Financial Mathematics FULL NOTES - PDF

Financial Mathematics Notes taken by Vittorio Peritore from the course of Financial Mathematics 0 Index Financial operations and Financial quantities.................................................................

Financial Mathematics Notes taken by Vittorio Peritore from the course of Financial Mathematics 0 Index Financial operations and Financial quantities.................................................................................................................................................................. 2 Common Financial Laws..................................................................................................................................................................................................... 5 Financial laws................................................................................................................................................................................................................ 7 Simple Accumulation Rule (Simple Interest Rate)...................................................................................................................................................... 8 Compound Accumulation Rule (Compound Discount)............................................................................................................................................... 9 Complex Financial Operations.......................................................................................................................................................................................... 14 Ordinary annuities...................................................................................................................................................................................................... 15 Accumulated value........................................................................................................................................................................................................... 15 Present value..................................................................................................................................................................................................................... 16 Annuity due.................................................................................................................................................................................................................. 18 Present value..................................................................................................................................................................................................................... 19 Accumulated value........................................................................................................................................................................................................... 19 Deferred annuity......................................................................................................................................................................................................... 19 Ordinary deferred annuity.............................................................................................................................................................................................. 19 Deferred annuity due....................................................................................................................................................................................................... 19 Perpetuities.................................................................................................................................................................................................................. 21 Unitary ordinary perpetuity........................................................................................................................................................................................... 21 Unitary perpetuity due..................................................................................................................................................................................................... 21 Deferred ordinary perpetuity......................................................................................................................................................................................... 21 Deferred perpetuity due.................................................................................................................................................................................................. 21 Financial Laws................................................................................................................................................................................................................... 21 Properties.................................................................................................................................................................................................................... 21 Continuation Accumulation Factor............................................................................................................................................................................ 23 Instantaneous force of interest.................................................................................................................................................................................. 24 Force of interest for simple accumulation rule............................................................................................................................................................ 25 Force of interest for compound accumulation rule..................................................................................................................................................... 25 Decomposability of a financial law............................................................................................................................................................................ 25 End Module I...................................................................................................................................................................................................................... 26 Market of Riskless Bond................................................................................................................................................................................................... 27 Zero coupon bond........................................................................................................................................................................................................ 27 Pure-discount Bond.......................................................................................................................................................................................................... 28 Term Structure of Interest Rates..................................................................................................................................................................................... 