102 Session 10

102 Session 10 Calculate par yield

© KSES Exam questions are copyright Faculty & Institute of Actuaries & are used with their permission Source: www.actuaries.org.uk 309

Jargon: What’s a par yield? It’s a kind of average yield. The par yield is p%, where a bond paying a coupon of p% pa is priced at par 200

What does “priced at par” mean? Priced at par means the price is 1, or 100%. Or, put another way, the price per £100 nominal is £100.

100

0

-100

Cost 100

Pay 100 (+p)

-200

E.g. suppose the 5 year par-yield is 7%. Then a 5 year bond paying a 7% annual coupon would be priced at £100 per £100 nominal. 310

Specimen Q3

311

Specimen Q3

Rough guess: par yield ≈ …. year spot yield ≈ average forward rate = 1/3 * (6% + 6½% + 7%) = ….% 0

1

2

3

Par yield is p: bond paying p is priced at par Ie …….. = pv(1) + pv(2) + (100+p)v(3) => 100 – …… v(3) = p[ v(1) + v(2) + v(3) ]. Interest rate isn’t constant: v(1) = 1/……. v(2) = v(1) / 1.065 v(3) = v(2) / ……. ⇒p = ……. per £100 nominal ⇒Par yield = ……..% (close to guess?)

1 "grows to"

1

1.06

1.06 * 1.065

1.06 * 1.065 * 1.07

312

Specimen Q3

Rough guess: par yield ≈ 3 year spot yield ≈ average forward rate = 1/3 * (6% + 6½% + 7%) = 6½% 0

1

2

3

Par yield is p: bond paying p is priced at par Ie 100 = pv(1) + pv(2) + (100+p)v(3) => 100 – 100 v(3) = p[ v(1) + v(2) + v(3) ]. Interest rate isn’t constant: v(1) = 1/1.06 V(2) = v(1) / 1.065 V(3) = v(2) / 1.07. ⇒p = 6.478 per £100 nominal ⇒Par yield = 6.478% (close to guess)

1 "grows to"

1

1.06

1.06 * 1.065

1.06 * 1.065 * 1.07

313

Specimen Q3

Model answer

314

Apr 2001 Q8(ii)(b)

315

Apr 2001 Q8(ii)(b)

0

1

2

Par yield is p: bond paying p is priced at par Ie …….. = pv(1) + (100+p)v(2) ⇒100 – …… v(2) = p[ v(1) + v(2) ]. Interest rate isn’t constant: v(1) = 1 / (1 + y….) v(2) = 1 / (1+y2)….. ⇒…….–100 * 1.042-2 = p [ 1.041-1 + 1.042-2] ⇒p% = Par yield = …….% (close to guess?)

Close guess: par yield ≈ ….. year spot yield = 0.04 + …/1000 = ……………% 1 "grows to"

1

1.041

1.042^2

316

Apr 2001 Q8(ii)(b)

0

1

2

Par yield is p: bond paying p is priced at par Ie 100 = pv(1) + (100+p)v(2) ⇒100 – 100 v(2) = p[ v(1) + v(2) ]. Interest rate isn’t constant: v(1) = 1 / (1 + y1) v(2) = 1 / (1+y2)2 ⇒100–100 * 1.042-2 = p [ 1.041-1 + 1.042-2] ⇒p% = Par yield = 4.198% (close to guess)

Close guess: par yield ≈ 2 year spot yield = 0.04 + 2/1000 = 0.042 = 4.2% 1 "grows to"

1

1.041

1.042^2

317

Apr 2001 Q8(ii)(b)

Model answer

318

Apr 2003 Q6(ii)

319

Apr 2003 Q6(ii) growth factors

1 at time 0 "grows to"

1 at time 1 "grows to"

1.118

1

1 at time 0 "grows to"

1 at time 1 "grows to"

1

1.058

1.14

1

1

1.118 / 1.058

1.118

1.14 x 1.118 / 1.058 320

Apr 2003 Q6(ii)

Rough guess: par yield ≈ 3 year spot yield = [………x 1.118 / 1.058 ) ]⅓ - 1 = ……..% 0

1

2

3

Par yield is p: bond paying p is priced at par Ie …….. = pv(1) + pv(2) + (100+p)v(3) => 100 – 100 v(3) = …..[ v(1) + v(2) + v(3) ]. Interest rate isn’t constant: v(1) = …….. / (1118 / 1.058) v(2) = 1000 / …………. v(3) = V(1) * ………. / 1140. ⇒Par yield = p% = 6.36% (near guess)

1 at time 0 "grows to"

1

1.118 / 1.058

1.118

1.14 x 1.118 / 1.058 321

Apr 2003 Q6(ii)

Rough guess: par yield ≈ 3 year spot yield = [1.14 x (1.118 / 1.058 ) ]⅓ - 1 = 6.4% 0

1

2

3

Par yield is p: bond paying p is priced at par Ie 100 = pv(1) + pv(2) + (100+p)v(3) => 100 – 100 v(3) = p[ v(1) + v(2) + v(3) ]. Interest rate isn’t constant: v(1) = 1000 / (1118 / 1.058) v(2) = 1000 / 1118 v(3) = V(1) * 1000 / 1140. ⇒Par yield = p% = 6.36% (near guess)

1 at time 0 "grows to"

1

1.118 / 1.058

1.118

1.14 x 1.118 / 1.058 322

Apr 2003 Q6(ii)

Model answer (assumes you already have calculated the spot rates i1, i2, i3)

323

Key question Understand April 2003 Q6(ii). It doesn’t matter how many times you see the answers. You understand when you can make it seem straightforward to someone else.

324

Next session: forward prices

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