1. FINC4101 Investment Analysis
Instructor: Dr. Leng Ling
Topic: Interest Rate Risk

2. Learning objectives Define interest rate risk for bonds.
Recognize maturity as a major determinant of interest rate risk.
Compute Macaulay’s duration and modified duration.
Identify the determinants of Macaulay’s duration.
Interest rate risk and interest rate sensitivity are synonymous.

3. Concept Map Foreign Exchange Derivatives Market Efficiency
Fixed Income
Equity
Asset
Pricing
Portfolio
Theory
FI400

4. Interest rate risk Why are changes in interest rates a risk for bonds?
Interest rate risk: Sensitivity of a bond’s price to changes in market interest rate (i.e., yield to maturity).
In our discussion, interest rate risk and interest rate sensitivity are synonymous.
Changes in bond price (capital gain/loss)
Uncertainty in return from bond investment
Changes in interest rates
Changes in bond price ultimately contribute to uncertainty in returns because one component of return is price changes (see HPR formula).
Why do bond prices respond to interest rate fluctuations? In a competitive market, all securities must offer investors fair expected rates of return. In a bond is issued with an 8% coupon when competitive yields are 8%, then it will sell at par value. If the market rate rises to 9%, however, the bond price must fall until its expected return increases to the competitive level of 9%. Conversely, if the market rate falls to 7%, the 8% coupon on the bond is attractive compared to yields on alternative investments. Investors eager for that return will respond by bidding the bond price above its par value until the total rate of return falls to the market level.
Once the definition of interest rate risk is given, tell students that we will use interest rate risk and interest rate sensitivity interchangeably.

5. Bond Pricing Relationships (1)
Figure shows the percentage changes in price corresponding to changes in YTM for four bonds that differ according to coupon rate, initial YTM, and time to maturity. All four bonds illustrate that bond prices decrease when yields rise, and that the price curve is convex, meaning that decreases in yields have bigger impacts on price than increases in yields of equal magnitude.

6. Bond Pricing Relationships (2)
For all 4 bonds, we observe that:
Bond prices and yields are inversely related.
As YTM increases, bond price falls
As YTM decreases, bond price rises
Price-yield relationship is convex.
An increase in YTM results in a smaller price change than a decrease in YTM of equal magnitude.
This property is called “convexity”.
Based on the Figure on the previous slide, we can say the following.
Bond prices and yields are inversely related: as yields increase, bond prices fall; as yield falls, bond prices rise.
We will come back to the convex nature of the price-yield relationship later on.

7. Bond Pricing Relationships (3)
Comparing bonds A and B, we see that:
Interest rate sensitivity increases with maturity.
Prices of long-maturity bonds tend to be more sensitive to interest rate changes than prices of short-maturity bonds.
Interest rate sensitivity increases at a decreasing rate as maturity increases.
A and B are identical except that A has a short maturity than B.
While bond B has six times the maturity of bond A, it has less than six times the interest rate sensitivity. Although interest rate sensitivity seems to increase with maturity, it does so less than proportionally as bond maturity increases.

8. Bond Pricing Relationships (4)
Comparing bonds B and C, we see that,
Interest rate sensitivity is inversely related to coupon rate.
Prices of high-coupon bonds are less sensitive to changes in interest rates than prices of low-coupon bonds.
Comparing bonds C and D, we see that,
Interest rate sensitivity is inversely related to the YTM at which the bond currently is selling.
B and C are identical except that C has a lower coupon rate than B.
C and D are identical except that D has a lower YTM.

9. Maturity and interest rate risk
A bond’s maturity is a major determinant of interest rate risk.
Maturity alone is not enough to measure interest rate risk.
Cash flows received before maturity affects the relationship between maturity and interest rate sensitivity.
Consider the following…

