Q1

c.

The price-yield curve of a bond is a convex one as shown above. And the bonds duration at a specific yield level is the slope of the curve at that interest rate (i.e. the slope of the tangent line to the curve at that interest rate level). When interest rate increases, the curve becomes flatter, and hence the duration becomes smaller. So when you use the duration rule to calculate price changes: P/P ~= -Dy/(1+y), the duration is that at the original interest rate (6%) which is bigger than the actual durations (which keeps changing, become smaller and smaller) over the course of interest rate increase. And thus the estimated (negative) price change is too big in magnitude. So it is clear from the discussion above that the approximation error is due to convex price-yield curve and thus negative duration-yield relationship. Now for your own exercise, please assume interest rate decreases from 6% to 5%, and redo a) and b). What do you find? And please follow my arguments above to explain the findings. d. P/P ~= -Dy/(1+y) + 0.5convexity(y)2 I assume everyone can do this calculation. e. From c we know that the approximation error of duration rule is due to the convex price-yield curve and the change in duration when interest rate changes. You can easily see from the price-yield curve above that the curve becomes flatter when interest rate is higher the curve is closer to a linear line (it becomes almost a flat line when interest rate is very high). And the flatter (or less convex) the curve is, the smaller the change in duration due to change in interest rate, and hence the smaller the approximation error if we use duration to estimate price change.

Q4 P/P ~= -Dy/(1+y). so the percentage price change depends on the duration. This question essentially tests the determinants of bond duration. We know from the lecture that, everything else equal, the bond duration: i) increases with maturity, but at a decreasing rate; ii) decreases with coupon rate; iii) decreases with interest rate (or more specifically yield). a) Bonds A, B and C have the same time-to-maturity. The coupon rate is 0% for bond A, 6% for bond B and 7% for bond C. Further bonds A and B have the same YTM of 6%, while bond C has a YTM of 7%. So based on the relationship between duration and coupon rate, we know that DA > DB > DC, if everything else equal. And actually bond C even has a higher YTM, which makes it even more obvious that its duration should be lowest. So the percentage price change should be largest for bond A, and lowest for bond C. Please verify this with your calculation. b) this is no-brainer the only difference between these two bonds is that bond D has a longer maturity, and thus a higher duration. c) everything else equal, duration increases with maturity at a decreasing rate. So DE > DD > DC, but DE - DD < DD - DC Hence the difference in percentage price change should be between bonds C and D. again, please verify it. d) the only difference between bonds E and F is the YTM bond F has a higher YTM (its YTM > 7%, where YTM of bend E = 7%). The answer is clear based on the relationship btw duration and YTM.

Selected end-of-chapter questions BKM16: 1-4, 7, 8a, 9, 12, 15, 22(a, b and c) 1. While it is true that short-term rates are more volatile than long-term rates, the longer duration of

the longer-term bonds makes their prices and their rates of return more volatile. The higher duration magnifies the sensitivity to interest-rate changes.

2. Duration can be thought of as a weighted average of the maturities of the cash flows paid to holders of the perpetuity, where the weight for each cash flow is equal to the present value of that cash flow divided by the total present value of all cash flows. For cash flows in the distant future, present value approaches zero (i.e., the weight becomes very small) so that these distant cash flows have little impact and, eventually, virtually no impact on the weighted average.

3. The percentage change in the bonds price is:

7.194 0.005 0.0327 3.27%,1 1.10

D yy

= = = +

or a 3.27% decline

4. a. YTM = 6%

(1) (2) (3) (4) (5)

Time until Payment (Years)

Cash Flow

PV of CF (Discount

Rate = 6%)

Weight

Column (1) Column (4)

1 $ 60.00 $ 56.60 0.0566 0.0566 2 60.00 53.40 0.0534 0.1068

3 1,060.00 890.00 0.8900 2.6700

Column sums $1,000.00 1.0000 2.8334

Duration = 2.833 years

b. YTM = 10%

(1) (2) (3) (4) (5)

Time until Payment (Years)

Cash Flow

PV of CF (Discount

Rate = 10%)

Weight

Column (1) Column (4)

1 $ 60.00 $ 54.55 0.0606 0.0606 2 60.00 49.59 0.0551 0.1102

3 1,060.00 796.39 0.8844 2.6532

Column sums $900.53 1.0000 2.8240

Duration = 2.824 years, which is less than the duration at the YTM of 6%.

7. Bond d. Investors tend to purchase longer term bonds when they expect yields to fall so they can capture significant capital gains, and the lack of a coupon payment ensures the capital gain will be even greater.

8. a. Bond B has a higher yield to maturity than bond A since its coupon payments and maturity are equal to those of A, while its price is lower. (Perhaps the yield is higher because of differences in credit risk.) Therefore, the duration of Bond B must be shorter.

9. a.

