SEEM 3590: Investment Science Lecture Notes 7: Term Structure of Interest Rates, Forward Rate, and Swaps YANG Chen Fall 2024 1 / 18 Interest Rates in the Market ▶ Various types of rates in the market ▶ Treasury rates: return rates of Treasury bills, Treasury notes, and Treasury bonds ▶ LIBOR (London Interbank Offered Rate): representing borrowing costs between AA-rated financial institutions ▶ Repo (repurchase agreement) rate: overnight borrowing cost ▶ LIBOR rates were usually used as short-term risk-free rates by derivative traders. ▶ The publication of U.S. LIBOR has been ceased since July 2023. Replacements of LIBOR include SOFR (Secured Overnight Financing Rate) and SONIA (Sterling Overnight Index Average). ▶ Nevertheless, other interbank offered rates are still used in the world, such as EURIBOR (for EU) and HIBOR (for Hong Kong). 2 / 18 Example: LIBOR Rates Figure: The quotes of LIBOR Rates on April 04, 2013. Source: Wall Street Journal. The rates are in percentage term and are quoted rates. For instance, the 0.28040 quote for three-month LIBOR means the interest rate in three months is 0.28040%/4. 3 / 18 Example: Treasury Rates Figure: Treasury rates on April 04, 2013. Source: https://www.treasury.gov/resource-center/data-chart-center/ interest-rates/pages/textview.aspx?data=yield. The rates are calculated from the price quotes of on-the-run Treasury bills, Treasury notes, and Treasury bonds. For the details of the calculation methodology, refer to http://www.treasury.gov/resource-center/data-chart-center/ interest-rates/Pages/yieldmethod.aspx 4 / 18 Term Structure of Interest Rates ▶ In the market, the interest rates for different maturities are different! ▶ Spot rates: the interest rates representing the borrowing cost in the period starting from today to some future time called maturity ▶ Example: the n-year spot rate refers to the interest rate of borrowing cost within n years. ▶ Spot rates are also known as zero rates because they represent the return rates of zero-coupon bonds. ▶ The dependence of spot rates on its term is called the term structure of interest rates 5 / 18 Example: Term Structure of Interest Rates 0 2 4 6 8 10 12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time to Maturity (months) Sp ot

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(% ) Figure: The Term Structure of LIBOR Rates on April 04, 2013. Source: Wall Street Journal. 6 / 18 Discount Factors and Short Rate ▶ The rates observed in the market are usually quoted rates. ▶ Denote “today” by t and “some future time” by T . Denote y0(t, T ) as the spot rate with continuous compounding from t to T . ▶ The discount factor, denoted by B(t, T ), from t to T , is the time-t value of $1 at time T . Therefore, B(t, T ) = e−y0(t,T )(T−t). ▶ The quantity r(t) := lim T↓t y0(t, T ) is called short rate, representing the instantaneous borrowing cost today. 7 / 18 Forward Rate Agreements ▶ Forward rate agreement (FRA) is an OTC agreement that a certain interest rate will apply to either borrowing or lending a certain principal during a specified future period of time: ▶ t: today ▶ T1: start date of the future period ▶ T2: end date of the future period ▶ y0(T1, T2): the spot rate observed in the market at time T1 for the period between T1 and T2 ▶ K: the rate agreed in the FRA (continuously compounded) ▶ N : the principal of the FRA ▶ Settlement date is T1; at this time, the payoff of the long position of FRA is[ N · ey0(T1,T2)(T2−T1) −N · eK·(T2−T1) ] · e−y0(T1,T2)(T2−T1). ▶ This time T1-payoff is equivalent to the following payoff at time T2 N · ey0(T1,T2)(T2−T1) −N · eK·(T2−T1). 8 / 18 Valuing FRA By Spot Rates ▶ Consider Strategy I: short NeK·(T2−T1) ·B(t, T2) dollar amount for the period [t, T2] at the spot rate y0(t, T2); invest N ·B(t, T1) dollar amount for the period [t, T1] at the spot rate y0(t, T1) and reinvest the principal plus interest at T1 for the period [T1, T2] at the spot rate y0(T1, T2). ▶ Initial cost of Strategy I: N ·B(t, T1)−NeK·(T2−T1) ·B(t, T2). ▶ Payoff at T2: N · ey0(T1,T2)(T2−T1) −N · eK·(T2−T1). ▶ No intermediate cash flow ▶ Consider Strategy II: buy and hold the FRA to the settlement date and invest the payoff for the period [T1, T2] at spot rate. ▶ Initial cost of Strategy II: value of FRA at t. ▶ Payoff at T2: N · ey0(T1,T2)(T2−T1) −N · eK·(T2−T1). ▶ No intermediate cash flow 9 / 18 Forward Rate ▶ By no arbitrage assumption, the value of the FRA at time t is N · [ B(t, T1)− eK·(T2−T1) ·B(t, T2) ] . ▶ The particular K making the value of FRA zero is called the continuously compounded forward rate today with the forward period [T1, T2]: K = T2 − t T2 − T1 y0(t, T2)− T1 − t T2 − T1 y0(t, T1) ▶ The instantaneous forward rate today with forward date T is defined as the forward rate when T1 = T , T2 = T +∆T , and ∆T ↓ 0. ▶ Try it yourself: show that the instantaneous forward rate is − ∂∂T lnB(t, T ). 