31 Forward Contract.............................................................................................................................................................................................................. 32 Yield Curve......................................................................................................................................................................................................................... 34 Amortization of a Loan...................................................................................................................................................................................................... 34 Internal Rate of Return (IRR)........................................................................................................................................................................................... 41 Adjusted Present Value (APV).......................................................................................................................................................................................... 43 Generalized Adjusted Present Value (GAPV)................................................................................................................................................................... 46 Financial Leverage............................................................................................................................................................................................................ 47 1  Financial operations and Financial quantities Financial calculus: Financial operation: every agreement concerning the exchange of money at different time dates. Simple financial operation: only two cash flows; Complex financial operation: several cash flows. Financial calculus is based on two types of processes: a. Accumulation b. Discount Accumulation Discount o Principal value 𝐶 paid at time 𝑡! o Present/discounted value 𝐶 cashed at o Final value 𝑀 cashed at time 𝑇 time 𝑡! o Nominal value 𝑀 paid at time 𝑇 Simple financial operation: This operation shows how much €1,000 today are worth in 1 month (€1,100). It is called ACCUMULATION. This is one side of the operation, from the point of view of agent 1. From the point of view of agent 2, it is a DISCOUNT operation: The difference (€100) is the price of money, the cost an economic agent has to pay to get €1,000 now instead of waiting 1 month, when they would be worth €1,100. This works under the assumption that the market is efficient (= fair). 2  (1) Accumulation process: Given 𝐶, 𝑡! , and 𝑇, we need to find the value of 𝑀 (2) Discount process: Given 𝑀, 𝑡! , and 𝑇, we need to find the value of C. The fnancial operation for which “C at time 𝑡! ” is equivalent to “M at time T”, defines a financial equivalence. This is possible thanks to the basic principle of financial calculus which essentially says that €1 today is better than €1 tomorrow! REMARK: in order to have a fair financial operation, the value of the engagement into the operation (left-hand side) must be equal or less than the value of the right-hand side. Important assumption: time value of money is non-negative (𝑀 ≥ 𝐶). The non-negative time value of money states that if 𝑀 available in 𝑇 > 0 instead of 𝐶 available in 0, then 𝑀 ≥ 𝐶 (€1 now is worth more (for sure not less) than €1 available tomorrow) ⟶ interest rates are always positive. Actually in 2014 the EU economy was so doing so bad that the ECB (Mario Draghi) had to cross the zero-lower-bound and impose negative interest rates (you got paid to borrow money), in order to provide a strong stimulus to the economy. It was an unknown territory, in which great arbitrage opportunities were available, but it worked. Terminology: C = starting value (time 0) M = accumulated value (time T) There’s a financial equivalence (~) between C (€1000 available now) and M (€1100 available at time T) because money has a time dimension (difference between financial mathematics and mathematics). 𝐶~𝑀 €1100 available at time T €1000 available now 3 Financial quantities: Accumulation factor: It provides the capital value of €1 available today at some future time T. The accumulation factor can be used to compute the value of money invested at t=0 in t=T (by multiplying the money available today by the accumulation factor you get the value of money at time T). If two operations have the same accumulation factor then they are fair, because the time value of money is the same. 𝑀 𝑓= =1+𝑖 𝐶 ""!! ##!! Ex: "!!! = 1,1 and #!!! = 1,1 are fair Discount factor: It tells you how much €1 available at future time (t=T) is worth now (t=0). Knowing the discount factor (you get it from checking other financial operations (in 1 month time)) allows you to know if the amount of money you are being offered today in exchange of a higher amount given in the future is fair or not. The discount is the inverse operation of the accumulation, in this case we say that the accumulation and discount factors are conjugate. 1 𝐶 𝜑= = 𝜑 𝜖 [0,1] 𝑓 𝑀 "!!! Ex: = 0,91 ""!! Interest and Interest rate: the interest rate is the interest (reward) you get for each unit of money invested. The interest 𝐼, when seen in the opposite way (in the discount process), can also be called discount (suffered to have money available now instead of having them in the future), if the operation considered is a discounting process. 𝐼 𝑀−𝐶 𝐷 = 𝐼 = 𝑀 − 𝐶 (𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡) → 𝑖 = (𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒) = =𝑓−1 𝐶 𝐶 Interest rates are usually expressed for time unit interval (day, month, year). It is very important to define the time unit. If not specified, we consider the year as the (default) unit time measure. Note that: 𝑖 (𝑡! , 𝑇 ) or 𝑖(0, 𝑇 − 𝑡! ) are different in general. Ex: 𝑖(0, 1) and 𝑖(1, 2) are interest rate both related to an interval of length 1 time unit (one year). However, 𝑖(0, 1) (the so called spot interest rate) represents the interest on a unit of capital invested today after one year. On the other hand, 𝑖(1, 2) (called forward interest rate) is the interest of a unit of capital invested for one year from one year from now. The two (spot and forward) interest rates, even if they refer to financial operation of the same length, are usually different. Discount rate: ratio between the discount suffered to have money available now instead of having them in the future and the amount of money available in T. 𝐷 𝐼 𝑀−𝐶 𝑑== = =1−𝜑 → 𝜑 =1−𝑑 𝑀 𝑀 𝑀 1 → =1−𝑑 REMARK: From the relation which links the discount rate 𝑓 1 𝑑 and the interest rate 𝑖: → 𝑓= 𝑖 1−𝑑 𝑑(𝑖) = 𝑑 1+𝑖 → 𝑖= 4 if 𝑖 increases, also 𝑑 rate increases. 