10. Coupon bond vs. zero coupon bond
Prices of 8% annual coupon bonds
YTM
T = 10 years
T = 20 years 8%
1000 9%
935.82
908.71
% price change
-6.42%
-9.13%
Prices of zero-coupon bonds
YTM
T = 10 years
T = 20 years 8%
463.19
214.55 9%
422.41
178.43
% price change
-8.80%
-16.84%
For both maturities beyond one year, the price of the zero-coupon bond falls by a greater proportional amount than the price of the 8% coupon bond. The observation that long-term bonds are more sensitive to interest rate movements than short-term bonds suggests that in some sense a zero-coupon bond represents a longer term investment than an equal-time-to-maturity coupon bond.
Key point: when you have coupon payments paid out before maturity, those payments shorten the “effective” maturity of the bond and thus lowers the interest rate sensitivity. The zero coupon bond has no coupon payments, thus, its “effective” maturity is exactly equal to its time to maturity.
Explain why coupon bond has a shorter effective maturity.
The times to maturity of the two bonds in this example are not perfect measures of the long- or short-term nature of the bonds. The 8% bond makes many coupon payments, most of which come years before the bond’s maturity date. Each payment may be considered to have its own “maturity date”. In this sense, it is often useful to view a coupon bond as a “portfolio” of coupon payments. The effective maturity of the bond should be measured as some sort of average of the maturities of all the cash flows paid out by the bond. The zero-coupon bond, by contrast, makes only one payment at maturity. Its time to maturity is a well-defined concept.
A high-coupon rate bond has a higher fraction of its value tied to coupons rather than payment of par value, and so the portfolio is more heavily weighted toward the earlier, short-maturity payments, which give it a lower “effective maturity”. This explains why price sensitivity falls with coupon rate.
A higher yield reduces the PV of all of the bond’s payments, but more so for more distant payments. Therefore, at a higher yield, a higher fraction of the bond’s value is due to its earlier payments, which have lower effective maturity and interest rate sensitivity. The overall sensitivity is thus lower.

11. Measuring interest rate risk
An acceptable measure of interest rate risk must account for:
Time to maturity.
Cash flows which are paid out during the life of the bond.
Such a measure is Macaulay’s duration.
An acceptable measure of interest rate risk must account for:
1) Time to maturity, since we have seen that interest rate sensitivity increases with time to maturity
2) Cash flows which are paid out during the life of the bond, since from the previous example, we see that cash flows paid out before maturity shortens the effective maturity and hence the interest rate sensitivity.
So what we want is a measure of effective maturity,

12. Macaulay’s duration (1)
A measure of interest rate sensitivity.
Macaulay’s duration is the weighted average of the times to each coupon or principal payment made by the bond.
Often referred to as ‘duration’.
The weight for each payment time is the PV of the payment divided by the bond price.
Coupon payments made prior to maturity make the effective (i.e., weighted average) maturity of the bond less than its actual time to maturity.

13. Macaulay’s duration (2)
Time until 4th cash flow
Weight of 4th cash flow
Formula for annual coupon payment bond.
For semi-annual payment bond, y is the semi-annual YTM, T is the no. of semi-annual periods to maturity, CFt is the semi-annual coupon payment, bond price is computed assuming semi-annual compounding.
Cash flow paid at time t

14. Examples: 3-year bond, 8%, annual coupon payments, YTM = 10%
Time until
Payment
Column (B)
Discounted x
(Years)
at 10%
Weight
Column (E)
A. 8% coupon bond 1 80
72.727
0.0765 2
66.116
0.0696
0.1392 3
1080
0.8539
2.5617
Sum:
1.0000
2.7774
B. Zero-coupon bond
0.000
0.0000
1000
3.0000
This slide shows the examples of computing maturity. Part A computes maturity for 8% coupon bond, Part B computes zero-coupon bond.
This example shows that duration is shorter when there are cash flows paid before maturity. This is exactly what we want and accords to what we saw earlier.
Also note that in general, the Macaulay of a zero coupon bond is exactly equal to its maturity.
Tell student that if coupons are paid semi-annually, then the duration you get is the semi-annual duration. To convert to annual duration, divide the semi-annual duration by 2.
Use the same example to compute convexity.

15. Macaulay’s duration measures interest rate sensitivity
Absolute percentage change in price when
Bond
YTM increases from 10% to 11%
YTM decreases from 10% to 9%
8% coupon bond
(duration =2.7774)
2.48%
2.57%
Zero coupon bond
(duration =3)
2.68%
2.78%
This example, based on the previous slide shows that Macaulay’s duration indeed tracks interest rate sensitivity. The higher the Macaulay’s duration, the more sensitivity is the bond price to changes in YTM. See that the 3-year zero has a higher duration and also experiences bigger percentage price changes when YTM either goes up or down. This turns out to be true in general. The higher the Macaulay’s duration, the higher the interest rate sensitivity.
YTM increases from 10% to 11%:
=====================
8% coupon bond
FV=1000, pmt=80, n=3, i/y = 11, pv =
Percentage price change = ( – )/ = or -2.48%
Zero coupon bond
FV=1000, pmt=0, n=3, i/y = 11, pv =
Percentage price change = ( – )/ = or -2.68%
YTM decreases from 10% to 9%:
FV=1000, pmt=80, n=3, i/y = 9, pv =
Percentage price change = ( – )/ = or 2.57%
FV=1000, pmt=0, n=3, i/y = 9, pv =
Percentage price change = ( – )/ = or 2.78%
Higher duration is associated with larger percentage price changes, i.e., greater interest rate sensitivity.

