and the convexity model predicts that D.P 1 + r P(r + Ar) = P(r) - Ar + 1 CX · P · (Ar)². · Consider the cash flow stream and r in (c). If r increases from 10% to 11%, what is the new present values predicted by the duration model and the convexity model, respectively? Which one of them predicts more accurately?
and the convexity model predicts that D.P 1 + r P(r + Ar) = P(r) - Ar + 1 CX · P · (Ar)². · Consider the cash flow stream and r in (c). If r increases from 10% to 11%, what is the new present values predicted by the duration model and the convexity model, respectively? Which one of them predicts more accurately?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:and the convexity model predicts that
P(r+ Ar) = P(r)
D·P
-Ar +
1+r
cx P (Ar)".
СХ Р.
Consider the cash flow stream and r in (c). If r increases from 10% to
11%, what is the new present values predicted by the duration model
and the convexity model, respectively? Which one of them predicts more
accurately?
![3. (Duration and Convexity for General Cashflow Streams) Equation (4) and (11)
in the Lecture Notes 2 give the duration and convexity for the coupon bonds.
This question illustrates duration and convexity calculations for general cash
flow streams. Consider the n-period cash flow stream (n > 1) depicted on Page
23 of Lecture Notes 2, with xo = 0, xx > 0 for k = 1, 2, ..., n – 1, and x, > 0.
Assume that all periods have equal length of 1 year (therefore, R= r). Denote
the present value of this cash flow stream as
%3D
X2
Xn
P = P(r) =
+
(1+r)²
+
(1+r)"'
(1)
1+r
Required precision: 4 digits after decimal point.
1+r dP Find out the expression for wk,
P dr
(a) The duration is defined as D =
k = 1, ...,n which satisfy the following conditions: wi+w2+·..+Wn = 1,
Wk 2 0, k = 1,..., n, and
D = w1 •1+ w2 · 2 + · · · + Wn · N.
EP. Find out the expression for wk,
1 ď²P
P dr² ·
(b) The convexity is defined as CX =
k = 1, ...,n which satisfy the following conditions: wi+w2+·..+Wn = 1,
Wk 2 0, k = 1,..., n, and
%3D
%3D
1
• [W1 •1· 2+ w2 · 2 · 3+ ..+ Wn · n· (n+1)].
CX
(1+r)²
(c) Now consider a 5-period cash flow stream x1
0, xz = 5, with r = 10%. Calculate its present value, duration and convex-
0, x4
0, x2
2, x3 =
ity.
(d) If the interest rate changes from r to r+Ar, the present value of the cash
flow stream changes from P(r) to P(r+Ar). The duration model predicts
that
D·P
P(r + Ar) = P(r)
1+r](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc8d80de-c6bf-4733-98b7-66f87108d889%2Fbeaf1d8e-3b96-452d-b651-600ef922c528%2F6vbw9jr_processed.png&w=3840&q=75)
Transcribed Image Text:3. (Duration and Convexity for General Cashflow Streams) Equation (4) and (11)
in the Lecture Notes 2 give the duration and convexity for the coupon bonds.
This question illustrates duration and convexity calculations for general cash
flow streams. Consider the n-period cash flow stream (n > 1) depicted on Page
23 of Lecture Notes 2, with xo = 0, xx > 0 for k = 1, 2, ..., n – 1, and x, > 0.
Assume that all periods have equal length of 1 year (therefore, R= r). Denote
the present value of this cash flow stream as
%3D
X2
Xn
P = P(r) =
+
(1+r)²
+
(1+r)"'
(1)
1+r
Required precision: 4 digits after decimal point.
1+r dP Find out the expression for wk,
P dr
(a) The duration is defined as D =
k = 1, ...,n which satisfy the following conditions: wi+w2+·..+Wn = 1,
Wk 2 0, k = 1,..., n, and
D = w1 •1+ w2 · 2 + · · · + Wn · N.
EP. Find out the expression for wk,
1 ď²P
P dr² ·
(b) The convexity is defined as CX =
k = 1, ...,n which satisfy the following conditions: wi+w2+·..+Wn = 1,
Wk 2 0, k = 1,..., n, and
%3D
%3D
1
• [W1 •1· 2+ w2 · 2 · 3+ ..+ Wn · n· (n+1)].
CX
(1+r)²
(c) Now consider a 5-period cash flow stream x1
0, xz = 5, with r = 10%. Calculate its present value, duration and convex-
0, x4
0, x2
2, x3 =
ity.
(d) If the interest rate changes from r to r+Ar, the present value of the cash
flow stream changes from P(r) to P(r+Ar). The duration model predicts
that
D·P
P(r + Ar) = P(r)
1+r
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