1−𝑑 If 𝑓, 𝜑, 𝑖, 𝑑 are related to the same simple financial operation that implies cash flows at t=0 and at t=T, or to any other financial operation having cash flows at t=0 and at t=T, then the following relationships among the basic financial quantities hold: Common Financial Laws Consider the two different simple financial operations: 1) 𝑀 110 𝑓= = = 1.1 → accumulation factor for this Ginancial operation 𝐶 100 𝑖 = 𝑓 − 1 = 1.1 − 1 = 0.1 = 10% → interest rate 𝐶 100 𝜑= = = 0. K90 KKK → discount factor 𝑀 110 𝑖 𝑑= KKKK → discount rate = 0. 09 1+𝑖 2) KKKK ; 𝑑 = 0. K09 𝑓 = 1.1 ; 𝑖 = 0.1 = 10% ; 𝜑 = 0. 90 KKK Those two financial operations seem the same. The second allows you to get the same interest rate but in 1 month time instead of 1 year time. The second financial operation is more remunerative than the first one; despite so, the interest rate (i), accumulation factor (f), discount factor (𝜑) and discount rate (d) are numerically the same for the two operations. This is due to the fact that the 4 financial quantities: i, f, 𝜑, and d, have a time dimension (very important, changes the conveniency of those financial quantities). The time period is different, this means that the interest rate (as the other quantities) has time dimensions. 10% is always the interest rate, but if the financial operation is over the year, that 10% 5 would be the annual interest rate (annual accumulation factor, annual discount factor and annual discount rate), while in the second case it is the monthly interest rate (monthly accumulation factor, monthly discount factor and monthly discount rate) which are very different. In the financial operation 1), we have that the annual interest rate (𝒊) is 10% which means that for any €1 invested now, we obtain €1 and €0.10 after 1 year. While in the financial operation 2), we have that the monthly interest rate (𝒊(𝟏𝟐) ) is 10%, which means that for any €1 invested now, we obtain €1 and €0.10 after 1 month (more convenient). Other financial operation is as follows: 3) The interest rate would still be 10% but it is a biannual interest rate, not annual. Each financial quantity has two dimensions: o Numerical o Temporal So, when indicating an interest rate or any other financial quantity, the unit of time to which it refers has to be specified. 4) The monthly interest rate in this 4th financial operation is 1%. We cannot say which one is better, if a 1% monthly or a 10% annual, since we would have to check the “rule” under which interests are computed: financial law which says how interests accumulate. 5) In this financial operation, the monthly interest rate is 0.7%. this can still be better than 10% annual, it depends on the “financial law”, just like before. 6  Financial laws A financial law is a rule that describes how interests are accumulated or accrued. Particularly, the financial law is needed to compare interest rates with different time dimensions (ex: monthly vs annual). A financial law can be mathematically represented as a two-variable function, one variable being time, and the other interest rate: 𝑓(𝑡; 𝑖) 𝑓(𝑡; 𝑖) is the value of €1 after 𝑡 units of time when invested at an interest rate for unit of time given by 𝑖. if 𝑡 are months, then 𝑖 has to be monthly interest rate, if 𝑡 are years, then 𝑖 has to be annual interest rate. Interest rate 𝑖 is the interest for unit of time and for unit of capital. 𝑓(1; 𝑖) is the interest accumulated after 1 unit of time based on an interest of €1. Any function 𝑓(𝑡; 𝑖) can be used as a financial law if certain conditions are satisfied. Those conditions are: 1) 𝑓(0; 𝑖) = 1 → the value at time 𝑡 of €1 invested at time 0 has to be 1. 2) 𝑓 must be a monotonically increasing function with respect to 𝑡. So going back to the previous examples with the simple financial operations, we can now say which interest rate is more remunerative. The annual interest rate is 10%, there are 12 months in a year, so let us imagine that every month we "!% earn 0.82% ( "# ). ) We are considering that "# = 𝑖 ("#) the monthly interest rate equivalent to the annual interest rate. 𝑖 1 + 𝑖 = 1 + 12 × 𝑖 ("#) = 1 + 12 × 12 $ 𝑖 (") = " ∨ 𝑖 = 𝑖 (") 𝑚 7  Simple Accumulation Rule (Simple Interest Rate) The monthly interest rate is 10%, to assume the interest after “x” months we would simply keep adding the interest rate until we reach the 𝑡 unit of time we are looking at. Assuming the rule of accumulation of the interest rate, 𝑓(𝑡; 𝑖) is made through: 1 × 𝑓(12; 0.1) = 2.2 1 × (1 + 12 × 0.1) = 2.2 The financial law that describes this type of simple accumulation rule for the interest rate is: amount of € invested. 𝑓(𝑡; 𝑖) = 1 + 𝑖𝑡 If I invest €𝐶 at time 0 for 𝑡 unites of time, then: 𝑀 = 𝐶𝑓(𝑡; 𝑖) = 𝐶(1 + 𝑖𝑡) = 𝐶 + 𝑖𝑡𝐶 The interest of the financial operation is proportional to the capital invested (𝐶) and to the amount of time in which the investment lasts (𝑡). The problem with this financial operation is that under this rule, interest rate does not participate in the accumulation of forward interest rate: the interest rate accumulated from time 0 to 1 month doesn’t take part in the computation of the interest rate from 1 month to 2 months. Interest does not generate further interest: interest gained each year only depends on initial conditions. This implies that stopping the operation and restarting it immediately after brings extra rewards (intermediate capitalisation is beneficial for the trader ⟶ principle for the compound accumulation rule). According to the simple accumulation rule, we have that the accrued capital 𝑀 evolves as follows: 𝑡 = 0 → 𝑀(0) = 𝐶 𝑡 = 1 → 𝑀(1) = 𝐶(1 + 𝑖) = 𝐶 + 𝑖𝐶 = 𝐶 + 𝐼" (interest accumulated after 1 unit of time) 𝑡 = 2 → 𝑀(2) = 𝐶(1 + 2𝑖) = 𝐶 + 𝑖𝐶 + 𝑖𝐶 = 𝐶 + 𝐼" + 𝐼# → where 𝐼# = 𝑖𝐶 As you can see, 𝐼" does not contribute to calculating 𝐼#. 8 Generally, &'( 𝑡 = 𝜏 → 𝑀(𝜏) = 𝐶(1 + 𝜏𝑖) = 𝐶 + @ 𝐼% + 𝐼& → SIMPLE ACCUMULATION RULE %)( so, interest does not contribute to generate further interest. The simple accumulation rule is relevant when considering mortgages, since interest has to be computed using this rule. The simplicity of the simple accumulation rule makes it easier for borrowers to understand how interest is calculated and how it impacts the overall cost of the mortgage. It provides a transparent way for individuals to estimate the total interest payments and plan their finances accordingly. Compound Accumulation Rule (Compound Discount) 𝑖 = interest rate for unit of time (for 1 period) = 10% If we have the following financial operation with investing €1 at time 𝑡 = 0 with an interest rate of 10%, assuming interest rate is accumulated following the simple accumulation rule, after period 1 we would obtain 𝑀(1) = 𝐶(1 + 𝑖) = 1 + 0,1 = 1,1. If we were to follow the simple accumulation rule, we would go on to the next period and get, at period 2, 𝑀(2) = 𝐶(1 + 2𝑖) = 1 + 0,2 = 1,2, and so on for the following periods. However, if we were to stop the financial operation at period 1 and start a new investment, the following would happen: The cash in would be €1,1 which is the capital available at period 1 Start a new investment of €1,1, immediately after at the interest rate of 10% according to the simple