16. Macaulay’s duration for semi-annual payment coupon bond
For a semi-annual coupon bond, use the same formula as for an annual coupon bond, but:
Each period is a semi-annual period
CFt : cash flow for semi-annual period t
Yt : semi-annual YTM
Result: semi-annual Macaulay duration.
Annual Macaulay duration
= Semi-annual Macaulay duration/2

17. Example: 3-year bond, 8%, semi-annual coupon payments, YTM = 10%
Time until
Payment
Column (B)
Discounted x
(6-months)
at 5%
Weight
Column (E)
A. 8% coupon bond 1 40
38.095
0.0401 2
36.281
0.0382
0.0764 3
34.554
0.0364
0.1092 4
32.908
0.0347
0.1387 5
31.341
0.0330
0.1651 6
1040
0.8176
4.9054
Sum
5.4349
2.7174
Semi-annual coupon payment bonds are very common. So on this slide, we look at the computation of Macaulay’s duration for the same 3-year bond but now it pays coupons semi-annually.
Be careful here, with semi-annual coupon bonds, the YTM must be the semi-annual yield, i.e., annual YTM/2. So cash flows are discounted at the semi-annual YTM of 5%. Each cash flow is the cash flow received per semi-annual period and the price is bond price assuming semi-annual compounding.
Now, suppose the coupons are paid semi-annually. We can also calculate Macaulay’s duration for semi-annual payment bonds, but what we get is the semi-annual Macaulay’s duration. To convert this number to the annual Macaulay duration, we divide it by the number of semi-annual periods, 2.
Therefore, to annualize the semi-annual Macaulay duration, divide by 2.
In this example, the semi-annual duration is six-month periods. The annualized duration is /2 = years (there is rounding off error).
Semi-annual duration
Annualized duration

18. Modified duration (1)
Turns out that we can transform Macaulay’s duration into a measure that estimates percentage price change for a given change in YTM.
This measure is called Modified duration.
Directly gauges the impact of interest rate risk because it approximates the change in price when yield changes (increases or decreases).

19. Modified duration (2) Modified duration, D* YTM (in decimal)
If Macaulay’s duration is annual, then modified duration is also annual.
If Macaulay’s duration is semi-annual, then modified duration is also semi-annual.
Modified durations are also in terms of years/semi-annual periods.
If the original compounding basis on the bond was semi-annual, the modified duration must first be calculated on a semi-annual basis and then annualized.
YTM (in decimal)

20. Using Modified duration to estimate percentage price change
Dollar change in bond price
Change in YTM in decimal
Initial price
Macaulay’s duration not only tells us the interest rate sensitivity, it can also be used to estimate bond price change for a given change in YTM. To do that, we compute modified duration, which estimates the proportional (percentage) change in bond price for a given change in YTM. We use Δy to represent the change in YTM
The equation says that bond price volatility (price change) is proportional to the bond’s duration. For a given change in yield, the bigger the duration, the bigger the price change and the more sensitive is the bond, vice versa. Thus, duration becomes a natural measure of interest rate exposure.
Why is there a negative sign? Because of the inverse relationship between bond price and interest rate. Duration is always positive. When yield increases, the second term on the RHS is positive. We need the negative sign to ensure that the number on the RHS is negative. This ensures consistency with the inverse price-yield relationship. Similarly, when yield decreases, the second term on the RHS is negative. This combines with the negative sign ensures that the price change is positive, again consistent with the price-yield relationship.
Tell student that this is just an approximation. The approximation is more accurate for small changes in YTM. Turns out that for big changes in yield, using duration will not give good approximation. We need to include another measure, called convexity.
Modified duration

21. Using Modified duration to estimate dollar price change
For a given change in yield, , the predicted price =
The higher the Macaulay's duration, the greater the interest rate sensitivity of a bond, i.e., the bigger the price change for a given change in YTM, vice versa.
The higher the modified duration, the greater the interest rate sensitivity of a bond, i.e., the bigger the price change for a given change in YTM, vice versa.

