Manual for SOA Exam FM/CAS Exam 2.

Transcription

1 Manual for SOA Exa FM/CAS Exa 2. Chapter 1. Basic Interest Theory. c Miguel A. Arcones. All rights reserved. Extract fro: Arcones Manual for the SOA Exa FM/CAS Exa 2, Financial Matheatics. Spring 2009 Edition, available at 1/24

2 Noinal rate of interest When dealing with copound interest, often we will rates different fro the annual effective interest rate. Suppose that an account follows copound interest with an annual noinal rate of interest copounded ties a year of i (), then $1 at tie zero accrues to $(1 + i () ) at tie 1 years. 2/24

3 Noinal rate of interest When dealing with copound interest, often we will rates different fro the annual effective interest rate. Suppose that an account follows copound interest with an annual noinal rate of interest copounded ties a year of i (), then $1 at tie zero accrues to $(1 + i () ) at tie 1 years. The 1 i () year interest factor is (1 + ). 3/24

4 Noinal rate of interest When dealing with copound interest, often we will rates different fro the annual effective interest rate. Suppose that an account follows copound interest with an annual noinal rate of interest copounded ties a year of i (), then $1 at tie zero accrues to $(1 + i () ) at tie 1 years. The 1 i () year interest factor is (1 + ). The ( 1 i () -year ) thly effective interest rate is. 4/24

5 Noinal rate of interest When dealing with copound interest, often we will rates different fro the annual effective interest rate. Suppose that an account follows copound interest with an annual noinal rate of interest copounded ties a year of i (), then $1 at tie zero accrues to $(1 + i () ) at tie 1 years. The 1 i () year interest factor is (1 + ). The ( 1 i () -year ) thly effective interest rate is ( $1 at tie zero grows to $ 1 + i (). ) in one year. 5/24

6 Noinal rate of interest When dealing with copound interest, often we will rates different fro the annual effective interest rate. Suppose that an account follows copound interest with an annual noinal rate of interest copounded ties a year of i (), then $1 at tie zero accrues to $(1 + i () ) at tie 1 years. The 1 i () year interest factor is (1 + ). The ( 1 i () -year ) thly effective interest rate is ( ). $1 at tie zero grows to $ 1 + i () in one year. ( $1 at tie zero grows to $ 1 + i () ) t in t years. 6/24

7 Noinal rate of interest When dealing with copound interest, often we will rates different fro the annual effective interest rate. Suppose that an account follows copound interest with an annual noinal rate of interest copounded ties a year of i (), then $1 at tie zero accrues to $(1 + i () ) at tie 1 years. The 1 i () year interest factor is (1 + ). The ( 1 i () -year ) thly effective interest rate is ( ). $1 at tie zero grows to $ 1 + i () in one year. ( ) t $1 at tie zero grows to $ 1 + i () in t years. ( ) t. The accuulation function is a(t) = 1 + i () 7/24

8 Exaple 1 Paul takes a loan of $569. Interest in the loan is charged using copound interest. One onth after a loan is taken the balance in this loan is $581. (i) Find the onthly effective interest rate, which Paul is charged in his loan. (ii) Find the annual noinal interest rate copounded onthly, which Paul is charged in his loan. 8/24

9 Exaple 1 Paul takes a loan of $569. Interest in the loan is charged using copound interest. One onth after a loan is taken the balance in this loan is $581. (i) Find the onthly effective interest rate, which Paul is charged in his loan. (ii) Find the annual noinal interest rate copounded onthly, which Paul is charged in his loan. Solution: (i) The onthly effective interest rate, which Paul is charged in his loan is i (12) 12 = = %. 9/24

10 Exaple 1 Paul takes a loan of $569. Interest in the loan is charged using copound interest. One onth after a loan is taken the balance in this loan is $581. (i) Find the onthly effective interest rate, which Paul is charged in his loan. (ii) Find the annual noinal interest rate copounded onthly, which Paul is charged in his loan. Solution: (i) The onthly effective interest rate, which Paul is charged in his loan is i (12) 12 = = %. (ii) The annual noinal interest rate copounded onthly, which Paul is charged in his loan is i (12) = (12)( ) = %. 10/24