accumulation rule. After 1 further period (𝑀(2)), you would get: 𝐼# = 1,1 + 1,1(0,1) = (𝐶 + 𝐼" )𝑖 = 𝑀(1)𝑖 > 0,1(= 𝑖𝐶) So 𝑀(2) = 𝐶 + 𝐼" + 𝐼# = 1,21 instead of 1,2 (what we would get following the simple accumulation rule) This rule would repeat for the following periods: 𝐼* = 𝑀(2)𝑖 > 0,1(= 𝑖𝐶) and so 𝑀(2) = 𝐶 + 𝐼" + 𝐼# + 𝐼* 9 𝑡 = 0 → 𝑀(0) = 𝐶 𝑡 = 1 → 𝑀(1) = 𝐶 + 𝐼" = 𝐶 + 𝑖𝐶S𝐶(1 + 𝑖)T 𝑡 = 2 → 𝑀(2) = 𝐶 + 𝐼" + 𝐼# = 𝑀(1) + 𝑖𝑀(1) = 𝑀(1)(1 + 𝑖) = 𝐶(1 + 𝑖)(1 + 𝑖) = 𝐶(1 + 𝑖)# 𝐶 + 𝐼" = 𝑀(1) ; 𝐼# = 𝑀(1)𝑖 𝑡 = 3 → 𝑀(3) = 𝐶 + 𝐼" + 𝐼# + 𝐼* = 𝑀(2) + 𝑀(2)𝑖 = 𝑀(2)(1 + 𝑖) = 𝐶(1 + 𝑖)# (𝑖 + 𝑖) = 𝐶(1 + 𝑖)* 𝐶 + 𝐼" + 𝐼# = 𝑀(2) Generally, 𝑡 = 𝜏 → 𝑀(𝜏) = 𝐶(1 + 𝑖)& → COMPOUND ACCUMULATION RULE Example: investment of €100, for 10 years, at the annual interest rate of 10%. Compute the final value of the investment. Under the simple accumulation rule, we have that: 𝑀(10) = 100S1 + 10(0,1)T = €200 Under the compound accumulation rule we have that: 𝑀(10) = 100(1 + 0,1)"! = €259,4 The difference between the two: 𝑀+ − 𝑀 , = €259,4 − €200 = €59,4 which is the extra interest generated by accumulated interest. Once we specify the unit of time: Simple accumulation rule Compound accumulation rule Tis financial law is defined by the This financial law is defined by the accumulation factor: accumulation factor: 𝑓* (𝑡; 𝑖) = 𝑓* (𝑡) = 1 + 𝑖* 𝑡 𝑓+ (𝑡; 𝑖) = 𝑓+ (𝑡) = (1 + 𝑖+ ), 𝑖* =simple interest rate 𝑖+ =compound interest rate 10 Conjugate discount factor: We can also define the same financial law using this discount factor. Whenever there is 𝑓(𝑡) (I accumulate money 𝐶 through the accumulation process 𝑓, obtaining 𝑀), if 𝑀 is discounted from time 𝑡 to time 0, I should return to money 𝐶. 𝐶𝑓(𝑡) = 𝑀 ⟶ 𝑀 𝜑(𝑡) = 𝐶 𝑓+/* (𝑡)𝜑+/* (𝑡) = 1 1 𝜑+ (𝑡) = (1 + 𝑖. ), 1 𝜑* (𝑡) = (1 + 𝑖* 𝑡) Simple financial law ⟶ 𝑓* (𝑡; 𝑖) All the financial laws obtained varying the interest rate 𝑖 are called simple regime. Examples of 2 different simple financial laws: 1 + 0,1𝑡 = 𝑓(𝑡; 0,1) ⟶ 𝑖 = 10% 1 + 0,05𝑡 = 𝑓(𝑡; 0,05) ⟶ 𝑖 = 5% For any financial law we have that the financial law at time 1 (𝑓(1)) is: 𝑓(1) = 𝑓(1; 𝑖) = 1 + 𝑖 1 𝑖 𝜑(1) = 𝜑(1; 𝑖) = = 1 − 𝑑 (discount rate for unit of time ⟶ 𝑑 = ) 1+𝑖 1+𝑖 Simple accumulation rule Compound accumulation rule 𝑓, (𝑡) = 𝑓, (𝑡; 𝑖) = 1 + 𝑖𝑡 𝑓+ (𝑡) = 𝑓+ (𝑡; 𝑖) = (1 + 𝑖)- 𝑓, (1) = 𝑓, (1; 𝑖) = 1 + 𝑖 𝑓+ (1) = 𝑓+ (1; 𝑖) = 1 + 𝑖 Example: Consider a new financial law, defined as: 𝑓(𝑡) = 1 + 𝛼𝑡 # = 1 + (𝛼𝑡)- What is the meaning of the parameter 𝛼? Since 𝑓(1) must be equal to 1 + interest rate for unit of time, and since 𝑓(1) is equal to 1 + 𝛼 , we have that 𝛼 is the interest rate for unit of time. Whenever there is a financial law represented by an accumulation factor 𝑓(𝑡), or by a discount factor 𝜑(𝑡), the interest rate or discount rate are computing by doing 𝒊 = 𝒇(𝟏) − 𝟏 and it gives 𝟏 us the interest rate (𝑖) or by doing 𝒅 = 𝟏 − 𝝋(𝟏) = 𝟏 − and it gives the discount rate (𝑑) 𝒇(𝒕) Example: Consider a financial law given by 𝑓(𝑡) = 𝑒.- , determine the interest rate for unit of time. 𝑖 = 𝑓(1) − 1 → 𝑖 = 𝑒 /, − 1 𝛿 = 1 + i → 𝛿 = ln (1 + 𝑖) which symbolizes the yield at 1 year. The economic meaning of 𝛿 then is that it is the logarithm return of the investment. 11  / The logarithm return is 𝑙𝑛(𝑀) − 𝑙𝑛(𝐶) = 𝑙𝑛 ( ) + In case 𝑀 = 1 + 𝑖 and 𝐶 = 1 we have that the logarithm / return is 𝑙𝑛 \ + ] = 𝑙𝑛(𝑀) − 𝑙𝑛(𝐶) = 𝑙𝑛(1 + 𝑖) − 𝑙𝑛 (1). The 𝑙𝑛(1) is equal to 0, so we are only left with 𝑙𝑛(1 + 𝑖) which is exactly 𝛿 Speed of Interest accumulation for the two processes: Simple accumulation rule Compound accumulation rule Tis financial law is defined by the This financial law is defined by the accumulation factor: accumulation factor: 𝑓* (𝑡; 𝑖) = 𝑓* (𝑡) = 1 + 𝑖* 𝑡 𝑓+ (𝑡; 𝑖) = 𝑓+ (𝑡) = (1 + 𝑖+ ), 𝑖* =simple interest rate 𝑖+ =compound interest rate For time between 0 and 1 I would prefer simple accumulation law since interest accumulates faster, for time periods greater than 1 I would prefer the compound accumulation law. at time exactly 1 I would be indifferent between the two laws. Properties of compound accumulation process: The only thing different from the simple accumulation process is the term 𝑰𝟏 𝒊 which is the interest rate 𝑡=0→𝑀=𝐶 accumulated over the past interest rate. 𝑡 = 1 → 𝑀 = 𝐶(1 + 𝑖) = 𝐶 + 𝐼" (𝐼" = 𝑖𝐶) 𝑡 = 2 → 𝑀 = 𝐶(1 + 𝑖)# = 𝐶(1 + 𝑖)(1 + 𝑖) = 𝐶 + 𝑖𝐶 + 𝐼" 𝑖 + 𝐼" = (𝐶 + 𝐼" )(1 + 𝑖) → at each period, the interest rate yielded up is added to the principle becoming new capital invested in the operation. This process is known as interest capitalization, and it will produce new interest in the subsequent periods. 12 When using the compound accumulation process, there is no advantage in doing intermediate capitalization. In compound accumulation there is no difference between stopping the operation after one period and restarting it immediately after, and keeping the operation going; this means that compound accumulation leaves no room for arbitrage possibilities. 𝑓+ (𝑡; 𝑖) = 𝑓+ (𝑡( + 𝑡3 ; 𝑖) = (1 + 𝑖),0 4,1 = (1 + 𝑖),0 (1 + 𝑖),1 = 𝑓+ (𝑡( ; 𝑖)𝑓+ (𝑡3 ; 𝑖) The same doesn’t apply to the simple accumulation process, since: 𝑓* (𝑡; 𝑖) = 𝑓* (𝑡( + 𝑡3 ; 𝑖) = 1 + (𝑡( + 𝑡3 )𝑖 < 1 + (𝑡( + 𝑡3 )𝑖 + 𝑡( 𝑡3 𝑖 3 = 𝑓* (𝑡( ; 𝑖)𝑓* (𝑡3 ; 𝑖) Floating interest rate: Under simple accumulation process, when interest rate changes over time (floating), when calculating the amount of money accumulated at the end of the period of 𝑡2 we would calculate: 6 𝑡5 = 𝑡5 − 𝑡5'( 𝑀 = 𝐶 W1 + @X𝑖5 𝑡5 YZ 5)( 𝐶𝑓* (𝑡; 𝑖( , 𝑖3 , 𝑖7 , 𝑖8 ) = 𝐶𝑓* (𝑡; 𝚤̅) → 𝚤̅ = 𝐚𝐯𝐞𝐫𝐚𝐠𝐞 𝐢𝐧𝐭𝐞𝐫𝐞𝐬𝐭 𝐫𝐚𝐭𝐞 Under compound accumulation process, when interest rate changes over time (floating), when calculating the amount of money accumulated at the end of the period of 𝑡2 we would calculate: Production sign 6 ,3 𝑡5 = 𝑡5 − 𝑡5'( 𝑀 = 𝐶 [X1 + 𝑖5 Y 5)( 𝐶𝑓* (𝑡; 𝑖( , 𝑖3 , 𝑖7 , 𝑖8 ) = 𝐶𝑓* (𝑡; 𝚤̅) → 𝚤̅ = 𝐚𝐯𝐞𝐫𝐚𝐠𝐞 𝐢𝐧𝐭𝐞𝐫𝐞𝐬𝐭 𝐫𝐚𝐭𝐞 13  Complex Financial Operations Any kind of operations that involves several investments made regularly to get the final amount + interest. Annuity is a sequence of payments with fixed frequency (annual or monthly fixed payments in the bank account), the amount could either be fixed about or change over time but have to be paid at fixed time periods. An annuity is something like a complex operation. If the inflows of money are made not in fixed frequencies of time (say some annual and some monthly), that is not an annuity (ex: I cash-in €10 at time 1, €10 at time 2, €10 at time 1,5 and €10 at time 3, this is not an annuity). The term “annuity” originally referred to annual payments but is now used also for fixed monthly payments. Among the class of all possible annuities, are those where the amount paid at every period is constant, called a certain annuity. Examples are mortgages when buying a house, since the bank makes all the calculations of the payments that must be done and split them in fixed amounts over fixed periods. Whenever the terms of payments depend on some other events (when in finance we talk about financial derivatives, which is a contract that inflows an exchange of money and whose value depends on how other economic variables behave), is called a