Tutorial - Bonds - solved problems PDF

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Tutorial – Fixed income

Question 1: You have estimated spot rates as follows: Year

Spot Rate

1 5.00% 2 5.40% 3 5.70% 4 5.90% 5 6.00% ____________________________ (a) What are the discount factors for each date (that is, the present value of $1 paid in year t)? (b) What are the forward rates for each period? (c) Calculate the PV of the following Treasury notes (with face value 100). Assume annual coupons: a. 5%, 2-year note b. 5%, 5-year note c. 10%, 5-year note (d) Explain intuitively why the yield to maturity on the 10% bond is less than that on the 5% bond. (e) What should be the yield to maturity on a 5-year zero-coupon bond? (f) Show that the correct yield to maturity on a 5-year annuity is 5.75%. For simplicity, assume that the annuity pays 1 every year (with the first payment starting in time 1!).

Question 2: The following table shows the prices of a sample of strips of UK gilts (government bonds) in December 2008. Each strip makes a single payment of 100 pounds at maturity (and no coupon payments). Maturity

Price

December 2010 December 2015 December 2016 December 2017 December 2018

90.826 73.565 70.201 67.787 65.528

(a) Calculate the annually compounded, spot interest rate for each year. (b) Is the term structure upward- or downward-sloping? (c) Would you expect the yield on a coupon bond maturing December 2018 to be higher or lower than the yield on the 2018 strip?

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(d) Calculate the annually compounded, one-year forward rate of interest for December 2015. Now do the same for December 2016.

Question 3: (a) Find the price of a 5% coupon bond that has three years until maturity, pays annual coupons, and has a par value of 100. Use the following prices for zero coupon bonds paying 1 at maturity: 1 year 2 years 3 years 4 years

0.9748 0.9493 0.9238 0.90

(b) What are the holdings of zero coupon bonds in the replicating portfolio that has the same cash flows as this coupon bond?

Question 4: (a) Today is 21st October 2009, what would happen to the value of the 12 5/8% Treasury Bond with face value $1000, maturing on 21th October 2014 if the market interest fell from 7.6% to 6%? Assume annual coupons. (b) What would happen if the value rose to 9%?

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Tutorial 4 – Answer Key Question 1: You have estimated spot rates as follows: Year

Spot Rate

6 5.00% 7 5.40% 8 5.70% 9 5.90% 10 6.00% ____________________________ (a) value of $1 paid

What are the discount factors for each date (that is, the present in year t)? 1 DF t = t ( 1+r t ) to compute DF for each date: We'll use this equation: Year 1

Spot rate 5.00%

2

5.40%

3

5.70%

4

5.90%

5

6.00%

Formula

DF

1/(1+0.050)

. 9524 . 9002 . 8468 . 7951 . 7473

1/ (1+0.054)^2 1/ (1+0.057)^3 1/ (1+0.059)^4 1/ (1+0.060)^5

(b) What are the forward rates for each period?

f (0 ,m , m+1 )= Using this equation: we can compute the forward rates: Year F(0,1,2 ) F(0,2,3 ) F(0,3,4

B(0,m ) .9524

B(0,m+1 ) .9002

.9002

.8468

.8468

.7951

B(0 , m) −1 B (0 , m+1 ) , Formula [.9524/.9002] – 1 [.9002/.8468] – 1 [.8468/.7951] –

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Forward rate 5.799% 6.306% 6.502%

) F(0,4,5 )

.7951

.7473

1 [.7951/.7473] – 1

6.396%

Note that these forward rates are already annualised as we are using annual spot rates to calculate them.

