Investment Valuation and Portfolio Management (E2093)
Vit Bubak
(Tutorial 2 and 3) Bond Valuation, DV01, and DUR, and Convexity
I. Bond Price Calculation (Revision)
We discount the future coupon payments (C) by the appropriate discount rate (r, YTM).
1) In general, the coupon rate (c = C/M) remains the same until the maturity (T). However, the interest
rate (r), may change over time. Assuming semiannual-coupon payments discounted on semi-annual
basis, we can then calculate the bond price as…
P=
n − mT
∑
t =1
C 2
M
+
r
r
( 1 + t )t ( 1 + t )n
m
m
2) If the interest rates do not change (a special case of a flat yield curve), then we may use…
P=
n − mT
∑
t =1
C 2
M
+
t
( 1 + r ) ( 1 + r )n
m
m
3) The last formula can be further simplified using the annuity present value factor (APVF) as…
P=
(
M
(1 + r 2 )
2×T
1 − 1 + r
2
C 
+ ⋅
r
2
2
)
−2×T


Q1) How much should each of the following bonds (a-e) sell for?
a) A 10% $100 bond with 10 years to maturity and yield to maturity of 12%.
Answer:
P=
P=
(
1 − 1 + r
2
C 
+ ⋅
r
2
2
M
(1 + r 2 )
2×T
(
100
(1 + 0.12 2 )
20
)
−2×T
1 − 1 + 0.12
2
C 
+ ⋅
0.12
2
2


)
− 20


= 31.18 + 57.35 = $88.53
Q2) HW Questions (Valuing the bonds)
How much should each of the following bonds (a-e) sell for?
b)
c)
d)
e)
A 10% $100 bond with 5 years to maturity and yield to maturity of 12%.
A 10% $100 bond with 10 years to maturity and yield to maturity of 8%.
A 5% $100 bond with 10 years to maturity and yield to maturity of 12%.
A 10% $100 zero-bond with 10 years to maturity and yield to maturity of 12%.
Q3) HW Questions (Comparing the bonds with different characteristics)
With regard to your answers in Q1:
a)
b)
c)
d)
Explain why the bond in (a) sells for less than its face value.
Explain why your answer in (b) is greater than the answer in (a).
Explain why your answer in (c) is greater than the answer in (a).
Explain why your answer in (d) is less than the answer in (a).
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Investment Valuation and Portfolio Management (E2093)
Vit Bubak
II. DV01 Calculation (Revision)
DV01 represents a dollar value of 1bp decrease (in interest rates). It is a measure of how much a
bond’s price will increase in response to a one bp decline in bond’s yield to maturity.
If the yield and interest rates are the same (i.e., in case of flat term structure of interest rates), DV01
can be useful in estimating the interest rate risk.
How do we calculate the DV01. Using derivative calculus, we can compute the DV01 as…
DV 01 = −0.0001
dP
dr
This can be rearranged as dP = 10,000 × dr = 10,000 × (∆bp ) , where the latter part of the equation
gives us a non-calculus equivalent (since 10,000dr is the yield change in number of basis points)
III. DUR Calculation (Revision)
Duration is a concept closely related to DV01. It is a weighted average of the waiting times (measures
in years) for receiving its promised future cash-flows. The weight on each time is proportional to the
discounted value of the cash-flow to be paid at that time, or…
 C2 
 C 
 C1 
1
⋅1 + " +  1  ⋅ T
⋅
+

 (1 + r ) 
2 
T
 PV (C t ) 
 (1 + r ) 