contingent annuity. Examples are life insurance policies. Valuing a contingent annuity requires using the use of probabilities. Terminology (of annuities): Payment period: time between two consecutive payments (in a monthly annuity, the payment period is the month). Mortgages are paid at the beginning of the payment period; salaries instead are paid at the end of the payment period. Term: the time from the beginning of the first payment period to the end of the last payment period. Ordinary Annuity: payments are made at the end of the payment period (salaries). Annuity Due: payments are made at the beginning of the payment period (mortgages). Deferred Annuity: the term of annuity is not to begin until some time in the future (could be ordinary or due). a) Annual Rent: the sum of payments made during 1 year. 14  We will refer to Ordinary Annuities in the next pages Ordinary annuities Accumulated value discount factor ⟶ the future payments made in The analysis of annuities relies on formula for geometric series/sums: the future need to be discounted, so this x is the discount factor, which is between 0 and 1, this is why we consider this formula ! " # * 4 𝑥 26" − 1 1 𝑥 + 𝑥 + 𝑥 + 𝑥 + ⋯+ 𝑥 = log 2→64 → when x ∈ (0; 1) we have = 𝑥−1 1−𝑥 Finite geometric sums: 2 𝑥 26" − 1 𝑥 ! + 𝑥" + 𝑥 # + 𝑥 * + ⋯ + 𝑥 2 = 1 + 𝑥 + 𝑥 # + 𝑥 * + ⋯ + 𝑥 2 = d 𝑥 2 = 𝑥−1 78! In our financial operations we have: 2 𝑥 26" − 1 𝐶𝑥 + 𝐶𝑥 + 𝐶𝑥 + 𝐶𝑥 + ⋯ + 𝐶𝑥 = 𝐶 d 𝑥 2 = 𝐶 ! " # * 2 𝑥−1 78! #!! :" If 𝑥 = 2 and 𝑛 = 10, we have 2! + 2" + 2# + ⋯ + 2"! = ∑"! 9 78! 2 = #:" = 2047. The closed form ; "#! :" solution 𝐶 makes it easier to compute complicated calculations. ;:" If we consider an ordinary annuity paying 1 unit of capital (€1) at the end of every period for 𝑛 periods. The accumulated value of this annuity at the end of the nth period, with a periodic interest of 𝑖, is called the amount of the accumulated value and is denoted with 𝑺𝒏|𝒊. Set the discount sector: 𝑢 = 1 + 𝑖, we have: 𝑆#|% > 𝑛 because 𝑖 > 0 𝑆#|% = 𝑛 Not defined for 𝑖 = 0, since the only if 𝑖 = 0 formula would become just a First €1 paid at time 1, capitalized for sum of 𝑛 1s. n-1 periods. The second €1 paid at time 2 would be capitalized for n-2 periods in order to capitalize it to n. 𝑆2|) = 1(1 + 𝑖)2:" + 1(1 + 𝑖)2:# + ⋯ + 1(1 + 𝑖)2:(2:") + 1 "$! 6?"$% 6⋯6?6" if using "u" instead of the accumulation factor → 𝑢 = (1 + 𝑖)? 2:" ! " 2:# 2:" 𝑢2 − 1 𝑢2 − 1 (1 + 𝑖)2 − 1 = 𝑢 + 𝑢 + ⋯+ 𝑢 +𝑢 = d 𝑢7 = = = = 𝑆2|) 𝑢−1 𝑖 𝑖 78! 𝑢 =1+𝑖 →𝑢−1=𝑖 If we have instalments of more than €1, then the accumulated value would have to be multiplied by the value of each instalment (R). 15 Example 1: Assume that 𝑖 = 0,1 and the annuity is structure as follow: ("6!,")!& :" The accumulated value would so be: 𝑆2|) = !," = €15,94 = 𝑆"!|!," 𝑛=1 𝑖 = 10% Note that the accumulated value cannot be, say, €9; this is because the sum of the pure installments is €10, so the accumulated value at the end of the annuity cannot be lower than €10. Example 2: Consider 𝑖 = 0,1 and an annuity as follow: The accumulated value would be 100 × 𝑆"!|!," = 100 × €15,94 = €1594,00 Present value Consider an ordinary annuity paying one unit of capital (€1) at the end of any period for 𝑛 periods. 𝑎#|% < 𝑛 The present value of this annuity at the beginning of the first period, with a periodic interest rate 𝑖, is because 𝑖 > 0 called the present value and is denoted with 𝒂𝒏|𝒊. " Set the discount factor: 𝜈 = "6), we have that: 1𝑣 + 1𝑣 # + ⋯ + 1𝑣 2:" + 1𝑣 2 = 𝑣 + 𝑣 # + ⋯ + 𝑣 2:" + 𝑣 2 = 𝑣(1 + 𝑣 + ⋯ + 𝑣 2:" + 𝑣 2:" ) 2:" = 𝑣(𝑣 ! + 𝑣 + ⋯ + 𝑣 2:" + 𝑣 2:" ) = 𝑣 d 𝑣 7 78! 2 2 𝑣 −1 1 𝑣 −1 1 𝑣2 − 1 = 𝑣m n= o p= o p 𝑣−1 1+𝑖 1 1+𝑖 1−1−𝑖 1+𝑖 − 𝑖 1+𝑖 1 𝑣= 1+𝑖 1 − 𝑣 2 1 − (1 + 𝑖):2 If we have instalments of more than €1, then the present value = = = 𝑎2|) 𝑖 𝑖 would have to be multiplied by the value of each instalment (R). Remark: Since the present value of the annuity is the value of the annuity at time 0, and the accumulated value of the annuity is the value of the annuity at time 𝑛, we could see the present value as being the 𝐶, and the accumulated value as being the 𝑀: 1 𝑀 = 𝐶𝑓(𝑡) → 𝐶 = 𝑀 𝑓(𝑡) 1 𝑎6|$ = 𝑆 (1 + 𝑖)6 6|$ 𝑆6|$ = ((1 + 𝑖)6 )𝑎6|$ 16  Example: Given an interest rate 𝑖 = 0,01, consider the following annuity: The present value would be: 1 − (1 + 𝑖):2 1 − (1,01):K! 20S𝑎2|) T = 20 r s = 20 r s = €656,69 𝑖 0,01 𝑛 = 40 𝑖 = 0,01 When the interest rate refers to a unit of time different from the payment period (time lapses between the payment of one instalment and the payment of the next instalment: Example: Here we have 20 installments (10 x 2), and the annual compound interest rate is 𝑖 = 5%. The present-value can be computed as: :2 1 − S1 + 𝑖 (#) T 1 − (1 + 0,0247)):#! 2(𝑎2|) (%) ) = 2 u v = 2r s = €31,27 𝑖 (#) 0,0247 This 𝑖, is not the annual compound interest rate 𝑖 = 0,05, since the installments are paid every 6 months. We have to use the semiannual interest rate in this example, 𝑖 (") means that we have 2 installments in 1 year. To find this we have to: "$ " +1 + 𝑖 ("). = (1 + 𝑖)$ → +1 + 𝑖 ("). = (1 + 𝑖) → 𝑖 (") = √1 + 𝑖 − 1 = 𝟎, 𝟎𝟐𝟒𝟕 Example: :2 1 − S1 + 𝑖 ("#) T 1 − (1 + 0,004074):#! 2(𝑎2|) (!%) ) = 2 u v = 2r s = €38,34 𝑖 ("#) 0,004074 Just like as before, this 𝑖, is not the annual compound interest rate 𝑖 = 0,05, since the installments are paid every month. We have to use the monthly interest rate, 𝑖 (%") which means that we have 12 installments in 1 year. To find this we have to: %"$ %" % +1 + 𝑖 (%"). = (1 + 𝑖)$ → +1 + 𝑖 (%"). = (1 + 𝑖) → 𝑖 (%") = (1 + 𝑖)%" − 1 = 𝟎, 𝟎𝟎𝟒𝟎𝟕𝟒 Equivalent interest rate for compound accumulation process 𝑖8 (#) → interest rate # )1 + 𝑖8 (#) , = (1 + 𝑖8 ) compounded 𝑛 times in # (9) 9 a year. ((3) )1 + 𝑖8 (#) , = /1 + 𝑖8 0 𝑖+ = monthly (once every month) interest rate (?) (@) 4𝑡 (>?) /1 + 𝑖8 0 − 1 𝑖 + 12𝑡𝑖< = /1 + 𝑖8 0 → 𝑖< = 12𝑡 As shown, the equivalence depends on time 𝑡. Annuity due: It is like an ordinary annuity, but the payment of each instalment is made at the beginning od the payment period. There are still 𝑛 instalments like in an ordinary annuity, but the first one is at time 𝑡 = 0, and the last one at time 𝑡 = 𝑛 − 1. The value of the annuity due at time 𝑡 = −1 is equal to the evaluation of the ordinary annuity at time 𝑡 = 0. Since in the annuity due we have the first payment at time 𝑡 = 0, and the last at time 𝑡 = 𝑛 − 1, the formulas to calculate the present and accumulated value change a bit. The formula to compute the present value of the annuity due: 𝑎̈ +𝑛=𝑖. = 𝑎+𝑛=𝑖. (1 + 𝑖) This formula is denoted as: 𝒂̈ +𝒏=𝒊. The same applied under the formula to compute the final value (accumulated value) of the annuity: 𝑠̈ +𝑛=𝑖. = 𝑠+𝑛=𝑖. (1 + 𝑖) This formula is denoted as: 𝒔̈ +𝒏=𝒊. 𝑠̈ +𝑛=𝑖. is 𝑠+𝑛=𝑖. capitalized for 1 period, on the other hand 𝑠+𝑛=𝑖. is 𝑠̈ O𝑛P𝑖Q discounted for 1 period. Same applied for the present value. 