Luenberger Exercise Chapter 2 
EXERCISES
35
EXERCISES
Suppose $1 were invested in l776 <It 31% interesl cOn1{Xlunded
L (A nice inherilance)
yearly
(0) Approximately how much would that investment be wonh today: $1,000, $10,000,
$100,000, or $1,000,000 '1
(b) Whm if the inlerest ratc were 669 '0?
2. (The 72 nile) The number ot years II required for an investment 1.11 imerest rate I to double in value musl satisfy (1 + I yl = 2 Using In 2 = 69 and the approximation In( I + I ) ~ I valid ror small" show that /I ~ 69/i, where i is the interest rate percentage (that is. i = 100,) Using the better approximation In( 1 +, ) ~ , - !/2, show that lor, ~ 08 there holds 11 ~ 72/i
3. (Effective nltes)
Find the corresponding effective rates for:
(a) 3% cOnlpounded monthly
(b) 18% compounded monlhly
(t) 18% compounded quarterly
4. (Newton 's method 0) The IRR is generally calculated using an iterMive procedure Sup~ pose that we define I (A) = -ao + {{I A+ 02A 2 +
+ {{nAil, where all {{I ' S arc positive and
11 > I Hefe is an iterative technique that generates a sequence AO, AI ' A2.
,Ak,
ot estimates that converges to the root X > 0, solving / (X) = 0 Start with <Iny AO > 0 close to the solution Assuming A~ has been colcl1lated, evaluate
t
(Ad = {{I
+ 2a2Ak + 3aJAi +
+ IlllnArl
and define
A
- A _ J(Ad
k+1 -
k
r(A~)
This is Newton 's method It is based on approximating the function / by a line tangent to its graph at Ab as shown in Figure 1. 4 Try the procedure on J(A) = -I + A + A1 Start with AO = I and compute four additional estimates fiGURE 2.4
/
/
°Exercises rollowed by
0
/
71
1
1
1
arc Inillhcnlatic;.lily more difficult thull nvcmgc
Newton 's method
36
Chapter 2
THE BASIC THEORY OF INTEREST
5. (A prize) A major lottery advertises thilt it pays the winner $10 miiHon However, this prize money is paid at the rate of $500,000 each year (with the first payment being immediate) fot a total of 20 payments What is the present value of this prize at 10% interest?
6, (Sunk costs) A
35
EXERCISES
Suppose $1 were invested in l776 <It 31% interesl cOn1{Xlunded
L (A nice inherilance)
yearly
(0) Approximately how much would that investment be wonh today: $1,000, $10,000,
$100,000, or $1,000,000 '1
(b) Whm if the inlerest ratc were 669 '0?
2. (The 72 nile) The number ot years II required for an investment 1.11 imerest rate I to double in value musl satisfy (1 + I yl = 2 Using In 2 = 69 and the approximation In( I + I ) ~ I valid ror small" show that /I ~ 69/i, where i is the interest rate percentage (that is. i = 100,) Using the better approximation In( 1 +, ) ~ , - !/2, show that lor, ~ 08 there holds 11 ~ 72/i
3. (Effective nltes)
Find the corresponding effective rates for:
(a) 3% cOnlpounded monthly
(b) 18% compounded monlhly
(t) 18% compounded quarterly
4. (Newton 's method 0) The IRR is generally calculated using an iterMive procedure Sup~ pose that we define I (A) = -ao + {{I A+ 02A 2 +
+ {{nAil, where all {{I ' S arc positive and
11 > I Hefe is an iterative technique that generates a sequence AO, AI ' A2.
,Ak,
ot estimates that converges to the root X > 0, solving / (X) = 0 Start with <Iny AO > 0 close to the solution Assuming A~ has been colcl1lated, evaluate
t
(Ad = {{I
+ 2a2Ak + 3aJAi +
+ IlllnArl
and define
A
- A _ J(Ad
k+1 -
k
r(A~)
This is Newton 's method It is based on approximating the function / by a line tangent to its graph at Ab as shown in Figure 1. 4 Try the procedure on J(A) = -I + A + A1 Start with AO = I and compute four additional estimates fiGURE 2.4
/
/
°Exercises rollowed by
0
/
71
1
1
1
arc Inillhcnlatic;.lily more difficult thull nvcmgc
Newton 's method
36
Chapter 2
THE BASIC THEORY OF INTEREST
5. (A prize) A major lottery advertises thilt it pays the winner $10 miiHon However, this prize money is paid at the rate of $500,000 each year (with the first payment being immediate) fot a total of 20 payments What is the present value of this prize at 10% interest?
6, (Sunk costs) A
References: Exercise 10 is a modification of an example in [6] Alexander, G J, W F Sharpe, and Y J Builey (1993), Fwu/amellla/ of /nveHment, 2nd cd, Prentice Hnll, Englewood Cliffs, NJ Bodie, H M, A Kane, nnd A J Maf{~I1s (1993), /lIl 'eHmenH, 2nd cd, ltwin, Home~ wood,IL Bre£liey, R, nnd S Meyers (1981), P/incip/e of Corporate Finance, McGHIw~Hill, New