22. Interpretation
As Macaulay’s duration ↑, interest rate sensitivity ↑ , vice versa.
As modified duration ↑, interest rate sensitivity ↑ , vice versa.
The higher the Macaulay's duration, the greater the interest rate sensitivity of a bond, i.e., the bigger the price change for a given change in YTM, vice versa.
The higher the modified duration, the greater the interest rate sensitivity of a bond, i.e., the bigger the price change for a given change in YTM, vice versa.

23. Problems (1)
A nine-year bond has a yield of 10% and a (Macaulay) duration of years. If the bond’s yield increases by 50 basis points, what is the percentage change in the bond’s price.
Note:
1 basis point = or 0.01%
100 basis points = 1%
BKM Q1. Compute percentage price change using modified duration.
The duration is annual duration.
Compute modified duration = 7.194/1.1 = 6.54
Percentage price change = x = or 3.27%

24. Problems (2)
A 30-year maturity bond making annual coupon payments with a coupon a rate of 12% has a duration of years. The bond currently sells at a yield to maturity of 8%. If the yield to maturity falls to 7%, compute:
The percentage price change.
The estimated dollar price change.
The predicted price.
Modified duration = /(1.08) =
1) Percentage price change = x = or , or 10.69%
2) First, Compute current price at 8% yield.
FV=1000, N=30, I/Y=8%, PMT=0.12 x 1000 = 120, CPT PV=
Current price is $
Estimated dollar price change = x ( x -0.01) =
3) The predicted price = dollar price change + current price = = $

25. Problems (3)
A six-year 6.1% semi-annual coupon bond has a yield to maturity of 10% and a semi-annual Macaulay duration of
What is the annual Macaulay duration?
What is the modified duration on a semi-annual basis?
What is the modified duration on an annual basis?
If the yield increases by 25 basis points, what is the percentage price change using the annual modified duration?
Show class how to tackle duration when you have a semi-annual payment bond.
If the original compounding basis on the bond was semi-annual, the modified duration must first be calculated on a semi-annual basis and then annualized. You cannot use the annual Macaulay Duration to calculate the Modified Duration.
Annual Macaulay duration = /2 = years
Divide semi-annual Macaulay duration by (1 + ytm/2)
Modified duration on semi-annual basis = /( ) =
3) To convert to annual modified duration, divide by no. of semi-annual periods.
Modified duration on annual basis = /2 =
4) Percentage price change = x = or -1.19%

26. Determinants of duration (1)
Rule 1: The duration of a zero-coupon bond equals its time to maturity.
Rule 2: Holding time to maturity and yield to maturity constant, a bond’s duration and interest rate sensitivity are higher when the coupon rate is lower.
Refers to Macaulay duration. Of course, since modified duration is just Macaulay duration divided by 1 + y, these rules also apply to modified duration. Now we talk about several rules that summarize most of the important properties of duration.
Rule 1: This happens because the zero-coupon bond has zero cash flows prior to maturity. Thus, all weights prior to maturity are zero. All weight (1) is placed on maturity, so duration of a ZCB equals time to maturity.
Rule 2: The lower the coupon, the less weight these early payments have on the weighted average maturity of all the bond’s payments and thus, the higher the duration. In other words, a lower fraction of the total value of the bond is tied up in the (earlier) coupon payments rather than the (later) repayment of par value. Compare the plots of the durations of the 3% coupon and 15% coupon bonds, each with identical YTM of 15%.

27. Determinants of duration (2)
Rule 3: Holding the coupon rate constant, a bond’s duration and interest rate sensitivity generally increase with time to maturity. Duration always increases with maturity for bonds selling at par or at a premium to par.
Rule 4: Holding other factors constant, the duration and interest rate sensitivity of a coupon bond are higher when the bond’s yield to maturity is lower.
Rule 3: while long-maturity bonds generally will be high-duration bonds, duration is a better measure of the long-term nature of the bond because it also accounts for coupon payments. Only when the bond pays no coupons is time to maturity an adequate measure; then maturity and duration are equal.
Rule 4: At lower yields the more distant payments have relatively greater present values and thereby account for a greater share of the bond’s total value. Thus, in the weighted-average calculation of duration, the distant payments receive greater weights, which results in a higher duration measure.
The intuition is that while a higher yield reduces the present value of all of the bond’s payments, it reduces the value of more distant payments by a greater proportional amount. Therefore, at higher yields a higher fraction of the total value of the bond lies in its earlier payments, thereby reducing effective maturity.