(1) (2) (3) (4) (5)

Time until Payment (Years)

Cash Flow

PV of CF (Discount Rate = 10%)

Weight

Column (1) Column (4)

1 $10 million $ 9.09 million 0.7857 0.7857 5 4 million 2.48 million 0.2143 1.0715

Column sums $11.57 million 1.0000 1.8572

D = 1.8572 years = required maturity of zero coupon bond.

b. The market value of the zero must be $11.57 million, the same as the market value of the obligations. Therefore, the face value must be:

$11.57 million (1.10)1.8572 = $13.81 million

12. a. PV of the obligation = $10,000 Annuity factor (8%, 2) = $17,832.65 (1) (2) (3) (4) (5)

Time until Payment (Years)

Cash Flow

PV of CF (Discount

Rate = 8%)

Weight

Column (1) Column (4)

1 $10,000.00 $ 9,259.259 0.51923 0.51923 2 10,000.00 8,573.388 0.48077 0.96154

Column sums $17,832.647 1.00000 1.48077

D = 1.4808 years

b. A zero-coupon bond maturing in 1.4808 years would immunize the obligation. Since the present value of the zero-coupon bond must be $17,832.65, the face value (i.e., the future redemption value) must be

$17,832.65 1.081.4808 = $19,985.26

c. If the interest rate increases to 9%, the zero-coupon bond would decrease in value to

92.590,17$09.1

26.985,19$4808.1 =

The present value of the tuition obligation would decrease to $17,591.11

The net position decreases in value by $0.19 If the interest rate decreases to 7%, the zero-coupon bond would increase in value to

99.079,18$07.1

26.985,19$4808.1 =

The present value of the tuition obligation would increase to $18,080.18

The net position decreases in value by $0.19 The reason the net position changes at all is that, as the interest rate changes, so does the duration of the stream of tuition payments.

15. a. The duration of the annuity if it were to start in one year would be

(1) (2) (3) (4) (5)

Time until Payment (Years)

Cash Flow

PV of CF (Discount

Rate = 10%)

Weight

Column (1) Column (4)

1 $10,000 $ 9,090.909 0.14795 0.14795 2 10,000 8,264.463 0.13450 0.26900

3 10,000

7,513.148 0.12227 0.36682

4 10,000 6,830.135 0.11116 0.44463

5 10,000 6,209.213

0.10105 0.50526

6 10,000

5,644.739 0.09187 0.55119

7 10,000 5,131.581 0.08351 0.58460

8 10,000

4,665.074 0.07592 0.60738

9 10,000 4,240.976 0.06902 0.62118

10 10,000 3,855.433 0.06275 0.62745

Column sums $61,445.671 1.00000 4.72546

D = 4.7255 years

Because the payment stream starts in five years, instead of one year, we add four years to the duration, so the duration is 8.7255 years.

b. The present value of the deferred annuity is

968,41$10.1

)10%,10(factor Annuity 000,104 =

Alternatively, CF 0 = 0; CF 1 = 0; N = 4; CF 2 = $10,000; N = 10; I = 10; Solve for NPV = $41,968.

Call w the weight of the portfolio invested in the five-year zero. Then

(w 5) + [(1 w) 20] = 8.7255 w = 0.7516

The investment in the five-year zero is equal to

0.7516 $41,968 = $31,543

The investment in the 20-year zeros is equal to

0.2484 $41,968 = $10,423

These are the present or market values of each investment. The face values are equal to the respective future values of the investments. The face value of the five-year zeros is

$31,543 (1.10)5 = $50,801

Therefore, between 50 and 51 zero-coupon bonds, each of par value $1,000, would be purchased. Similarly, the face value of the 20-year zeros is

$10,425 (1.10)20 = $70,123

22. a. The price of the zero-coupon bond ($1,000 face value) selling at a yield to maturity of 8% is $374.84 and the price of the coupon bond is $774.84.

At a YTM of 9%, the actual price of the zero-coupon bond is $333.28 and the actual price of the coupon bond is $691.79.

Zero-coupon bond:

Actual % loss %09.111109.084.374$

84.374$28.333$==

= loss

The percentage loss predicted by the duration-with-convexity rule is:

Predicted % loss [ ] [ ] %06.111106.001.03.1505.001.0)81.11( 2 ==+= loss Coupon bond:

Actual % loss$691.79 $774.84 0.1072,or10.72%

$774.84

= = loss

The percentage loss predicted by the duration-with-convexity rule is:

Predicted % loss [ ] 2( 11.79) 0.01 0.5 231.2 0.01 0.1063,or10.63% = + = loss

b. Now assume yield to maturity falls to 7%. The price of the zero increases to $422.04, and the price of the coupon bond increases to $875.91.

Zero-coupon bond:

Actual % gain$422.04 $374.84 0.1259,or12.59%

$374.84

= = gain

The percentage gain predicted by the duration-with-convexity rule is:

Predicted % gain [ ] 2( 11.81) ( 0.01) 0.5 150.3 0.01 0.1256, 12.56%or = + = gain Coupon bond:

Actual % gain$875.91 $774.84 0.1304,or13.04%

$774.84

= = gain

The percentage gain predicted by the duration-with-convexity rule is:

Predicted % gain [ ] 2( 11.79) ( 0.01) 0.5 231.2 0.01 0.1295,or 12.95% = + = gain

c. The 6% coupon bond, which has higher convexity, outperforms the zero regardless of whether rates rise or fall. This can be seen to be a general property using the duration-with-convexity formula: the duration effects on the two bonds due to any change in rates are equal (since the respective durations are virtually equal), but the convexity effect, which is always positive, always favors the higher convexity bond. Thus, if the yields on the bonds change by equal amounts, as we assumed in this example, the higher convexity bond outperforms a lower convexity bond with the same duration and initial yield to maturity.