10 / 18 Example: Continuously Compounded Forward Rate Assume the spot interest rate with 3-month maturity is 6.0% and the spot interest rate with 6-month maturity is 6.2% (both are continuously compounded). What is the forward interest rate with continuous compounding for the second quarter (that is, the period from 3 months later to 6 month later)? We use the equation for continuously compounded forward rate K on page 10. According to the question, T1 − t = 3/12 = 0.25, T2 − t = 6/12 = 0.5, T2 − T1 = 0.25, y0(t, T1) = 6.0%, and y0(t, T2) = 6.2%. Therefore, K = 6.4%. 11 / 18 Forward Rate ▶ In practice, the rate agreed in FRA can also be quoted with a compounding frequency that reflects the length of the period to which they apply. For instance, if the quoted rate is L for the period [T1, T2], then the interest is L · (T2 − T1). ▶ The value of FRA is modified accordingly to N · [B(t, T1)− (1 + L · (T2 − T1)) ·B(t, T2)] . 12 / 18 Interest Rate Swaps ▶ Interest rate swaps can be regarded as swaps of different interest payments between two parties at multiple future times. ▶ In a vanilla interest rate swap: ▶ One party agrees to pay the other party semiannual cash flows equal to interest at a predetermined fixed rate on a notional principal until the maturity of the swap. ▶ In return, it receives semiannual cash flows equal to interest at a floating rate on the same notional for the same period of time. ▶ In a vanilla interest rate swap on LIBOR, the floating rate in each half-year period is chosen to be the LIBOR rate at beginning of the period with maturity to be the end of the period. The LIBOR is quoted with semiannual compounding. ▶ Interest rate swaps on other interbank offered rates (e.g. EURIBOR, HIBOR) work similarly. 13 / 18 Example: A Vanilla Interest Rate Swap Microsoft entered into a $100 million 3-year interest rate swap with a financial institution, where a fixed rate of 5% per annum is paid and LIBOR is received. Rates are quoted with semiannual compounding. The contract was signed on Mar. 5, 2007. The following table shows the cash flow streams (in millions of dollars) to Microsoft during this 3-year period. Date Six-month Floating cash Fixed cash Net cash LIBOR (%) flow received flow paid flow Mar. 5, 2007 4.20 Sep. 5, 2007 4.80 +2.10 −2.50 −0.40 Mar. 5, 2008 5.30 +2.40 −2.50 −0.10 Sep. 5, 2008 5.50 +2.65 −2.50 +0.15 Mar. 5, 2009 5.60 +2.75 −2.50 +0.25 Sep. 5, 2009 5.90 +2.80 −2.50 +0.30 Mar. 5, 2010 +2.95 −2.50 +0.45 14 / 18 Pricing Interest Rate Swaps ▶ The party paying interest at the floating rate is called floating-rate payer and the party paying interest at the fixed rate is called fixed-rate payer. ▶ Next, we find the value of a vanilla interest rate swap with fixed rate S, notional N , and maturity T . ▶ Denote by T1 < T2 < . . . Tn = T the interest exchange dates in the swap. ▶ Consider the following strategy: invest N dollars at six-month LIBOR rate, extract the interest payment after six months, invest the principal at six-month LIBOR rate for the next six-month period, and keep rolling until the maturity of the interest rate swap; short N · S2 ·B(t, Ti) dollars at the spot rate with maturity Ti, i = 1, . . . , n and short N ·B(t, T ) dollars at the spot rate with maturity T . 15 / 18 Pricing Interest Rate Swaps ▶ The cash flow stream of the strategy is same as the cash flow stream of the interest rate swap, so, by no arbitrage, it has same value as the swap. ▶ Thus, the value of the swap (from the fixed-rate payer’s perspective) is[ 1− ( B(t, T ) + S 2 n∑ i=1 B(t, Ti) )] N. 16 / 18 Example: Pricing an Interest Rate Swap Consider the swap entered by Microsoft in the previous example. The discount factors observed when the contract was signed are Maturity Date Discount factor Sep. 5, 2007 0.9778 Mar. 5, 2008 0.9541 Sep. 5, 2008 0.9291 Mar. 5, 2009 0.9048 Sep. 5, 2009 0.8781 Mar. 5, 2010 0.8479 How much did Microsoft pay to or receive from the financial institution when the contract was signed? The value of the swap from the fixed-rate payer is[ 1− ( 0.8479 + 5% 2 (0.9778 + · · ·+ 0.8479) )] × 100 = $1.4811 million. Therefore, Microsoft, as the fixed-rate payer, paid $1.4811 million to the floating-rate payer when signing the contract. 17 / 18 References ▶ The relevant parts in the textbooks are: [L] Chapter 4: Sections 1-3 18 / 18 51作业君版权所有