 (1 + r ) 
DUR =
t
=∑
C1
C2
CT
P 
t =1 
+
+"+
(1 + r ) (1 + r )2
(1 + r )T
Note that the weights,
PV (Ct )
, always sum to 1.
P
IV. Linking DV01 and DUR
We already know that duration and DV01 is related to the derivative of a bond’s price with respect
to interest rates. Such a derivative also will enable us to relate duration to DV01. To illustrate this,
consider a 2-year zero coupon bond with a face value of $100 and a continuously compounded (CC)
yield to maturity of r. The bond’s price is P = $100e-2r and its duration is two years. Using calculus,
we percentage change in the bond’s price for a change in yield to maturity is …
(
)
dP P dP 1
1
=
= $100 − 2e − 2 r ×
= −2
dr
dr P
$100e − 2 r
Thus, –2, the negative bond’s duration, is the percentage sensitivity of the bond’s price to changes in
the bond’s CC yield.
This derivative property can be generalized as P =
T
∑C e
t ´=1
… and the % change in P with respect to CC yield is
t
− rt
,…
dP P
1 T

= −  ∑ tC t e − rt  = − DUR .
dr
P  t´=1

(II.1)
(II.2)
The relationship between duration and DV01 is straightforward if the “01” in DV01 is defined as a CC
rate. In this case, DV01 is the product of –0.0001 and the derivative of the value of the bond w/ respect
to a shift in the bond’s CC yield, that is DV 01 = −0,0001
dP
dr
(II.3)
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Investment Valuation and Portfolio Management (E2093)
Realizing that (II.2) can be in fact written as DUR = −
DV01= DUR × P × 0.0001
Vit Bubak
1 dP
, it follows that DV01 can be written as...
P dr
(II.4)
In other words, we cas see that DUR and DV01 can be considered as equivalent tools both for
measuring interest rate risk as well as for hedging purposes.
Note: Realizing that the duration is not usually based on continuous compounding, but more often on
the rates being compounded m-times per year, we should modify the formula (II.4) to the following…
r



DV 01 ⋅ 1 + 
 DUR 
m

… it follows that DV01 = 
 × P × 0.0001 , or that DUR=
P ⋅ 0.0001
1 + r 
m 

(II.5)
Q4) Without performing any calculations which of the bonds in Q1 would you think is the most
sensitive to interest rate changes? Least sensitive? Explain why.
Q5) The DV01 of a portfolio of bonds
Assume that the DV01 of a 30-year U.S. Treasury bond is $0.10 per $100 face, the DV01 of a 5year U.S. Treasury note is $0.04 per $100 face amount, and the DV01 of a 10-year U.S. Treasury
note is $0.06 per $100 face amount. Compute the DV01 of a portfolio that has $5mil face value of
30-year bonds, $8mil face value of 5-year notes, and –$16mil face value of 10-year notes.
Answer:
Sum the products of the face amounts per $100 and the DV01s per $100. The result is:
DV 01 =
$16mil
$8mil
$5mil
× ($0.10) +
× ($0.04) −
× ($0.06) = −$1,400
100
100
100
Note the negative value of the DV01 for the whole portfolio. Negative means that the position’s
value increases when the interest rates rise.
Q6) Using DV01s to form perfect hedge portfolios
Assume that changes in the yields of various maturity bonds are indentical. How much of a 7-year
bond, with a computed DV01 of $0.05 per $100 of face value, should you purchase to perfectly
hedge the interest rate risk of the portfolio in Q5 (which has a DV01 of –$1,400)?
Answer:
The portfolio to be hedged has a DV01 of –$1,400. The DV01 of the portfolio if we buy $x face
amount of the 7-year bond is:
 $x

− $1,400 + 
× 0.05 
 100

This equals zero when x is $2.8mil. Thus, a $2.8mil face amount of 7-year bonds hedges the
portfolio against interest rate risk.
Q7) HW Question (Using DV01s to form imperfect hedges)
The 7- year bond from the last example has a DV01 of $0.05 per $100 face value. Assuming that
changes in the yields of various maturities are indentical, how much of the 7-year bond should be
bought to change the corporate risk exposure, measured currently as DV01 of –$1,400, to a risk
exposure with a DV01of –$400 ? And to +$400? State one reason why we might like to do it.
Hint: Portfolio DV01s are additive.
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Investment Valuation and Portfolio Management (E2093)
Vit Bubak
V. Convexity
Convexity measures how much DV01 changes as the yield of a bond or bond portfolio changes. A
portfolio with a DV01 will be insensitive to small interest rate movements and less sensitive to large
interest rate movements the smaller the convexity. In other words, the DV01 remains close to zero as
interest rates change if convexity is close to zero.
IF DV01 can be viewed as a derivative of the bond price with respect to its yield to maturity, then
CONV is the second derivative of the portfolio’s price with respect to its yield in percent (not its basis
point shift!).
 $1,000,000 
CONV (r ) = 
 × [DV 01(r ) − DV 01(r + 0.0001)]
P