28. Determinants of duration (3)
Rule 5: The duration of a perpetuity is
E.g., at a 15% yield, the duration of a perpetuity that pays $100 once a year forever will equal 1.15/0.15 = 7.67 years, while at an 8% yield, it will equal 1.08/0.08 = 13.5 years.
The equation makes it obvious that maturity and duration can differ substantially. The maturity of the perpetuity is infinite, while the duration of the instrument at a 15% yield is only 7.67 years. The PV weighted cash flows early on in the life of the perpetuity dominate the computation of duration.

29. Graphical summary
Rule 1
Rule 4
Rule 2
Rule 3

30. Conceptual problems (1)
Rank the following bonds in order of descending duration.
Bond
Coupon
Time to Maturity
Yield to Maturity A
15%
20 years
10% B 15 10 C 20 D 8 E
Rule 1 tells us that C has a duration of 20 years.
Next look at the other 20-year maturity bonds. Rule 2 says that duration is higher when the coupon rate is lower. This implies that bond D has a longer duration than bond A.
Now, look at the 15year bonds. can these bonds have longer durations than the 20-year bonds? No because the maximum duration is 15 years if the bond is a zero coupon bond. Since bonds B and E are coupon bonds, Rule 2 implies that these bonds must have durations less than 15 years. Therefore these bonds must have lower durations than A, C and D.
B has a lower YTM than E. So rule 4 implies that B has a longer duration than E.
So the ranking is C, D, A, B, E.
Solutions manual’s explanation:
C: Highest maturity, zero coupon
D: Highest maturity, next-lowest coupon
A: Highest maturity, same coupon as remaining bonds
B: Lower yield to maturity than bond E
E: Highest coupon, shortest maturity, highest yield of all bonds.

31. Conceptual problems (2)
Which set of conditions will result in a bond with the greatest price volatility?
A high coupon and a short maturity
A high coupon and a long maturity
A low coupon and a short maturity
A low coupon and a long maturity
Look for the condition giving rise to the highest duration.
Answer is D. Low coupon -> higher duration. Plus long maturity, so the duration will be highest.

32. Conceptual problems (3)
An investor who expects declining interest rates would be likely to purchase a bond that has a ________ coupon and a ________ term to maturity.
Low, long
High, short
High, long
Zero, long
If you expect interest rates to decline, then you expect bond prices to rise. Then you want to buy a bond with the greatest interest rate sensitivity so that you will have the biggest possible price appreciation when interest rates decline. So the task is to choose the bond characteristics that imply the highest interest rate sensitivity.
Generally, long time to maturity corresponds to high durations. So that narrows the field to A, C and D. We know that the lower the coupon rate, the higher the duration, so that eliminates C.
Comparing A and D, we know from rule 1 that the zero coupon bond’s duration = time to maturity. This is higher than the duration of A. Therefore, D has the highest duration and the highest interest rate sensitivity.
Therefore, Answer is D

33. Conceptual problems (4)
With a zero-coupon bond:
Duration equals the weighted average term to maturity.
Term to maturity equals duration.
Weighted average term to maturity equals the term to maturity.
All of the above.
Answer: D.
A, B and C are just different ways of stating the same thing.

34. Summary Interest rate fluctuations are a key risk to bonds.
Maturity is a major determinant of interest rate risk.
Compute Macaulay’s duration and modified duration.
Determinants of Macaulay’s duration.

35. Practice 7
Chapter 11: 4,8,9,12,14.

36. Homework 7
1. A 10% semi-annual coupon bond currently sold at $950 will mature after 20 years. Investors expect that the firm will be able to make good on the remaining interest payments but that at the maturity date bondholders will receive full $1000 par value with 0.5 probability and 70% of par value with 0.5 probability. Compute the expected bond equivalent YTM.
2. Suppose you are a money manager who manages a portfolio of two bonds. A bond is a 5% semi-annual coupon bond with 5 years to maturity. Its par value is $1000. The YTM is 7%. There are 10,000 unit of A bonds hold in the portfolio. B bond is a 3-year 6% semi-annual coupon bond of $500 par value and its YTM is 8%. You hold 5,000 unit of B bond. (You must keep 5 decimal places in all steps, eg. $ , otherwise you get 0)
(1) What is the duration of A and B bond? (you can use excel)
(2) If the YTM for both bonds increase by 50 basis points, how much is the dollar amount change in the portfolio value ? You MUST use modified duration method to solve this question.