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(c) Calculate the PV of the following Treasury notes (with face value 100): a. 5%, 2-year note A 5% 2 year note (with face value 100) will pay a coupon of 5 in year 1 and a coupon of 5 plus principal of 100 in year 2. Since the discount factor calculates the PV of cash flows, the discount factors as calculated above for each year can be used to calculate the PV of the cash flows of this note. Specifically, for year 0: PVo = CFo*DFo = 0*1 = 0 Year 1: PV1 = CF1*DF1 = 5*.9524 = 4.76 Year 2: PV2 = CF2*DF2 = 105*.9002 = 94.52 The PV of the bond is the total of the PVs of each CF, or 99.28. A summary of the result is as follows: Time Cash Flow Discount Factor PV Cash Flows PV of Bond

0 0 1.0000 0.00 99.28

1 5 0.9524 4.76

2 105 0.9002 94.52

b. 5%, 5-year note Time Cash Flow Discount Factor PV Cash Flows PV of Bond

0 0 1.0000 0.00 95.93

1 5 0.9524 4.76

2 5 0.9002 4.50

3 5 0.8468 4.23

4 5 0.7951 3.98

5 105 0.7473 78.47

1 10 0.9524 9.52

2 10 0.9002 9.00

3 10 0.8468 8.47

4 10 0.7951 7.95

5 110 0.7473 82.20

c. 10%, 5-year note Time Cash Flow Discount Factor PV Cash Flows PV of Bond

0 0 1.0000 0.00 117.14

(d) Explain intuitively why the yield to maturity on the 10% bond is less than that on the 5% bond. The YTM is a weighted average of the different annual spot rates. In a 10% bond the weight to the early period is bigger than in a 5% bond (e.g., the first payment on the 5% bond represents 4% of the total payments whereas for the 10% bond it represents 6.67% of the payments). Since the term structure is upward sloping, the early periods spot rates are lower than the late periods and therefore the YTM on a 10% bond is lower than the YTM on a 5% bond. (e) What should be the yield to maturity on a 5-year zero-coupon bond? The YTM on a five-year zero will be 6%. This is because the YTM for a ZCB is the spot rate. (f) Show that the correct yield to maturity on a 5-year annuity is 5.75%. For simplicity, assume that the annuity pays 1 every year (with the first payment starting in time 1!). To calculate the YTM on a 5-year annuity it is necessary to start by calculating the PV of all cash flows from the annuity according to the term structure of interest rates within which we are working. Applying the discount factors as calculated above to

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these payments yields the PVs. (i.e. PV1 = CF1* DF1 = 1*.9524 and so on for each period as above). Summing the individual PVs gives the PV of the Annuity (4.24). We then proceed as above (in part (d)). Start with the PV and known stream of cash flows. Then determine the YTM that will yield the correct discount factors to produce the ‘target’ PV. A summary of the result is as follows: Value Using Term Structure of Interest Rates Time 0 1 2 Cash Flow 0 1 1 Discount Factor 1 0.9524 0.9002 PV Cash Flows 0.00 0.95 0.90 PV of Annuity 4.24

3 1 0.8468 0.85

4 1 0.7951 0.80

5 1 0.7473 0.75

Solving for YTM with Cash Flows YTM 5.75% Time 0 1 Cash Flow 0 1 Discount Factor 1 0.9456 PV Cash Flows 0.00 0.95 PV of Annuity 4.24 Target PV 4.24

3 1 0.8456 0.85

4 1 0.7996 0.80

5 1 0.7561 0.76

2 1 0.8942 0.89

Question 2: The following table shows the prices of a sample of strips of UK gilts (government bonds) in December 2008. Each strip makes a single payment of 100 pounds at maturity. Maturity

Price

December 2010 December 2015 December 2016 December 2017 December 2018

90.826 73.565 70.201 67.787 65.528

(a) Calculate the annually compounded, spot interest rate for each year. The strips are essentially ZCBs. As such, the prices represent the PVs of the cash flow of each strip. Since the discount factor can be computed as price/principal value, the discount factors for each year can be easily computed here (just divide each price by 100).