11 Two rates of interest or discount are said to be equivalent if they give rise to sae accuulation function. Since, the accuulation function under an annual effective rate of interest i is a(t) = (1 + i) t, we have that a noinal annual rate of interest i () copounded ties a year is equivalent to an annual effective rate of interest i, if the rates and a(t) = ( 1 + i () a(t) = (1 + i) t ) t agree. This happens if and only if ( ) 1 + i () = 1 + i. 11/24

12 Exaple 1 John takes a loan of 8,000 at a noinal annual rate of interest of 10% per year convertible quarterly. How uch does he owe after 30 onths? 12/24

13 Exaple 1 John takes a loan of 8,000 at a noinal annual rate of interest of 10% per year convertible quarterly. How uch does he owe after 30 onths? Solution: We find ( ) = 8000 ( ) 10 = In the calculator, we do: 8000 PV 2.5 I/Y 10 N CPT FV. 13/24

14 The calculator TI BA II Plus has a worksheet to convert noinal rates of interest into effective rates of interest and vice versa. To enter this worksheet press 2nd ICONV. There are 3 entries in this worksheet: NOM, EFF and C/Y. C/Y is the nuber of ties the noinal interest is converted in a year. The relation between these variables is C/Y 1 + EFF 100 = 1 + NOM. 100 C/Y You can enter a value in any of these entries by oving to that entry using the arrows: and. To enter a value in one entry, type the value and press ENTER. You can copute the corresponding noinal (effective) rate by oving to the entry NOM ( EFF ) and pressing the key CPT. It is possible to enter negative values in the entries NOM and EFF. However, the value in the entry C/Y has to be positive. 14/24

15 Exaple 2 If i (4) = 5% find the equivalent effective annual rate of interest. 15/24

16 Exaple 2 If i (4) = 5% find the equivalent effective annual rate of interest. Solution: We solve 1 + i = ( ) 4 4 and get i = %. In the calculator, you enter the worksheet ICONV and enter: NOM equal to 5 and C/Y equal to 4. Then, go to EFF and press CPT. To quit, press 2nd, QUIT. 16/24

17 Exaple 3 If i = 5%, what is the equivalent i (4)? 17/24

18 Exaple 3 If i = 5%, what is the equivalent i (4)? ( ) 4 Solution: We solve 1 + i (4) 4 = we get that i (4) = 4 ( ( ) 1/4 1 ) = %. In the calculator, you enter the worksheet ICONV and enter: EFF equal to 5 and C/Y equal to 4. Then, go to NOM and press CPT. To quit, press 2nd, QUIT. 18/24

19 The noinal rate of discount d () is defined as the value such that 1 unit at the present is equivalent to 1 d() units invested 1 years ago, i.e. {1 d () { units at tie 0} 1 unit at tie 1 }. This iplies that {1 unit at tie 0} { 1 1 d() units at tie 1 The accuulation function for copound interest under a the noinal ( rate of discount ) d () convertible ties a year is t. a(t) = 1 d() We have that ( ) ( ) 1 + i = 1 + i () = (1 d) 1 = 1 d (). }. 19/24

20 In the calculator TI BA II Plus, you ay: given i (), find i, by entering i () NOM and C/Y, then in EFF press CPT. given i, find i (), by entering i EFF and C/Y, then in NOM press CPT. given d (), find d, by entering d () NOM and C/Y, then in EFF press CPT. d appears with a negative sign. given d, find d (), by entering d EFF and C/Y, then in NOM press CPT. d () appears with a negative sign. given i, find d, by using the forula i = 1 1 d 1. given d, find i, by using the forula d = i. 20/24

21 Exaple 4 If d (4) = 5% find i. 21/24

22 Exaple 4 If d (4) = 5% find i. ( Solution: We solve 1 d(4) 4 ) 4 = 1 + i to get d = % and i = %. In the calculator, in the ICONV worksheet, we enter 5 in NOM, 4 in C/Y and we find that EFF is %, then we do % + 1 = 1/x 1 = to get i = %. 22/24

23 Exaple 5 If i = 3% find d (2). 23/24

24 Exaple 5 If i = 3% find d (2). ( Solution: We solve for d (2) in 1 d(2) 2 that d = % doing 3 % + 1 = 1/x 1 = ) 2 = 1 + i. First we find Then, using the ICONV worksheet, we get that d (2) = %. 24/24