18  𝑖 periodic interest rate Annuity due as follow: Period = payment period Present value is computed by: 1 1 1 1 1 # 1 2:" 𝑅 +𝑅m n+𝑅m n + ⋯ + 𝑅 m n = 𝑅(1 + + m n + ⋯ + m n 1+𝑖 (1 + 𝑖)# (1 + 𝑖)2:" 1+𝑖 1+𝑖 1+𝑖 " ! " Note that 1 = \"6)] and 𝑣 = "6) 1 2 1 − 𝑣 2 1 − \1 + 𝑖 ] 1 − (1 + 𝑖):2 = 𝑅(𝑣 ! + 𝑣 " + 𝑣 # + ⋯ + 𝑣 2:" ) = 𝑅 =𝑅 =𝑅 1−𝑣 1 𝑖 1−1+𝑖 1+𝑖 1 − (1 + 𝑖):2 = 𝑅(1 + 𝑖) r s 𝑖 = 𝑅(1 + 𝑖)𝑎2|) = 𝑅𝑎̈ O𝑛P𝑖 Q Accumulated value is: 𝑅(1 + 𝑖)2 + 𝑅(1 + 𝑖)2:" + ⋯ + 𝑅(1 + 𝑖)" = 𝑅(𝑢2 + 𝑢2:" + ⋯ + 𝑢" ) Note that 𝑢 = 1 + 𝑖 1 − 𝑢2 1 − (1 + 𝑖)2 (1 + 𝑖)2 − 1 = 𝑅𝑢(𝑢2:" + 𝑢2:# + ⋯ + 𝑢! ) = 𝑅𝑢 m n = 𝑅(1 + 𝑖) r s = 𝑅(1 + 𝑖) r s 1−𝑢 1 − (1 + 𝑖) 𝑖 = 𝑅(1 + 𝑖)𝑠2|) = 𝑅𝑠̈O𝑛P𝑖 Q Deferred annuity: There are two types of deferred annuities, they could be of an ordinary time or annuity due. The annuity starts after the deferred time “𝑚”. The present value of the two types of deferred annuities is the following: Ordinary deferred annuity The value of the ordinary deferred annuity at time 𝑚 is: 𝑎2|) (1 + 𝑖):N ⟶ which is the present value discounted for 𝑚 periods. This calculation is denoted as = S|T ('((4$)VS m𝒂𝒏|𝒊 = ((4$) U = $((4$)U Deferred annuity due While the value of a deferred annuity due at time 𝑚 is 𝑎̈ O𝑛P𝑖 Q(1 + 𝑖):N ⟶ which is equal to the present value discounted in order to get it at time 0. This calculation is denoted as =̈ S|T = ('((4$)VS m𝒂̈ 𝒏|𝒊 = ((4$) U = ((4$)S|TUV0 = $((4$)UV0 19  An annuity due deferred of 𝒎 periods is equal to an ordinary annuity deferred of 𝒎 − 𝟏 periods. % m𝑎!|# = 𝑎!|# 𝑣 $ → 𝑣 = %&# Accumulated value: Deferred ordinary annuity Ordinary annuity ("6))" :" = 𝑅" (1 + 𝑖)2:" + 𝑅# (1 + 𝑖)2:# + ⋯ + 𝑅2 (1 + 𝑖)2 − 1 m𝑠2|) = 𝑠2|) = = 𝑅" (1 + 𝑖)2:" + 𝑅# (1 + 𝑖)2:# + ⋯ + 𝑅2 ) 𝑖 𝑠6|$ is the value of 𝑅" = 𝑅# = ⋯ = 𝑅2 capitalized at time 𝑛. So 𝑅" is capitalized for (𝑛 − 1) periods etc. m𝑠!|# = 𝑠!|# m𝑠̈ '𝑛 (𝑖 ) = 𝑠̈ '𝑛(𝑖 ) Deferred annuity due Annuity due m𝑠̈ O𝑛P𝑖 Q = 𝑠̈ O𝑛P𝑖Q = 𝑠2|) (1 + 𝑖) = 𝑎2|) (1 + 𝑖)26" (1 + 𝑖)2 − 1 𝑠̈ O𝑛P𝑖 Q = = 𝑠2|) (1 + 𝑖) = 𝑎2|) (1 + 𝑖)26" 𝑖 1+𝑖 Example: Annual compound interest rate 𝑖 = 0,05. Consider the following annuity (ordinary): There are 𝑛 = 20 installments, and the annuity lasts 10 years. " 𝑠𝑒𝑚𝑖𝑎𝑛𝑛𝑢𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 ⟶ 𝑖 (#) = (1 + 0,05)# − 1 = 0,0247 2𝑎2|) = 2𝑎#!|!,!#KW = €31,27 Assume that the same annuity is deferred of 3 periods (first installments paid at time 𝑡 = 3.5 years ⟶ be careful that in this case our 𝑚 would be 6 not 3, since we are using the semiannual interest rate). X %&|&,&%01 m𝑎2|) = 6𝑎#!|!,!#KW = ("6!,!#KW) 2 20 Perpetuities: Perpetuities are annuities continuing forever. Unitary ordinary perpetuity ⟶ present value: ":("6))$" " lim 𝑅𝑎2|) = lim 𝑅 = 𝑅 = 𝑅𝑎64|) 2→64 2→64 ) ) Unitary perpetuity due ⟶ present value: ":("6))$" " "6) lim 𝑅𝑎̈ 2|) = lim 𝑅 ) (1 + 𝑖) = 𝑅 ) + 1 = 𝑅 ) = 𝑅𝑎̈ 64|) 2→64 2→64 Deferred ordinary perpetuity ⟶ present value (at time 0 (discounted for 𝒎 periods)) "|3X ":("6))$" " lim 𝑅m𝑎2|) = lim 𝑅 ("6)) 4 = lim 𝑅 = 𝑅 )("6))4 2→64 2→64 2→64 )("6))4 Deferred perpetuity due ⟶ present value (at time 0 (discounted for 𝒎 periods)) 𝑎̈ 𝑛|𝑖 𝑎𝑛|𝑖 1−(1+𝑖)−𝑛 1 lim 𝑅m𝑎̈ 2|) = lim 𝑅 (1+𝑖)𝑚 = lim 𝑅 (1+𝑖)𝑚−1 = lim 𝑅 𝑖(1+𝑖)𝑚−1 = 𝑅 𝑖(1+𝑖)𝑚−1 2→64 2→64 2→64 2→64 The accumulated value of a perpetuity DOESN’T EXIST, since it is an annuity that goes forever. Financial Laws Financial laws are (in the more general form) functions of 3 variables and depend on a parameter. 𝐹(𝐶; 𝑡D ; 𝑇; 𝛽) = 𝑀 𝐹 establishes a Ginancial equivalence between 𝐶 available in 𝑡! and 𝑀 available in 𝑇. Properties: In order to be considered a financial law, 𝐹 must satisfy 3 principles: a) Must be homogeneous of 1st degree with respect to C. 𝐹(𝐶; 𝑡! ; 𝑇; 𝛽) = 𝐶𝐹(1; 𝑡! ; 𝑇; 𝛽) 𝑓(𝑥) = 𝑥𝑓(1) Otherwise (if 𝐹 is not of degree 1), the market will not be efficient. I have the possibility to invest in 𝑡! of €100 and cash-in €200 in 𝑇 (financial operation 1), and if I go to a bank and deposit €10 and get €15 at time 𝑇 (financial operation 2). In the end, there is 1 financial operation (the second) that is more expensive than the other (the first one). 21  The aim is to have €200 at time 𝑇, it is cheaper to undergo financial operation 1 than opening #!! "[ = 13,3 financial operation 2. In the financial market, treaders always buy the cheaper operation and sell the expensive one, this creates arbitrage opportunities. To buy the 1st financial operation (go long) I would have to spend €100 and get €200 at time 𝑇. To spend those €100 I need to go short (sell) 10 times the 2nd financial operation (getting 10 times the €10 initial capital required for financial operation 2). To sell those 10 financial operations 2 I would have to give out 10 times the €15 of 2nd financial operation to the treader(s) that bought those 10 financial operations 2 at the beginning. Making the calculations, I would finally spend €150 (€15x10) and getting €200 at time 𝑇, earning €50. You always go long (buy) the cheapest operation and go short (sell) the more expensive one. At maturity, you would have to pay back the treader(s) that bought the operation initially. This arbitrage opportunity creates more demand, and so higher prices for those 2 financial operations, over time those €50 of profit will be eaten out. So 𝐹(1; 𝑡! ; 𝑇; 𝛽) is usually defined by 𝐹(𝑡! ; 𝑇; 𝛽) and 𝐹(𝐶; 𝑡! ; 𝑇; 𝛽) is usually defined by 𝐶𝐹(𝑡! ; 𝑇; 𝛽) b) 𝑭(𝒕𝟎 ; 𝒕𝟎 ; 𝜷) = 𝟏 ⟶ the financial operation starting at 𝑡! and ending at 𝑡! gives back the same amount invested. c) Must be monotonically increasing with respect to 𝑻.](-& ;_;`) If F is differentiable with respect to 𝑇, this condition can also be stated as._ >0 If 𝐹(𝑡! ; 𝑡! ; 𝛽) is not always monotonically increasing in T, there is arbitrage opportunity. Suppose €100 are invested in 𝑡! , in 𝑇 €150 will be given back (financial operation 1). In financial operation 2, at time 𝑡! €100 are invested but at 𝑇" , (smaller than 𝑇) you can cash in €200. The 2nd financial operation is cheaper than the 1st one. 100 𝑥 𝐹(𝑡! ; 𝑇; 𝛽) = 150 100 𝑥 𝐹(𝑡! ; 𝑇" ; 𝛽) = 200 In order for this to be possible (have both equations satisfied) it must be that 𝐹(𝑡! ; 𝑇" ; 𝛽) < 𝐹(𝑡! ; 𝑇; 𝛽) with 𝑇 < 𝑇". So 𝐹 is decreasing in 𝑇, leaving arbitrage opportunity. 22 Example % Is 𝐹(𝑡! ; 𝑇; 𝛽) = 𝑒 `(_:-&) a financial law? Check if this operation is similar to a simple or compound accumulation law (in this case it is not). Check the 3 properties: o The operation does not depend on C, so the 1st property is automatically satisfied. o Property 2: check if 𝐹(𝑡! ; 𝑡! ; 𝛽) = 1 % % This property is satisfied since: 𝐹(𝑡! ; 𝑡! ; 𝛽) = 𝑒 `(-&:-& ) = 𝑒 `(!) = 𝑒 ! = 1 o The 3rd property implies that 𝐹 must be monotonically increasing in 𝑇: not that %.](-& ;_;`) % 𝑒 `(_:-& ) is differentiable in T, then._ = 2 𝛽(𝑇 − 𝑡! )𝑒 `(_:-&) > 0 % Moreover, 𝐹(𝑡! ; 𝑇; 𝛽) = 𝑒 `(_:-&) is a financial law that depends only on 𝑡 = 𝑇 − 𝑡! so it can be reduced to a financial law in one single value: 𝐹(𝑡! ; 𝑇; 𝛽) = 𝐹(𝑡; 𝛽). Remark: with simple accumulation rule 𝐹(𝑡! ; 𝑇; 𝑖, ) = 1 + 𝑖, (𝑇 − 𝑡! ) which becomes = 1 + 𝑖, 𝑡 = 𝐹(𝑡; 𝑖, ) if 𝑡 = 𝑇 − 𝑡!. Same applies with the compound accumulation rule 𝐹(𝑡! ; 𝑇; 𝑖+ ) = (1 + 𝑖+ )_:-& which becomes = (1 + 𝑖+ )- = 𝐹(𝑡; 𝑖+ ) if 𝑡 = 𝑇 − 𝑡!. Continuation Accumulation Factor: 𝑓(𝑥) = 𝑓(𝑡" ) = accumulated value at time 𝑡" = 𝑥 𝑓(𝑦) = 𝑓(𝑡# ) = accumulated value at time 𝑡# = 𝑦 𝑓(𝑥)𝑓(𝑦 − 𝑥) = 𝑓(𝑡" )𝑓(𝑡# − 𝑡" ) = accumulated value at time 𝑡# = 𝑦 when the Ginancial operation is stopped in t" = 𝑥 and started again immediately after at the same conditions. The problem with this is that sometimes, 𝑓(𝑡# ) is different than 𝑓(𝑡" )𝑓(𝑡# − 𝑡" ). How should I modify in 𝐭 𝟏 the way in which interests are accumulated from 𝐭 𝟏 to 𝐭 𝟐 in order to have that €1 invested for 𝐭 𝟐 years generates the same amount of money of €1 invested till 𝐭 𝟏 and reinvested from 𝐭 𝟏 to 𝐭 𝟐 ? This is guaranteed by a two-variable financial law 𝐹(𝑥; 𝑦) defined as follows: 𝑓(𝑥)𝐹(𝑥; 𝑦) = 𝑓(𝑦) 𝑓(𝑦) 𝐹(𝑥; 𝑦) = → 𝐜𝐨𝐧𝐭𝐢𝐧𝐮𝐚𝐭𝐢𝐨𝐧 𝐚𝐜𝐜𝐮𝐦𝐮𝐥𝐚𝐭𝐢𝐨𝐧 𝐟𝐚𝐜𝐭𝐨𝐫 𝑓(𝑥) 23 Example Consider the simple accumulation rule from 0 to 𝑡#. 