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Forwards and swaps interest rates
- 1. Forwards and Swaps: Interest Rates
- 2. 2 Learning Objec-ves ¨ Understand and manage interest rate risk via forward and swap agreements ¨ Understand the rela-onship between discount rates, swap rates, zero coupon rates, forward rates, and bond yields
- 3. Interest Rate Risk 3 t0=0 q tS=0.5 tL=1.0 yrs yrs yrs f • A firm requires a $10,000,000 loan over a period from 6 to 12 months from present • Over the period 0.5 ≤ t ≤ 1.0 • The firm’s treasurer believes that the interest rate offered will rise over the next 6 months i.e., interest expense will be greater in the near future • Assume that the company can borrow and deposit funds at LIBOR. • Current 6 month LIBOR is 4.28363%, q (simple annual rate) • Current 12 month LIBOR is 4.51863%, r (simple annual rate) r • The firm might borrow for 12 months, but loan the funds for the first 6 months leaving an effec-ve ‘forward’ rate, f (1 q t ) (1 f (t t )) (1 r t ) + ⋅ S ⋅ + ⋅ L − S = + ⋅ L ⎤ (1 r t ) f 1 = 1 ⎥⎦ ⎡ ⎢⎣ − + ⋅ + ⋅ − L (1 q t ) (t t ) S L S f 1 4.65395% ⎤ 1 (1 + .0451863 ⋅ 1.0) (1 .0428363 0.5) (1.0 0.5) = ⎥⎦ ⎡ ⎢⎣ − + ⋅ − =
- 4. Forward Rate Agreement 4 qB-qO fB-fO rB-rO t0 tS tL FRA term loan term Similar to foreign exchange risk and ‘money market’ hedges, banks have a product called a ‘FRA’ forward rate agreement which packages the interest rate hedge The actual loan interest rate will be set at tS while the actual interest will be paid at tL The FRA will be executed at t0 and settled at tS The effective loan or forward rate is set at t0, but the relative benefit of the FRA and cost of the loan are not known The FRA includes a ‘notational principal’, and is cash settled
- 5. Forward Rate Agreement 5 FRA buyer • is the loan borrower and takes the long position in the FRA • believes that interest rates may rise so seeks to hedge its interest rate risk exposure • becomes a fixed rate payer instead of a floating rate payer as it is initially • Equivalently makes the following transactions • Borrows at the long offer rate rO over term t0 to tL • Lends at short bid rate qB over term t0 to tS • Locks in the forward offer rate fO over term tS to tL FRA seller • is often a financial intermediary such as bank and takes the short position, but most likely will ‘lay off’ its risk • takes the short position in the FRA and becomes a floating rate payer • Equivalently makes the following transactions • Lends at the long bid rate rB over term t0 to tL • Borrows at short offer rate qO over term t0 to tS • Locks in the forward bid rate fB over the term tS to tL ⎤ (1 r t ) f 1 = 1 ⎥⎦ ⎡ ⎢⎣ − + ⋅ B L + ⋅ − (1 q t ) (t t ) O S L S ⎤ (1 r t ) f 1 = 1 B ⎥⎦ ⎡ ⎢⎣ − + ⋅ O L + ⋅ − (1 q t ) (t t ) B S L S O
- 6. 6 t0 = 0 tS = .5 tL = 1.0 (1 .0439363 1.0) f 1 B = ⎡ + ⋅ 4.4092% ⎤ 1 (1 .0428363 .5) 0.5 ⎥⎦ ⎢⎣ − + ⋅ = (1 .0451863 1.0) f 1 O = ⎡ + ⋅ 4.7793% ⎤ 1 (1 .0415863 0.5) 0.5 ⎥⎦ ⎢⎣ − + ⋅ = • If the treasurer buys a FRA with notational principal of $10M and forward offer (borrowing) rate of 4.7793% • Treasurer effectively locks in the forward offer rate for a six month loan (tS < t ≤ tL) with principal $10M commencing in 6 mo. at tS. • The FRA is actually settled in cash at FRA expiry which we assume here is also the time of loan commencement. • Note that the FRA and loan are two completely separate agreements and transactions and that a party can buy or sell a FRA for speculation and not only to hedge a natural interest rate risk. qB = 4.15863% rB =4.39363% qO = 4.28363% rO =4.51863%
