a forward contract that is initially priced at its no-arbitrage value of $F_0(T) = S_0(1+R_f)^T$. At initiation, the value of such a forward is:
$V_0(T) = S_0 - F_0(T)(1+R_f)^{-T} = 0$
During the life of the contract, at any time $t$, the value of the forward to the buyer is:
$V_t(T) = S_t - F_0(T)(1+R_f)^{-(T-t)}$
This is simply the current spot price of the asset minus the present value of the forward contract price. This value can be realized by selling the asset short at $S_t$ and investing $F_0(T)(1+R_f)^{-(T-t)}$ in a pure discount bond at $R_f$. These transactions end any exposure to the forward; at settlement, the proceeds of the bond will cover the cost of the asset at the forward price, and the asset can be delivered to cover the short position.
At expiration (time $T$), the value of the forward to the buyer is:
$V_T(T) = S_T - F_0(T)(1+R_f)^{-(T-T)} = S_T - F_0(T)$
The long buys an asset valued at $S_T$ for the forward contract price $F_0(T)$, gaining if $S_T > F_0(T)$ and losing if $S_T < F_0(T)$. If the buyer has a gain, the seller has an equal loss, and vice versa.
In the more general case, when there are costs and benefits of holding the underlying asset, the value of a forward to the buyer at time $t < T$ is:
$V_t(T) = [S_t + PV_t(\text{costs}) - PV_t(\text{benefits})] - F_0(T)(1+R_f)^{-(T-t)}$
遠期合約的無套利定價為 $F_0(T) = S_0(1+R_f)^T$。
- 合約成立時：合約價值為零，$V_0(T) = S_0 - F_0(T)(1+R_f)^{-T} = 0$。
- 合約存續期間：在 $t$ 時點，多頭的合約價值＝當前現貨價格減去遠期價格的現值，$V_t(T) = S_t - F_0(T)(1+R_f)^{-(T-t)}$。要實現此價值，可在 $t$ 時點以 $S_t$ 賣空該資產，並以 $R_f$ 投資 $F_0(T)(1+R_f)^{-(T-t)}$ 於零息債券；到期時債券本利和正好支付遠期買價，並用於交割空頭部位。
- 到期時：$V_T = S_T - F_0(T)$。若 $S_T > F_0(T)$，多頭獲利、空頭等額損失；反之則相反。
當持有標的資產存在持有成本與收益時，多頭在 $t$ 時點的合約價值為：$V_t(T) = [S_t + PV_t(\text{成本}) - PV_t(\text{收益})] - F_0(T)(1+R_f)^{-(T-t)}$。
Explain how forward rates are determined for interest rate forward contracts and describe the uses of these forward rates.
Forward rates are yields for future periods. The rate of interest on a 1-year loan to be made two years from today is a forward rate.
The notation for forward rates must identify both the length of the loan period and how far in the future the money will be loaned (or borrowed). For example:
- 1y1y or $F_{1,1}$ — rate for a 1-year loan one year from now.
- 2y1y or $F_{2,1}$ — rate for a 1-year loan two years from now.
- 3y2y or $F_{3,2}$ — rate for a 2-year loan three years from now.
For money market rates the notation is similar: 3m6m denotes a 6-month rate three months in the future. Spot rates are zero-coupon rates; the YTM (annual compounding) on a zero-coupon bond maturing in $n$ years is denoted $Z_n$.
| Notation | Description | Period covered |
|---|---|---|
| $F_{1,1}$ | One-year forward 1-year rate | Year 1 → Year 2 |
| $F_{1,2}$ | One-year forward 2-year rate | Year 1 → Year 3 |
| $F_{1,3}$ | One-year forward 3-year rate | Year 1 → Year 4 |
| $F_{2,2}$ | Two-year forward 2-year rate | Year 2 → Year 4 |
| $F_{3,1}$ | Three-year forward 1-year rate | Year 3 → Year 4 |
An implied forward rate is the forward rate for which the following two strategies have the same yield over the total period:
- Investing from $t=0$ to the forward date, and rolling over the proceeds for the period of the forward.
- Investing from $t=0$ until the end of the forward period.