𝑓(𝑡# ) = 1 + 𝑖, 𝑡# If this operation is stopped at 𝑡" , the amount available at 𝑡" is 𝑓(𝑡" ) = 1 + 𝑖, 𝑡". If I then reinvest until 𝑡# , I obtain 𝑓(𝑡" )𝑓(𝑡# − 𝑡" ) = (1 + 𝑖, 𝑡" )(1 + 𝑖, (𝑡# − 𝑡" )) which is clearly different (greater) than 𝑓(𝑡# ). 𝑓(𝑡# ) 1 + 𝑖, 𝑡# 𝑓, (𝑡" ; 𝑡# ) = = → continuation accumulation factor for simple accumulation rule 𝑓(𝑡" ) 1 + 𝑖M 𝑡" Is 𝒇𝑺 (𝒕𝟏 ; 𝒕𝟐 ) a financial law? The continuation accumulation factor is a financial law: o The operation does not depend on C, so the 1st property is automatically satisfied. "6)5 -! o 𝑓, (𝑡" ; 𝑡" ) = =1 "6)6 -!.b5 (-! ;-% ) )6 o = "6) >0.-% 6 -! It is a financial law that is not reducible to a one single variable financial law ⟶ 𝑡 = 𝑡# − 𝑡". It is used to redetermine how interest is accumulated from 𝑡" to 𝑡# , in order to preserve the amount of money that would have been initially accumulated in 𝑡#. You would have to accumulate until 𝑡" using the simple accumulation financial law, while from 𝑡" to 𝑡# (if operation is stopped at 𝑡" ) has to follow the continuation accumulation financial law. Under the compound accumulation rule, there is no need to change the financial law if operation is stopped at 𝑡" and reinvested until 𝑡#. 𝑓+ (𝑡) = (1 + 𝑖+ )- 𝑓+ (𝑡# ) (1 + 𝑖+ )-% 𝐹+ (𝑡" ; 𝑡# ) = = = (1 + 𝑖+ )-%:-! = (1 + 𝑖+ )- = 𝑓+ (𝑡) 𝑓+ (𝑡" ) (1 + 𝑖+ )-! Instantaneous force of interest: Any financial law 𝑓(𝑡) such that 𝐹(𝑡" ; 𝑡# ) = 𝑓(𝑡# − 𝑡" ) is a decomposable financial law. That is, stopping the operation and starting immediately after does not impact the final value. b(-6c):b(-) " b(-6c):b(-) This is because the interest rate accumulates at a constant rate (lim b(-)c = b(-) lim c = c→! c→! b 7 (-). b(-) =.- ln 𝑓(𝑡) = 𝜹𝒕 → 𝐢𝐧𝐬𝐭𝐚𝐧𝐭𝐚𝐧𝐞𝐨𝐮𝐬 𝐟𝐨𝐫𝐜𝐞 𝐨𝐟 𝐢𝐧𝐭𝐞𝐫𝐞𝐬𝐭). 24 Let 𝑓(𝑡) be the accumulation factor that describes the financial law (generic). Let 𝐹(𝑡, 𝑡 + ℎ) be the continuation accumulation factor: b(-6c) 𝐹(𝑡, 𝑡 + ℎ) = b(-) b(-6c):" The force of interest is 𝑖(𝑡, 𝑡 + ℎ) = c → 1 is the 𝑓(𝑡) ](-,-6c):" b(-6c):b(-) " b(-6c):b(-) b 7 (-). lim 𝑖(𝑡, 𝑡 + ℎ) = lim = lim = lim = = ln 𝑓(𝑡) = 𝜹𝒕 c→! c→! c c→! b(-)c b(-) c→! c b(-).- In case 𝑓 is differentiable The instantaneous force of interest indicates the speed at which interest is Force of interest for simple accumulation rule: gets accumulated. 𝑓, (𝑡) = 1 + 𝑖, 𝑡. b 7 (-) ) 𝛿, (𝑡) =.- ln 𝑓, (𝑡) = b5 (-) = "6)5 - → force of interest depends on time. 5 5 The speed at which interest accumulates is maximum at the beginning, and declines over time, so it’s more efficient to stop the operation and start it again before it starts to decline. Force of interest for compound accumulation rule: 𝑓+ (𝑡) = (1 + 𝑖+ )-. b 7 (-) ("6)8 )9 de ("6)8 ) 𝛿+ (𝑡) =.- ln 𝑓+ (𝑡) = b8 (-) = ("6)8 )9 = ln (1 + 𝑖+ ) → force of interest doesn’t depend on time. 8 Not convenient to withdraw money at time (for example) 𝑡" , since if I reload money in the bank account for a further period, the interest I earn will be the same (the speed at which interest accumulates remains the same. Decomposability of a financial law: Consider the financial law described by 𝑓. Moreover, consider two instants of time (𝑡" , 𝑡# ) such that 𝑡# > 𝑡". Then the financial law is decomposable when 𝑓(𝑡" )𝑓(𝑡# − 𝑡" ) = 𝑓(𝑡# ). So, stopping the operation and restarting it until 𝑡# is equivalent to keeping it going until 𝑡#. Theorem: a one-variable financial law is decomposable if an only if its force of interest is constant (it doesn’t depend on time). From the theorem above, we have that the compound accumulation rule is a decomposable financial law, while the simple accumulation rule is not. 25  Theorem: any financial law is identified in a unique way by either an accumulation factor 𝑓(𝑡) or by its force of interest. 𝛿(𝑡) ⟺ 𝑓(𝑡) The force of interest has a relation 1 to 1 with the accumulation factor. Indeed, / 𝑓(𝑡) = 𝑒 L6M(,) = 𝑒 ∫/,L6M(,)O, = 𝑒 ∫ /(,)O, The compound accumulation rule is the unique one-variable financial law with constant force of interest, so it’s the unique one-variable financial law that is decomposable. Example Determine the accumulation factor related to the financial law described by the force of interest 𝛿(𝑡) = 𝑙𝑛 (1 + 𝑖+ ). 9 𝑡 𝑓(𝑡) = 𝑒∫ 𝛿(𝑡)𝑑𝑡 → 𝑓(𝑡) = 𝑒∫ g2 ("6)8)𝑑𝑡 = 𝑒[g 2("6)8 )]- = 𝑒g 2("6)8 ) = S𝑒lnO1+𝑖𝐶 Q T = (1 + 𝑖𝐶 )𝑡 → Compound accumulation rule (it was predictable, since the force of interest given didn’t depend on time 𝑡, and There is no 𝑡 inside the logarithm, so the solution the compound accumulation is just that constant multiplied by 𝑡 rule is the only law whose force of interest doesn’t depend on 𝑡) Compound accumulation rule defined by means 𝑒 /, of force of interest (𝛿 = ln(1 + 𝑖! )) 𝑓+ (𝑡) = Compound accumulation rule defined as a (1 + 𝑖+ )- function of the interest rate for unit of time. Example Determine the accumulation factor related to the financial law described by the force of interest )5 𝛿(𝑡) = "6)5 - ) ∫"6)5 -𝑑𝑡 𝑓(𝑡) = 𝑒∫ 𝛿(𝑡)𝑑𝑡 → 𝑓(𝑡) = 𝑒 5 = 𝑒lnO1+𝑖𝑆𝑡Q = 1 + 𝑖𝑆 𝑡 → Simple accumulation rule Example Determine the accumulation factor related to the financial law described by the force of interest 𝛿(𝑡) = 2𝛽𝑡 % 𝑓(𝑡) = 𝑒∫ 𝛿(𝑡)𝑑𝑡 = 𝑒∫ #`-𝑑𝑡 = 𝑒`- End Module I 26  Market of Riskless Bond Generally, when you buy a financial instrument (bond) saying that it is riskless means that there is no credit risk. Zero coupon bond: It is a legal contract between a party which issues the coupon bond and the party that lends money. This type of coupon bond is characterized by: Face value (FV): amount of money that the party who issue the bond promises to pay back at maturity. Maturity (T) Time at which ZCB is issued (t) 𝑇 − 𝑡 = time to maturity. We usually set 𝑡 = 0, so maturity and time to maturity are the same in this case. The timeline to indicate a ZCB is the following: Note that: 𝐹𝑉 − 𝑃(!) (𝑡; 𝑇) = the interest offered by the bond. (D) 𝐹𝑉 > 𝑃P+Q (𝑡; 𝑇) How to compute the annual compound interest rate implicit in the price of a ZCB: (.) 𝑃*+, (0; 𝑇)(1 + 𝑖+ )0 = 𝐹𝑉 % 0 𝐹𝑉 𝑖+ = 3 (.) 4 −1 𝑃*+, (0; 𝑇) There are 2 types of ZCB issued by the government: BOT (buoni ordinari del tesoro) ⟶ traded in the MTO (mercato telematico obbligazioni) market. Classical Italian ZCB. Stripped BTP: takes the coupons out of the bond. It is a coupon bond from which you take away the coupons. They are not issued by the government but are usually traded over the counter. The problem with trading over the counter is liquidity (difficult to liquidate the bond since the everything is exchanged between private traders with no regulations). 