Credit duration under static credit curve

Thanks to the Magician, I solved most of the questions.
One remains outstanding - see under 1.
How were the coupon (benchmark yield) of 187.500 and 275.000 calculated? :hot_face: :woozy_face: :woozy_face:

Adding Credit Duration under a Static Credit Curve

A Sydney-based investor notes the following available option-free bonds for an A rated Australian issuer:

The 5-year, 10-year, and 15-year Australian government bonds have YTMs and coupons of 0.50%, 0.75%, and 1.10%, respectively, and both corporate and government bonds have a semiannual coupon. As an active manager who expects stable benchmark yields and credit spreads over the next six months, the investor decides to overweight (by AUD50,000,000 in face value) the issuerā€™s 15-year versus 10-year bond for that period. Calculate the return to the investor of the roll-down strategy in AUD and estimate the returns attributable to benchmark yield versus credit spread changes.

Solution:

To estimate credit curve roll-down returns, we must solve for the first two return components from Coupon income +/āˆ’ Roll-down return and separate the impact of benchmark yield versus credit spread changes.

1 Solve for the respective 5-year, 10-year, and 15-year bond credit spreads. Yield spread and G-spread are reasonable approximations because the bonds are option-free, with maturities closely aligned to par government securities.

5-year spread: 0.50% (= 1.00% āˆ’ 0.50%)

10-year spread: 0.50% (= 1.25% āˆ’ 0.75%)

15-year spread: 0.85% (= 1.95% āˆ’ 1.10%)

2 Solve for 6-month expected returns of the 10-year versus 15-year bond:

a Incremental coupon income = $162,500 (= (2.00% āˆ’ 1.35%)/2 Ɨ $50 million)

How were the coupon (benchmark yield) of 187.500 and 275.000 calculated? :hot_face: :woozy_face: :woozy_face:

Divide incremental coupon into benchmark and credit spread components:

Income due to benchmark yields: $87,500 = $275,000 āˆ’ $187,500

Income due to credit spreads: $75,000 = $225,000 āˆ’ $150,000

b Price appreciation is determined by the bondā€™s price today and in six monthsā€™ time based on unchanged benchmark rates. In six months, the 10-year and 15-year positions will be 9.5-year and 14.5-year bonds, respectively, at a yield and yield spread point along the curve. Estimate all-in YTMs and yield spreads using interpolation to arrive at the following results:

Calculate price appreciation using the difference between current bond prices and those in six months using the Excel PV function (= āˆ’PV(rate, nper, pmt, FV, type)) where ā€œrateā€ is the interest rate per period (0.01225/2), ā€œnperā€ is the number of periods (19), ā€œpmtā€ is the periodic coupon (1.35/2), ā€œFVā€ is future value (100), and ā€œtypeā€ (0) involves payments made at the end of each period.

10-year: Initial price: 100.937

Price in six months: 101.118 (= āˆ’PV (0.01225/2, 19, 1.35/2, 100, 0))

Price appreciation: $89,660 (= (101.118 āˆ’ 100.937)/100.937 Ɨ $50 million)

Because the yield spread curve is flat at 0.50%, the full $89,660 price change in the 10-year is benchmark yield curve roll down.

15-year: Initial price: 100.648

Price in six months: 101.517 (= āˆ’PV (0.0188/2, 29, 1, 100, 0))

Price appreciation: $431,700 (= (101.517 āˆ’ 100.648)/100.648 Ɨ $50 million)

Because the 0.07% decline in YTM is estimated to be equally attributable to benchmark yield and yield spread changes, each is assumed equal to $215,850.
3 Incremental income due to price appreciation is therefore $342,040 (=$431,700 āˆ’ $89,660), of which $215,850 is attributable to credit spread changes. In total, the incremental roll-down strategy generates $504,540 (=$342,040 + 162,500), of which $290,850 (= $215,850 + $75,000) is estimated to be due to credit spread curve roll down.

What was the YTM on the 15-year bond? Whatā€™s the YTM on the (now) 14Ā½-year bond?

Thatā€™s half of the 431,700 price change. Whatā€™s the change in the benchmark yield? How about in the spread?

Incremental coupon income and income from credit spreads, respectively. You have those numbers near the top of your post.

ā€œWhat was the YTM on the 15-year bond? Whatā€™s the YTM on the (now) 14Ā½-year bond?ā€

Got it, however, how do we calculate the all-in yields (for all maturities given, 5y, 10y, 15y, 9.5y and 14.5 y) and the benchmark yields for 9.5 y and 14.5 years? How did they get to the 1,88% figure, for instance? Thanks!

Got me.

On the real exam theyā€™d have to give you that number; the point of this question isnā€™t for you to do some fancy interpolation to get it.

@S2000magician can you explain how the coupon (Credit spread) is calculated? I thought it was the (credit spread for the respective time period/2)$5mm? I.e. for the 15 year spread itā€™d be (.0085/2)$5mm however this isnā€™t jibing with the figures shown above.

Pick up coupon credit bond
Difference between coupons on 15 y and 10y on single A bond
$50m x (2.0% - 1.35%) x 0.5 = 162,500

Similar calculation on gvt bond. What we have got with same trade in gvt bond
$50m x (1.1% - 0.75%) x 0.5 = 87,500

Remaing gain due to credit 162,500 - 87,500 = 75,000

Calcs just on 15 year bonds
Credit coupon $50m x (2.0%) x 0.5 = 500,000
Gvt coupon $50m x (1.1%) x 0.5 = 275,000
Difference - coupon(credit spread) 225,000

Or in one go
10yr
coupon(credit spread) = $50m x (2% - 1.1%) *0.5 = 225,000
Whole postion
$50m x [ (2.0% - 1.35%) - (1.1% - 0.75%) ] x 0.5 = 75,000
position x [ relative pick up in coupon on corp bond versus gvt bond ] x half a year = 75,000

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Thank you @MikeyF

In this example, Iā€™m struggling to understand :

  1. what ā€œthe investor decides to overweight (by AUD50,000,000 in face value) the issuerā€™s 15-year versus 10-year bond for that period.ā€ means (Iā€™m assuming this means that the 15y position is higher than the 10y position by AUD50,000,000 in face value?)

  2. and why ā€œIncremental coupon incomeā€ is the difference between the two coupons (and not just the coupon on the AUD50,000,000 in face value of the 15-year instead).

Can somebody please explain ?

  1. what ā€œthe investor decides to overweight (by AUD50,000,000 in face value) the issuerā€™s 15-year versus 10-year bond for that period.ā€ means (Iā€™m assuming this means that the 15y position is higher than the 10y position by AUD50,000,000 in face value?)

THey go long the 15 year bond and short the 10 year bond

  1. and why ā€œIncremental coupon incomeā€ is the difference between the two coupons (and not just the coupon on the AUD50,000,000 in face value of the 15-year instead).

Because they are long the 15 year and short the 10 year

That makes sense. Thank you.