102 Session 14

102 session 14 Convexity and immunisation

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What is convexity?

VALUE

Convexity is the “bendiness” of the value of a portfolio with respect to changes in yield

YIELD 405

Why care about convexity?

VALUE

If (solid line) assets are more convex than (dashed line) liabilities (and if other assumptions hold), then portfolio can profit whenever yields change.

Gain i (current) Gain

YIELD 406

Q: How can we get more convexity? Suppose our liability is a payment in 10 year’s time. The only available assets are cash, 10 year bonds, and 20 year bonds. The dashed line (below) shows the present value of the 10 year bond if the 10-year yield changes immediately.

VALUE

What kind of a portfolio do we want?

YIELD

407

A: How can we get more convexity? Our liability is a payment in 10 year’s time.

VALUE

We want a portfolio that will keep more of its value (become cash-like) as yields go high (so portfolio must contain cash), and also rise to higher value (become long-bond-like) as yields fall (so contain some long bonds). We assume that yields are the same at all terms.

Overall, we want assets spread over a larger term than the liabilities

YIELD

408

How do we profit from convexity?

VALUE

The duration of the asset portfolio (a mix of cash and a long bond) rises as yields fall (because the value of the long bond rises, so the portfolio’s duration becomes weighted more to the long bond). As yields fall, the assets (solid line) rise in value more rapidly than the 10 year liability (because the portfolio’s sensitivity rises). As yields rise, the cash in the portfolio holds its value but the 10 year liability value (the dashed line) collapses to zero.

YIELD

409

How do we measure convexity? Convexity measures the “bendiness” of value with respect to yield. If the value-yield curve is convex, ie (like the solid line below), it starts off steep and ends up flat, (hardly changing) that means the sensitivity decreases as the yield rises. Ie d(sensitivity)/di < 0 or d(-sensitivity)/di > 0

VALUE

As it happens, convexity is the measure of d(-sensitivity)/di.

YIELD

410

How do we calculate convexity? Convexity = d(-sensitivity)/di = d (dValue/di / Value) / di = d2Value / di2 / Value

VALUE

(We could just as usefully define convexity as d2Value / dδ2 / Value)

YIELD

411

Can we increase convexity to whatever we want? Yes, but sensitivity gets increased too! If we measure convexity as Convexity = d2Value / dδ2 / Value So convexity of n-year zero coupon bond = d2( e-nδ / dδ2 / e-nδ = n2 Similarly, for a series of cashflows, convexity = average (weighted by cashflow value / total value) of term2 So by borrowing more, and buying more non-zero term assets, you increase the convexity of your assets, (but you increase their duration/sensitivity too). 412

What’s immunisation formula? Immunisation involves locking in any free lunch from small changes in yields – ie it means spreading your assets over a larger term than your liabilities. The immunisation formula is based on Taylor series. You can learn the Taylor series, or just recreate it by considering that when we linearly interpolate, we basically treat curves as straight lines. Ie we say s(x+h) ≈ s(x) + h s’(x), where s(x) is some smooth function of x

h s’(x)

s(x)

x

413

x+h

Taylor series The Taylor series extends to say that, not only s(x+h) ≈ s(x) + h s’(x), but s(x+h) = s(x) + h s’(x) + ½ h2 s’’(x) + … higher powers of h ½ h2 s’’(x)

h s’(x)

s(x)

x

414

x+h

Immunisation formula For any function s(i) s(i+h) = s(i) + h s’(i) + ½ h2 s’’(i) + … higher powers of h Say that s(i) is the size of the surplus when the yield at all terms is i. Say that s(i) = 0 ie value of assets = value of liabilities, and sensitivity of assets = sensitivity of liabilities Ie -d(value assets)/di = -d(value liabilities)/di value assets value liabilities Ie (since value assets = value liabilities) d(value assets)/di = d(value liabilities)/di Ie d(value assets – value liabilities)/di = 0 Ie d(surplus)/di = s’(i) = 0 So s(i+h) = s(i) + h s’(i) + ½ h2 s’’(i) + … higher powers of h

415

Recap: immunisation conditions If 4 conditions hold 1. Yield curve is always flat 2. PV (assets) = PV (liabilities) 3. Sensitivity of assets = sensitivity of liabilities 4. Convexity of assets is greater than convexity of liabilities Then (by 1) the surplus can be represented by s(i) s(i) can be represented as Taylor series s(i+h) = s(i) + h s’(i) + ½ h2 s’’(i) + higher powers of h..ignored if h is small (By 2) s(i) = 0 and (by 3) s’(i) = 0 So s(i+h) = ½ h2 s’’(i) > 0 (by 4) Ie whether h is positive or negative (whether yields rise or fall), ½ h2 is always positive so the surplus rises. 416

Is this a free lunch? Looks like it. E.g. it looks like a flat yield curve would mean that everyone (???) could make free lunches by selling 10 year liabilities and buying 20 year bonds (and cash). So the price of 10 year liabilities will fall (10 year yield will rise) and price of 20 year bonds will rise (long bond yield will fall).

VALUE

So… flat yield curves can’t stay flat for long.

YIELD

417

Specimen Q16(ii)b&c

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A: Specimen Q16(ii)b&c

419

Sep 2000 Q13(iii)

420

A: Sep 2000 Q13(iii)

421

Apr 2001 Q10(ii)

422

A: Apr 2001 Q10(ii)

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Sep 2001 Q12(iii)

424

A: Sep 2001 Q12(iii)

425

Apr 2003 Q5

426

A: Apr 2003 Q5

427

Sep 2003 Q11(ii)

428

A: Sep 2003 Q11(ii)

429

Key question Get 100% on April 2003 Q5 Cover the solution up & do it again until you can explain the solution to someone else.

430

Next session: Random rates END

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