27  A Zero-Coupon Bond (ZCB) with face value FV and maturity T is a financial instrument. Financial instruments are classified in two categories: fixed-income financial instruments (what the ZCB is ⟶ know exactly what I will get at maturity) and non-fixed income financial instruments. A ZCB gives you the right to have €FV at the future time (maturity) T. if the ZCB is sold before maturity, it is not a fixed-income financial instrument. Pure-discount Bond: is a ZCB with unitary face value, so its face value is equal to €1. When this happens, its current price is also denoted by 𝑉 (!) (0; 𝑇) which is the current market price of €1 available at time T (market discount factor). Example: The market quotes the following ZCBs: 𝑇 (*) 𝑃+&, (0; 𝑇) 𝐹𝑉 𝑉 (*) (0; 𝑇) (!) 𝑃o+p (0; 𝑇) 𝑉 (!) (0; 𝑇) = 𝐹𝑉 You would first put the different market discount factors computed in a graph, to derive the market discount curve (curve which puts in relations maturities with market discount factors). This curve helps us make estimations of market discount factors for bonds with different maturities. The curve is downward sloping in an arbitrage-free market (if it was the opposite, there would be arbitrage opportunities). ARBITRAGE STRATEGY Let’s assume that from 𝑡# to 𝑡* the market discount curve is not downward sloping, but upward sloping, and see how we can make arbitrage. Arbitrage tip: if a bond with longer maturity is more expensive than a bond with shorter maturity, go long on the cheap one (buy) and go short on the expensive one (sell). 28  A ZCB is a fixed-income financial instrument in the sense that if you keep it in your portfolio until maturity, you will have a guaranteed return of: % 0 𝐹𝑉 𝑖+ = 3 (.) 4 −1 𝑃*+, (0; 𝑇) If you sell it before maturity, the return is unknown (non-fixed income instrument). Consider a ZCB with maturity 10 years and face value €1. At the current time, its price is €0,90. (-) Therefore, 𝑃o+p (𝑡; 2034). If you wait 10 years, the price of the bond must go back to €1. Example: Determine the current value of a coupon bond paying coupon rates of 5% with annual frequency, maturity is 5 years, and face value (FV) €100. Knowing that the market is made of the following ZCBs: ZCB Maturity Current Price Face Value A 1 €98 €100 B 2 €950 €1000 C 3 €0,92 €1 D 4 €90 €100 E 5 €0,88 €1 F 6 €850 €1000 The difference between a coupon bond and a ZCB is that a coupon bond, between the initial payment and the maturity, pays coupons. Coupon Rate x FV Coupon (c) = Frequency of Coupon Payments What is the arbitrage-free price of the following coupon bond? A coupon bond as above is equivalent to a portfolio of ZCB [(1) maturity 1, FV 5; (2) maturity 2, FV 5; (3) maturity 3, FV 5; (4) maturity 4, FV 5; (5) maturity 5, FV 105]. A portfolio like this makes the same cash flow as the above coupon bond (their prices must be the same) (!) 𝑃+p (0; 5) = [𝑣 (!) (0; 1)]5 + [𝑣 (!) (0; 2)]5 + [𝑣 (!) (0; 3)]5 + [𝑣 (!) (0; 4)]5 + [𝑣 (!) (0; 5)]105 29 Current arbitrage-free price of a ZCB with maturity 1 and face value €5  Noting that: (t) Rqrs (D;T) 𝑣 (D) (0; 𝑇) = UV So, by taking numbers from our table, we compute the single market discount factors (we will then need to multiply those by the FV of each of our problem’s ZCBs): 98 𝑣 (!) (0; 1) = = 0,98 100 950 𝑣 (!) (0; 2) = = 0,95 (Y) 1000 𝑃8X (0; 5) = [𝑣 (Y) (0; 1)]5 + [𝑣 (Y) (0; 2)]5 + [𝑣 (Y) (0; 3)]5 + [𝑣 (Y) (0; 4)]5 + [𝑣 (Y) (0; 5)]105 0,92 𝑣 (!) (0; 3) = = 0,92 1 90 (Y) 𝑣 (!) (0; 4) = = 0,90 𝑃8X (0; 5) = 0,98(5) + 0,95(5) + 0,92(5) + 0,90(5) + 0,88(105) = €111,15 100 0,88 𝑣 (!) (0; 5) = = 0,80 1 Can two ZCBs with the same maturity and different face values have a different market price to face value ratio? ZCB Maturity Current Price Face Value A 1 (Y) €1 𝑃Z (0; 1) B 1 (Y) €100 𝑃X (0; 1) ($) ($) [" (Y;>) [& (Y;>) So, can > ≠ >YY , the answer is NOT IN AN ARBITRAGE-FREE MARKET (otherwise there would be the possibility to create arbitrage). (Y) (Y) 𝑃Z (0; 1) 𝑃X (0; 1) = 1 100 ($) ($) [" (Y;>) [& (Y;>) If it happens that > > >YY then the bond A would be more expensive (go short (sell)), (Y) (Y) and the cheapest would be B (I would go long (buy)). So, (100)𝑃Z (0; 1) − 𝑃X (0; 1) > 0. 30  Term Structure of Interest Rates It is very easy to transform prices of ZCBs into interest rates (annual compound interest rate): (!) 𝑃o+p (0; 𝑡) 1 Market Discount Factor → 𝑣 (!) (0; 𝑡) = = - 𝐹𝑉 S1 + 𝑖 (!) (0; 𝑡)T ( ( ) (') 1 ) 𝐹𝑉 𝐌𝐚𝐫𝐤𝐞𝐭 𝐂𝐨𝐦𝐩𝐨𝐮𝐧𝐝 𝐀𝐧𝐧𝐮𝐚𝐥 𝐈𝐧𝐭𝐞𝐫𝐞𝐬𝐭 𝐑𝐚𝐭𝐞 (at 𝑡 years) → 𝑖 (0; 𝑡) = P (') S − 1 = U (') Y −1 𝑣 (0; 𝑡) 𝑃 (0; 𝑡) *+, Example: Consider a market described by the following ZCBs: ZCB Maturity Current Price Face Value Market Discount Factor A 1 €0,99 €1 𝑣 (Y) (0; 1) = €0,99 B 2 €98 €100 𝑣 (Y) (0; 2) = €0,98 C 3 €95 €100 𝑣 (Y) (0; 3) = €0,95 D 4 €184 €200 𝑣 (Y) (0; 4) = €0,92 E 5 €900 €1000 𝑣 (Y)(0; 5) = €0,90 F 6 €0,88 €1 𝑣 (Y) (0; 6) = €0,88 Derive the market discount factors curve: Derive the market compound annual interest rate curve: % 1 % 𝑖 (#) (0; 1) = w y − 1 = 0,01 𝑣 (#) (0; 1) % ( (#) (0; 𝐹𝑉 𝑖 2) = { (#) − 1 = 0,01015 𝑃&!' (0; 2) 𝑖 (#) (0; 3) = 0,0172447 𝑖 (#) (0; 4) = 0,021 "!! "!! The value of €100 available in 2 years is ⟶ = % it is NOT ≠ % u"6) (&)(!;#)v u"6) (&)(!;")vu"6) (&)(!;#)v This is because those interest rates are the so called: “spot interest rates” from 0 to 2 ⟶ 𝑖 (!) (0; 2). While 𝑖 (!) (1; 2) is the “forward interest rate” from 1 to 2. The equation presented above can be corrected by using forward interest rates: 1 (DD (DD W(4$ (t) (D;3)X (D) The value of €100 available in 2 years is ⟶ = = 1 →𝑖 (1; 2) = —1 W(4$ (t)(D;()XW(4$ (t)((;3)X W(4$ (t) (D;3)X (4$ (t)(D;() Generally, to compute a certain forward interest rate having spot interest rates we compute: ( ,1 ,1 ',0 }1 + 𝑖 (D) (0; 𝑡3 )~ (D) 𝑖 (𝑡( ; 𝑡3 ) = W ,0 Z —1 X1 + 𝑖 (D) (0; 𝑡( )Y 31  Forward Contract Consider a ZCB with maturity 𝑇 and Face Value 𝐹𝑉, then the definition of a forward contract is the following: it is a contract where two counter parties agree to exchange the ZCB at a future time 𝑡 at a price agreed at the current time 0 (therefore agreed before exchange). This price is called the forward price and is denoted by 𝑃(!) (𝑡; 𝑇) from which we can compute the forward market discount factor: 𝑃(!) (𝑡; 𝑇) 𝑉 (!) (𝑡; 𝑇) = 𝐹𝑉 Example: Consider the following market: ZCB Maturity (𝑻) Current Price (𝑷(𝟎) (𝟎; 𝑻)) Face Value Market Discount Factor A 1 €97 €100 𝑣 (Y) (0; 1) = €0,97 B 2 €9,4 €10 𝑣 (Y) (0; 2) = €0,94 C 3 €900 €1000 𝑣 (Y) (0; 3) = €0,90 D 4 €0,87 €1 𝑣 (Y)(0; 4) = €0,87 (𝟎) What would be the forward price of the ZCB “B” exchanged at time 1 (𝑷𝑩 (𝟏; 𝟐))? The answer comes from the assumption of an arbitrage-free market, in which it must be that (Y) (Y) 𝑉 LM MNM(0;MO 1) × 𝑉 LM MNM(1;MO 2) = 𝑉 (Y) LM MNM(0;MO 2) \]^_ `abcd_ efgh^ij_ kah_^b l^bmabe `abcd_ efgh^ij_ kah_^b \]^_ `abcd_ efgh^ij_ kah_^b Therefore, the forward price is given by: (!) 𝑃p (1; 2) 𝑷𝑩 (𝟏; 𝟐) ⟶ 𝑉 0 (0; 1) × 𝑉 0 (1; 2) = 𝑉 0 (0; 2) ⟶ 𝑉 0 (0; 1) × = 𝑉 0 (0; 2) (𝟎) ( ) ( ) ( ) ( ) ( ) 𝐹𝑉 𝑉 0 (0; 2) ( )