Bài giảng Toán tài chính: Chương 5 - TS. Nguyễn Ngọc Anh

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Bài giảng Toán tài chính - Chương 5: Cấu trúc kỳ hạn của lãi suất và phòng ngừa rủi ro, cung cấp cho người học những kiến thức như Lãi suất tức thời và lãi suất tương lai; Lý thuyết về cấu trúc kỳ hạn của lãi suất; Thời lượng của dòng tiền; Lý thuyết REDINGTON & Phòng ngừa rủi ro lãi suất. Mời các bạn cùng tham khảo!

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Nội dung Text: Bài giảng Toán tài chính: Chương 5 - TS. Nguyễn Ngọc Anh

Mathematics of Finance Chapter 5: The Term Structure of Interest Rates and Immunisation Doctor. Nguyen Ngoc Anh Banking Faculty 1 Banking Faculty 05/10/24
Chapter 5 – Cấu trúc kỳ hạn của lãi suất và phòng ngừa rủi ro 1. Lãi suất tức thời và lãi suất tương lai 2. Lý thuyết về cấu trúc kỳ hạn của lãi suất 3. Thời lượng của dòng tiền 4. Lý thuyết REDINGTON & Phòng ngừa rủi ro lãi suất 05/10/24 2 Banking Faculty
Lãi suất tức thời & lãi suất tương lai Lãi suất giao ngay và LS kỳ hạn  Spot Rates (n-year spot rate: yn)  we consider two zero coupon bonds. Bond Ais a one-year bond and bond B is a two-year bond. Both have face values of $1,000. The one-year interest rate, r1, is 8 per-cent. The two-year interest rate, r2, is 10 percent. These two rates of interest are examples of spot rates. 05/10/24 3 Banking Faculty
Spot Rates
Spot Rates 1  Spot rate z(n) facevalue n y 1 z ( n) p
forward rates  Assume the following set of rates  What are the forward rates over each of the four years?  The forward rate over the first year is, by definition, equal to the one-year spot rate. Thus, we do not generally speak of the forward rate over the first year. The forward rates over the later years are:
forward rates  An individual investing $1 in the two-year zero coupon bond receives $1.1236 *$1*(1.06)^2 at date 2  He can be viewed as receiving the one-year spot rate of 5 percent over the first year and receiving the forward rate of 7.01 percent over the second year. An individual investing $1 in a three-year zero coupon bond receives $1.2250 *$1*(1.07)^3 at date 3  She can be viewed as receiving the two-year spot rate of 6 percent over the first two years and receiving the forward rate of 9.03 percent over the third year
Yield curve
The Expectations Hypothesis  Expectations Hypothesis : forward rate 2 =Spot rate expected over year 2  Equation says that the forward rate over the second year is set to the spot rate that people expect to prevail over the second year. This is called the expectations hypothesis.It states that investors will set interest rates such that the forward rate over the second year is equal to the one-year spot rate expected over the second year
Liquidity Preference Hypothesis  forward rate 2 >Spot rate expected over year 2  That is, to induce investors to hold the riskier two-year bonds, the market sets the forward rate over the second year to be above the spot rate expected over the second year. Equation above is called the liquidity preference hypothesis. rate expected over year 2
Segmented markets theory  This theory states that the market for different- maturity bonds is completely separate and segmented.  The interest rate for a bond with a given maturity is determined by the supply and demand for bonds in that segment with no effect from the returns on bonds in other segments.
The discounted mean term of a project (Macaulay’s Duration)  Duration is a weighted average of the time until the expected cash flows from a security will be received, relative to the security’s price k C t (t) t t =1 (1 + i) D= k Ct t =1 (1 + i) t
Macaulay’s Duration  Example  What is the duration of a bond with a $1,000 face value, 10% annual coupon payments, 3 years to maturity and a 12% YTM? The bond’s price is $951.96. 100 1 100 2 100 3 1,000 3 1 + 2 + 3 + (1.12) (1.12) (1.12) (1.12) 3 2,597.6 D 3 = 2.73 years 100 1000 951.96 t + t =1 (1.12) (1.12) 3
Macaulay’s Duration Example  What is the duration of a bond with a $1,000 face value, 10% coupon, 3 years to maturity but the YTM is 5%?The bond’s price is $1,136.16. 100 * 1 100 * 2 100 * 3 1,000 * 3 1 + 2 + 3 + (1.05) (1.05) (1.05) (1.05) 3 3,127.31 D = 2.75 years 1136.16 1,136.16
Macaulay’s Duration  Example  What is the duration of a zero coupon bond with a $1,000 face value, 3 years to maturity but the YTM is 12%? 1,000 * 3 (1.12) 3 2,135.34 D = 3 years 1,000 711.78 (1.12) 3  the duration of a zero coupon bond is equal to its maturity
Volatility  Consider an investement , suppose that the net present value is currently determined on the basis of an annual force of interest 0, but that this may change at very sort notice to another value 1, the proportionate change in the net present value of the projet would be  We define the volatility of the projet at force interest 0 to be
The matching of assets and liabilities  The matching of assets and liabilities requires that the company’s asset be chosen as far as possible in such a way as to make the assets and liabilities equally responsive to the influences which affect them both  We define the net liability –outgo (Lt) at time t to be Lt=St-Pt  Where : St denote the liability at time t, Pt be the money received by company at time t  Absolutely matching require that At=Lt for all value of t
Redington's Theory of Immunization  Consider a fund with asset cash-flow A and liability cash-flow L. Let VA and VL be their present values. We say that at interest rate i0 the fund is immunised against small movements in the interest rate if VA(i 0) = VL(i0)  The theory relies on a small change in interest rates. The fund may not be protected against large changes. In practice, this is not usually a problem as the theory is fairly robust; only large changes and strange liabilities may lead to problems at this point; rebalancing helps when interest rates change gradually rather than abruptly