- 7. 7 Now consider a sequence of future lending requirements – semi-annual for 3 years Zero coupon rates, zk 1 ≤ k ≤ 6 Forward rates, fk Δt = .5 Discount rates, dk 2 2 (1 z ) z1·Δt f2·Δt f3·Δt f4·Δt f5·Δt f6·Δt k 0 1 2 3 4 5 6 tk 0.0 0.5 1.0 1.5 2.0 2.5 3.0 d1 d2 d3 d4 d5 d6 t2 2 z t + ⋅ = t6 6 (1+ z ) t5 5(1+ z ) t4 4 t3 (1+z ) 3(1+ z ) t2 2(1+ z )
- 8. 8 C C 1 0 + ⋅ (1 z Δt) 1 = tk k k C C 0 (1 + z ) = ⎞ ⎟⎠ 1 1 z ⎛ + ⎜⎝ = d 1 + ⋅Δ = m (1 z t) 1 1 1 k (1 z ) tk k 1 d + = m 1 (d -‐1) 1 1 = (d -‐1) z 1 t 1 1 Δ = 1 1 k = − d z tk k
- 9. 9 d 1 1 + ⋅ (1 z Δt) 1 = d 1 2 + ⋅ (1 f Δt) d 2 = d k 1 = − k + ⋅ (1 f Δt) d k ⎞ ⎟⎟⎠ 1 m ⎛ d d f 1 = − − k − 1 1 − = ⎟⎟⎠ ⎜⎜⎝ ⎞ ⎛ ⎜⎜⎝ d k 1 d Δt k k k 7% 6% 5% 4% 3% 2% 1% Year Zero Coupon Rate Discount Factor Forward Rate 1 4.5000% 0.95694 4.5000% 2 5.0126% 0.90681 5.5276% 3 5.2723% 0.85715 5.7936% 4 5.4027% 0.81020 5.7948% 5 5.4671% 0.76633 5.7255% 0% 0 1 2 3 4 5 Rates t years
- 10. 10 Discount factors dk Zero coupon rates zk Forward rates fk ‘Boot-‐ strapping’ Yields for coupon bonds yj 1 1 k = − d z tk k k (1 z ) tk k 1 d + = d k 1 = − k + ⋅ (1 f Δt) d k ⎞ ⎟⎟⎠ ⎛ d f 1 = − −1 ⎜⎜⎝ k 1 d Δt k k
- 11. Interest Rate Swaps 11 Firm Swap Dealer Bank LIBOR +2% SWAP Rate LIBOR Net interest rate = LIBOR – (LIBOR +2%) – swap rate = - (swap rate +2%)
- 12. 12 Year Zero Coupon Rate Discount Factor Forward Rate Swap Rate Floating Cash Flow Fixed Cash Flow Net Flow to Swap Buyer 1 4.5000% 0.95694 4.5000% 4.5000% $ 450,000 $ 543,750 $ (93,750) 2 5.0126% 0.90681 5.5276% 5.0000% $ 552,764 $ 543,750 $ 9,014 3 5.2723% 0.85715 5.7936% 5.2500% $ 579,359 $ 543,750 $ 35,609 4 5.4027% 0.81020 5.7948% 5.3750% $ 579,479 $ 543,750 $ 35,729 5 5.4671% 0.76633 5.7255% 5.4375% $ 572,549 $ 543,750 $ 28,799 Present Value $ 2,336,729 $ 2,336,729
- 13. 13 P F c d F c c = ⋅ ⋅ + ⋅ ⋅ d + ... + F ⋅ ⋅ + ⋅ k d F d k k k 2 k 1 m m m F F s d F s s = ⋅ ⋅ + ⋅ ⋅ d + ... + F ⋅ ⋅ + ⋅ k d F d k k k 2 k 1 m m m 1 s d s s = ⋅ + ⋅ d + ... + ⋅ + k d d k k k 2 k 1 m m m Σ= 1-‐d = s k j 1 j d k k m 1-‐d = k Σ= j 1 k j k d m s ⎞ ⎟⎠ 1 s d s d ... d s k = ⋅ + ⋅ + + ⋅⎛ +1 ⎜⎝ m m m 2 k k 1 k
- 14. 14 ⎞ ⎟⎠ d s d ⎛ + ⋅ + = Σ − 1 s k ⎜⎝ = 1 m m k k 1 j 1 j k d ⎞ ⎟⎠ − s ⎛ + ⎜⎝ = Σ − = 1 m m 1 s d k k 1 j 1 j k k Year Zero Coupon Rate Discount Factor Forward Rate Swap Rate 1 4.5000% 0.95694 4.5000% 4.5000% 2 5.0126% 0.90681 5.5276% 5.0000% 3 5.2723% 0.85715 5.7936% 5.2500% 4 5.4027% 0.81020 5.7948% 5.3750% 5 5.4671% 0.76633 5.7255% 5.4375%
- 15. 15 Discount factors dk Zero coupon rates zk Forward rates fk k = − k (1 z ) Yields for coupon bonds yj ‘Boot-‐ strapping’ 1 1 d z tk k tk k 1 d + = d k 1 = − k + ⋅ (1 f Δt) d k ⎞ ⎟⎟⎠ ⎛ d f 1 = − −1 ⎜⎜⎝ k 1 d Δt k k 1-‐d k d Swap rates sk = k Σ= j 1 j k m s d ⎞ ⎟⎠ − s ⎛ + ⎜⎝ = Σ − = 1 m m 1 s d k k 1 j 1 j k k
- 16. 16 CME begins clearing interest rate swaps CHICAGO/NEW YORK Mon Oct 18, 2010 (Reuters) - CME Group Inc said on Monday that it had begun providing clearing to the $400 trillion interest-rate swaps market, the largest of the opaque markets that lawmakers are forcing onto more transparent venues. CME Information