V.1. Estimating price sensitivity to yield
Investors often use the convexity to improve upon the earlier (DV01 or DUR-based) estimate of the
change in the price of a bond portfolio for a given change in yield. Specifically, they can use…
∆P = −100 ⋅ DV 01 ⋅ ∆r + 0.5 ⋅
( )
P
⋅ CONV ∆r 2
100
Q8) Compute the change in the price of a bond portfolio with a $200 market value for a 25bp increase
in yield. At the current yield to maturity, the bond has a DV01 of $0.15 and a convexity of 1.2
(convexity is computed per $100 of market value!).
− 100 ⋅ ($0.15) ⋅ (0.25) + 0.5 ⋅
(
)
$200
⋅ 1.2 ⋅ 0.25 2 = −$3.675
100
Hence, the new price will be $196.325.
Q9) Assume you manage a $10mil (mkt value) portfolio comprising 46% of bond (b) and the
remainder in bond (e). Calculate DV01, duration, modified duration and convexity for the
portfolio. What is the face value of the debt held?
Answer:
a) Duration =
∑ w DUR
i
i
= 0.46 × 4,01 + 0.54 × 10 = 7.2446
i
b) Modified Duration =
∑w
i
mod DURi = 0.46 × 3.78 + 0.54 × 9.43 = 6.83%
i
c) Convexity =
∑ w CONV
i
i
= 0.46 × 0.18142 + 0.54 × 0.9345 = 0.58808
i
d) DV01 can be estimated (as usually) as follows…
DV 0 =
DUR
7.2446
× P × 0.0001 =
× 10,000,000 × 0.0001 = $6,834.53
r
1.06
1+
m
e) Face value of bond (b) =
100
× 4.6M = $4,965,458
92.64
f) Face value of bond (e) =
100
× 5.4M = $17,318,794
31.18
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Investment Valuation and Portfolio Management (E2093)
Vit Bubak
g) Alternative calculation for DV01
$4,965,458
$17,318,794
× 0.03505 +
× 0.0294 = $6,832.12
100
100
Q10) With regard to the portfolio from the previous question, you are concerned about the possibility
of a 50bp parallel upward shift in the yield curve over the next quarter. By taking a short position
in bond (a), construct a hedge that will minimize the price impact of a 50bp interest rate increase.
How effective is the hedge if rates increase by 50bp?
The ratio of the duration (DUR) of the short position to the duration of the bond portfolio should
be inversely equal to their prices. That is…
DUR S PP
6.31
10,000,000
=
→
=
→ PS = $11,481,141. So we need to také a short
7.2246
DUR B PS
PS
position in bond (a) with market value $11,481,141 to hedge our exposure.
If rates increase by 50bp the value of our bond portfolio will drop by an estimated…
50 × 6,834.53 = $341.726
At the same time, the value of our short position will increase by an estimated…
12,968,644
× 0.05267 × 50 = $341.542
100
Note that 0.05267 is DV01 and 12,968,644 is the face value of our short position.
However, as we have just seen, the actual increase in our short position will be $334,718 and the
actual fall in the value of our portfolio will be $334,445 which is still a very effective hedge.
Q11) HW (Summary questions)
For each bond in Q1, calculate:
a)
b)
c)
d)
e)
f)
g)
DV01
Duration
Modified duration
Convexity
Based on DV01, the estimated impact of a 50bp change in interest rates
Based on convexity, the estimated impact of a 50bp change in interest rates.
The actual price impact of a 50 bp change in interest rates.
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