To go from the discount factor to the spot rate we can use the formula: For the first spot, face = 100 price = 90.826 df = price / face => 0.9083 t = 2010-2008=> 2 r = (1/df)^(1/t)-1 in % => 4.9288%

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1 1t −1 rt= DF t

( )

The table below shows all spot rates

(b) Is the term structure upward- or downward-sloping? The term structure is downward sloping since the spot rates are decreasing in maturity, with the exception of a small “hump” for the 8-year spot rate. (c) Would you expect the yield on a coupon bond maturing December 2018 to be higher or lower than the yield on the 2018 strip? I would expect the yield to be higher since the term structure is downward sloping, so less weight is given to cash flows accruing later. (d) Calculate the annually compounded, one-year forward rate of interest for December 2015. Now do the same for December 2016. Using the same formula from Q.1 we get: f(0,7,8) = [73.565/70.201] –1 = 4.79% f(0,8,9) = [70.201/67.787] –1 = 3.56%

Question 3: (a) What is the fair price of a 5% coupon bond with three years maturity paying annual coupons, knowing that the set of zero coupon bond prices is: 1 year 2 years 3 years 4 years

0.9748 0.9493 0.9238 0.90

The fair price of a coupon bond can be calculated using the prices of a set of ZCB due to the concept of replication. The idea is that a coupon bond can be replicated by a series of ZCBs (i.e. each payment is represented by a ZCB) such that the prices of the ZCBs, which represent the discount factors for each year, can be applied to the cash flows of the coupon bond. These discount factors are used to calculate the PV of cash flows for the coupon bond. The sum of the PV of cash flows is then the fair price. Specifically, PV1 = 5*.9748 = 4.874 PV2 = 5*.9493 = 4.747 And so on. The Sum of the PVs = 106.620 A summary of the result is as follows:

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(b) What is the replicating portfolio of zero coupon bonds that is equivalent to this bond? As stated above, the coupon bond can be represented by a series of ZCBs. The cash flow the coupon bond pays each period represents the face amount of ZCBs maturing in each period that you need to buy. Since the par value of the ZCBs are 100, we must divide the face amount we need each period by the 100 to get the number of ZCBs we need to buy for each period. Therefore, we need $5 in year 1, so $5/$100 = 0.05 ZCB maturing in year 1. $5 in year 2, so $5/$100 = 0.05 ZCB maturing in year 2. $105 in year 3, so $105/$100 = 1.05 ZCB maturing in year 3. The PV of each cash flow of the coupon bond represents the present value of the replicating ZCB position for that year. Therefore,

PV of ZCB1 = .9745*5 = 4.874 PV of ZCB2 = .9493*5 = 4.747 And so on for each of the years.

A summary of the result is as follows:

Question 4: (a) Today is 21st October 2009, what would happen to the value of the 12 5/8% Treasury Bond with face value $1000, maturing on 21th October 2014 if the market interest fell from 7.6% to 6%? Assume annual coupons. (b) What would happen if the value rose to 9%? Today Initial discount rate Discount rate 1

Oct-09 7.60% 6.00%

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Discount rate 2 Date Time Period Coupon Principal Total cash flows Discount factor initial Discount Factor 1 Discount Factor 2 PV CV Initial PV CF 1 PV CF 2 PV of Bond Intial PV of Bond 1 PV of Bond 2

9.00% Oct-09 0 0 0 0 1 1 1 0 0 0 1202.767 1279.069 1141

Oct-10 1 126.25 0 126 0.9294 0.9434 0.9174 117.33 119.10 115.83

Oct-11 2 126.25 0 126.25 0.8637 0.8900 0.8417 109.05 112.36 106.26

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Oct-12 3 126.25 0 126.25 0.8027 0.8396 0.7722 101.34 106.00 97.49

Oct-13 4 126.25 0 126.25 0.7460 0.7921 0.7084 94.19 100.00 89.44

Oct-14 5 126.25 1000 1126.25 0.6933 0.7473 0.6499 780.86 841.60 731.99...