For example, lending for two years at $Z_2$ would produce the same ending value as lending for one year at $Z_1$ and, at $t=1$, lending the proceeds for one year at $F_{1,1}$. That is:
$(1+Z_2)^2 = (1+Z_1)(1+F_{1,1})$
When this condition holds, $F_{1,1}$ is the implied (no-arbitrage) forward rate.
Consider two zero-coupon bonds, one that matures in two years and one that matures in three years, with $Z_2 = 2\%$ and $Z_3 = 3\%$. Calculate the implied 1-year forward rate two years from now, $F_{2,1}$.
Answer:
Lending for three years at $Z_3$ should be equivalent to lending for two years at $Z_2$ and then for the third year at $F_{2,1}$:
$(1+Z_3)^3 = (1+Z_2)^2 \cdot (1+F_{2,1})$
Lending \$100 for two years at 2% gives $\$100(1.02)^2 = \$104.04$ at $t=2$. Lending \$100 for three years at 3% gives $\$100(1.03)^3 = \$109.27$. Therefore:
$F_{2,1} = \dfrac{109.27}{104.04} - 1 = 5.03\%$
| Strategy | Cash flow at $t=3$ (per \$100 invested at $t=0$) |
|---|---|
| Invest 3 years at $Z_3 = 3\%$ | $100(1.03)^3 = \$109.27$ |
| Invest 2 years at $Z_2$, then 1 year at $F_{2,1}$ | $100(1.02)^2(1+F_{2,1}) = \$104.04(1+F_{2,1})$ |
| No-arbitrage condition | $F_{2,1} = 5.03\%$ |
An example of an interest rate derivative is a forward rate agreement (FRA). An FRA is a method for locking in an interest rate today for a loan that will begin in the future. Typically, a borrower cannot lock in a future interest rate directly with a lender, but with an FRA they can achieve the same result.
In an FRA, a "long" counterparty (the FRA buyer) will pay a fixed rate (the forward rate in the contract) on a notional amount of principal at a future date, and a "short" counterparty (the FRA seller) will pay the MRR at that date on the same amount of notional principal. In practice, only the net amount is exchanged.
Consider a 3-month forward on a 6-month MRR ($F_{3m,6m}$) with a notional principal of \$1 million and a fixed rate of 1%. At settlement in three months, the buyer receives or pays the present value of $(\text{realized 6m MRR} - 1\%)/2 \times \$1{,}000{,}000$. If the MRR is greater than 1%, the FRA buyer receives a net payment, which offsets the higher interest cost of the loan. If the MRR is less than 1%, the FRA buyer makes a net payment, but it is offset by the lower interest cost of the loan.
| Time | Event |
|---|---|
| $t=0$ | Fixed rate (forward rate) determined = 1% |
| $t=3$ months | FRA expires; 6-month MRR discovered; payoffs computed; PV of payoff settled. Long pays/receives $(\text{MRR} - 1\%) \times \tfrac{180}{360} \times \$1\text{M}$ (PV). |
| $t=9$ months | End of underlying borrowing/lending period |
Assume that the current 3-month MRR is 1.0% and the 9-month MRR is 1.2%. Adjusting for periodicity, the no-arbitrage condition for the value of $F_{3m,6m}$ is:
$1 + 0.012\left(\dfrac{9}{12}\right) = \left[1 + 0.01\left(\dfrac{3}{12}\right)\right]\left[1 + F_{3m,6m}\left(\dfrac{6}{12}\right)\right]$
Solving for the implied annualized forward rate:
$F_{3m,6m} = \left[\dfrac{1 + 0.012(9/12)}{1 + 0.01(3/12)} - 1\right] \times \dfrac{12}{6} = 1.3\%$
FRA payoff:
Now consider the payoff to the fixed-rate payer in an $F_{3m,6m}$ FRA with notional principal of \$1 million when the 6-month MRR three months from now is 1.5%. Because the realized 6-month MRR is greater than the forward rate, the fixed-rate payer (floating-rate receiver) has a gain. The payment is the present value (discounted at the 6-month MRR) of the interest differential between two 6-month loans, one at 1.3% and one at 1.5%:
$\$1{,}000{,}000 \times \dfrac{(0.015 - 0.013)(6/12)}{1 + 0.015/2} = \$992.56$
FRAs are used primarily by financial institutions to manage the volatility of their interest-sensitive assets and liabilities. FRAs are also the building blocks of interest rate swaps over multiple periods — an FRA is equivalent to a single-period swap. Multiple-period swaps are used primarily by investors and issuers to manage interest rate risk.
