102 Session 11

102 Session 11 Calculate forward price

© KSES Exam questions are copyright Faculty & Institute of Actuaries & are used with their permission Source: www.actuaries.org.uk 326

Jargon: What’s a forward price? It’s a price fixed in advance. Once you agree a forward, you are committed to buy or sell. When you buy a house, you agree a forward price (you don’t renegotiate the price on the day of completion). You could call the redemption payment on a 20 year bond the “20-year-forward price”.

327

Gossip: Was there a fuss about this? Once upon a time, we confused forward prices with expected prices.

BUY ASSET

SELL FORWARD

But then some people pointed out what actually happens. (Coupons, transaction, tax, storage costs etc aside), if you buy a risky asset, and sell it forward one month you end up with a predictable amount of cash in exactly one month. Your counter-party is the market clearing house so you run next to no risk. (Your risky asset has become a risk-free asset.) Or, if you sell at today’s price (and make your risky asset a risk-free asset like cash), and lend the cash for one month at the one-month risk-free rate, you will also end up with a predictable amount of cash in exactly one month. You lend the cash “risk-free” so you run next to no risk. Déjà vu?

LEND CASH

GET PAID BACK

Credit risk etc aside, would you prefer one form of predictable cash to the other? Will the market (ie other people) give you more one way or the 328 other?

Jargon: No arbitrage “No arbitrage” assumes that things that pay the same cost the same. (Equivalently, it assumes that you can’t make risk-free profits (aka “free lunches”) by trading.

LEND CASH

BUY ASSET

GET PAID BACK

SELL FORWARD

e.g. transaction costs aside, if you want a risk free return in one month, then it costs the same whether you arrange it by lending cash at a known rate, or by buying a risky asset which you make risk-free by selling it forward for a known forward price. If, someday, things that pay the same don’t cost the same, buy the cheap one, sell the dear one and pocket the difference. Do no work and retire somewhere pleasant. Trouble is, you’ll drive up the price of the cheap one and push down the price of the 329 dear one. So, soon the free lunch is all gone.

So what’s new? Nothing, We’ve been assuming no arbitrage all along, when we wrote down formulae like Price = 10v(1) + 10v(2) +

110v(3),

we assumed that there was only one price, Ie we assumed that no one could make riskfree profits by selling at one price and buying at another.

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Specimen Q16(iii)

Cash if held to redemption 120 100 80 60 40 20 0

1

1.5

2

…

19 19.5 20

What is price at time 1? Upper bound: If price were 110, net yield would be (1 – 25%) * 10% / 110 = 6.8%. But net redemption yield (10%) >> 6.8% ⇒Must be capital gain ⇒ price << redemption amount = 110 Guess duration = 10 Guess 1% fall in yield increases price by about 10%. ⇒Guess price = 110 / {1 + (10% - 6.8%) * 10} = 83. Exact price at time 1 = 3.75 a38 + 110 v38 @ √1.10 – 1 = 3.75 * 17.138 + 110 * 0.1635 = 82.254%

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Specimen Q16(iii)

Cash-flow if buy a bond and sell it forward = -Current price + 3.75 a4(2) + Forward v4 = -82.254 + 3.75 a4(2) + Forward v4

Value if held to redemption 120 100 80 60

The risk free rate is 10% ie 82.254 = 3.75 a8 + Forward v8 @ √1.10 – 1 = 3.75 * 6.4944 + Forward * 0.68301

40 20 0

1

1.5

2

…

19 19.5 20

Value if sold forward 140 120 100 80 60 40 20 0

⇒Forward = (82.254 – 3.75 * 6.4944) / 0.68301 = 84.77% If forward price were higher than 84.77, borrow at 10%, buy the bond, sell it forward and in four years, pocket a risk-free profit.

1

1.5

2

…

3.5

4

If forward price were less than 84.77, pocket a risk-free profit NOW by selling (or shorting) the bond, lending some of the332 proceeds at 10% and buying the bond forward.

Specimen Q16(iii)

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Apr 2000 Q9

80

Roughly: dividend yield is 4/60 in just 7 months (ie more than 7% pa).

60 40 20 0 -20

0

1

2

3

4

-40

5

6

7

So make a capital loss (to get risk-free 7%) Ie forward price < 60.

-60 -80

Exactly, 60 =

2v3/12 +

2v6/12 + Forward v7/12

60 = 2e-(7% x 3/12) + 2e-(7% x 6/12) + Forward e-(7% x 7/12) 60 = 2 * 0.98265 + 2 * 0.965605 + Forward * 0.959989 ⇒Forward = £58.44 i.e.

334

Apr 2000 Q9

335

Sep 2000 Q3

336

Sep 2000 Q3

F

120 “grows” to 120 * 1.05(91/365)

120

F -120

Equivalently, (just using a different picture) 120 = Forward * 1.05-(91/365) ⇒ Forward = 120 * 1.05(91/365) 337

Sep 2000 Q3

338

Apr 2001 Q4

339

Apr 2001 Q4(ii)

200 100 0 -100

0

1

2

3

Roughly: Ignore dividends and interest About £30 of value is the special dividend So expect £30 in April and £120 in May Making total value £150

-200

Ie guess forward = £120 Assuming dividends don’t increase: 150 = 3% * 150 ā3/12¬ + 30 v(2/12) + Forward v(3/12) @ δ = 5% = 3% * 150 * 0.24844 + 30 * 0.99170 + Forward * 0.987578 => Forward = £120.63 Assuming dividends increase continuously at 2% pa 150 = 3% * 150 ā3/12¬ @ δ = 3% + 30 v(2/12) @ δ 5% + Forward v(3/12) @ δ 5% = 150 (1 – e-(3% * 3/12) ) + 30 e-(5% * 2/12) + Forward e-(5% * 3/12) => Forward = 150 e[ (5%- 3%) * 3/12] - 30 e(5% * 1/12) = £120.63 (again)

340

Apr 2001 Q4

341

Sep 2001 Q3

342

Apr 2002 Q1

343

Sep 2002 Q2

344

Sep 2003 Q4

345

Key question Get 100% on Apr 2000 Q9. It doesn’t matter how many times you see the answer. Cover the answer up & do it again until someone you explain it to goes “Aaah!”

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Next session: forward rates

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