遠期利率（Forward rate）是未來某期間的殖利率。例如「兩年後一年期貸款」的利率即為遠期利率。標記法須同時表明：①貸款期長度，②起算的未來時點：
- $F_{1,1}$（1y1y）：一年後的 1 年期利率。
- $F_{2,1}$（2y1y）：兩年後的 1 年期利率。
- $F_{3,2}$（3y2y）：三年後的 2 年期利率。
- 貨幣市場利率類似：3m6m 表示「3 個月後的 6 個月利率」。
$Z_n$ 表示 $n$ 年期零息債券的年化複利到期殖利率（即即期利率/spot rate）。
隱含遠期利率（Implied forward rate）使下列兩種策略殖利率相同：①先投資至遠期起算日，再展期至遠期到期日；②直接投資至遠期到期日。例如：$(1+Z_2)^2 = (1+Z_1)(1+F_{1,1})$。
例題：$Z_2 = 2\%$、$Z_3 = 3\%$，求兩年後的 1 年期遠期利率 $F_{2,1}$。$\$100(1.02)^2 = \$104.04$；$\$100(1.03)^3 = \$109.27$；故 $F_{2,1} = 109.27/104.04 - 1 = 5.03\%$。
遠期利率協議（FRA）是鎖定未來借款利率的工具。多頭（買方）支付固定利率，空頭（賣方）支付未來實際 MRR。實務上僅交換淨額。
教授提醒：貨幣市場利率以 30 日月、360 日年計算，故用簡單除法。結算發生在借款期初，但利息差別在期末才實現，故需折現至結算日。
例題：3 個月 MRR = 1.0%、9 個月 MRR = 1.2%，由無套利條件解出 $F_{3m,6m} = 1.3\%$。若三個月後實際 6 個月 MRR = 1.5%，則固定利率支付方獲利：$\$1{,}000{,}000 \times \dfrac{(0.015-0.013)(6/12)}{1+0.015/2} = \$992.56$。
FRA 主要用於金融機構管理利率敏感資產與負債的波動，並作為多期利率交換（swap）的基本單元；單期 swap 即等同於一個 FRA。
- A. the buyer.
- B. the seller.
- C. neither the buyer nor the seller.
- A. 2 years from now and ends 3 years from now.
- B. 2 years from now and ends 5 years from now.
- C. 3 years from now and ends 5 years from now.
- A. 1-year forward 1-year rate.
- B. 2-year forward 1-year rate.
- C. 2-year forward 2-year rate.
The value of a forward contract at initiation is zero.
During its life, the value to the buyer is the spot price of the asset minus the present value of the forward contract price; the value to the seller is the present value of the forward contract price minus the spot price of the asset.
At expiration, the value to the buyer is the spot price minus the forward price; the value to the seller is the forward price minus the spot price.
An implied forward rate is the forward rate for which the following two strategies have the same yield over the total period:
- Investing from $t=0$ to the forward date, and rolling over the proceeds for the period of the forward.
- Investing from $t=0$ until the end of the forward period.
In a forward rate agreement (FRA), the fixed-rate payer (long) will pay the forward rate on a notional amount of principal at a future date, and the floating-rate payer will pay a future reference rate times that same amount of principal. FRAs are used primarily by financial institutions to manage the volatility of their interest-sensitive assets and liabilities.
【LOS 70.a】遠期合約成立時價值為零；存續期間，多頭價值＝現貨價 − 遠期價現值，空頭反之；到期時，多頭價值 = $S_T - F_0(T)$，空頭反之。若標的有持有成本/收益，須在現貨價中加減其現值。
【LOS 70.b】隱含遠期利率使「先短期投資再展期」與「直接長期投資」兩種策略終值相等，由無套利條件決定。FRA 鎖定未來利率：多頭付固定利率、空頭付浮動利率（MRR），僅淨額交割（並折現至結算日）。FRA 是單期 swap，主要用於金融機構管理利率風險。