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102 Session 13 Calculate duration

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with their permissionSource: www.actuaries.org.uk

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DMT (roughly speaking)(Vaguely) discounted mean term describes the average term of the cashflows.

We’ve been describing DMT for a while in our rough estimates of values.

Eg. we’ve been saying things like “a 6 year annuity is like paying £6 in a lump sum all at year 3.”

We’ve gone on to assert that ā6¬ is approximately 6v3.”

Look at the graph on the right and see if you think (for i = 5%) ān¬ really is approximately nv(n/2)

interest rate of 5%

02468

101214

1 3 5 7 9 11 13 15 17 19

Term

ān¬

nv^(n/2)

≈

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DMT (exactly)

Discounted mean term is (exactly) the average term of the cashflows, weighted by their value.

So, discounted mean term changes as yields change. E.g. if yields rise (or at least if long yields rise), then the longer term cashflows will have less weight, so the average discounted mean term will fall.

Cashflowsunweighted

(= weighted at zero yield)

Longer cashflowsgiven less weight

(at high yield)

Discounted mean term = ∑ Term to cashflow * value of cashflow / total value

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Why care about DMT?DMT is related to sensitivity to changes in interest rates. If interest rates rise (at all durations) then our long cashflows will fall much further in value than our short ones.

Investment consultants may reasonably feel this isn’t saying much. We seem to be saying that “if long prices fall fast (so long yields rise a bit) then long prices fall fast”.

Term

Pric

e =

valu

e

Low yieldHigher yield

To liability consultants it may mean more! Liability valuations may appear to rely on yields (or assumptions) taken “out of the blue” and so we want to know how sensitive liability values are to changes in these assumptions.

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Does DMT measure sensitivity?

As asset-liability consultants, we care about how surplus or deficit values are sensitive to changes in yields.

Ie if we denote:the value of a portfolio (which may include liabilities as well as assets) as P the change in value as ΔP, the change in yield as Δi, We want to know ΔP / P per Δi

Ie if yields rise by 1%, by how much will values fall?

Up to now, we’ve used rough DMT as our rough guess of sensitivity to interest rates.

E.g. we’ve said that for, say ā6¬, where the “average cashflow” is at about year 3,the change in P (ΔP / P) ≈ the change in yield (Δi) * the term (3).Ie we’ve maintained that ΔP / P per Δi is about 3.

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What is the exact measure of sensitivity?

We’re looking for ΔP / P per Δi

P (the present value of the portfolio based on current yields) is known, so we are looking for (ΔP per Δi ) / P

Ie (for very small Δi) we are looking for dP/di / P

It turns out that duration is defined as minus dP/di / P

Ie if yields go up “a little”, prices fall by “a little” * the duration.

Ie “Duration” is the exact measure of sensitivity = minus dP/di / P.

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How do we calculate duration (sensitivity) ?

Duration is defined as minus dP/di / P (= percentage change in total value per small change in yield)

Exactly, P = total value = ∑ value of each cashflow

= ∑ cashflow * (1 + i)^-term of cashflow

So dP / di = ∑ cashflow * d{ (1 + i)^- term of cashflow }/di

= ∑ cashflow * -term of cashflow * (1 + i)^(-term of cashflow – 1)

= - v * ∑ cashflow * term of cashflow * (1 + i)^(-term of cashflow)

= -v * ∑ term of cashflow * value of cashflow

So duration = minus dP/di / P = v * ∑ term of cashflow * value of cashflow / total value

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DMT and duration

DMT is the average term of the cashflows, so long cashflows => high DMT.

Duration is the sensitivity of values to changes in yields.

Long term liability values are more sensitive to yields than short-term values. (Cash payable next week is more or less indifferent to interest rates, but how about cash payable in 50 years?)

So long term cashflows => high duration (as well as high DMT).

Exactly, duration = v * ∑ term of cashflow * value of cashflow / total value.

Discounted mean term = ∑ Term to cashflow * value of cashflow / total value.

So duration = v * discounted mean term.

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Jargon reminderDuration or (effective duration) is the sensitivity of values to changes in yields.

Discounted mean term(or Macauley duration) is the average term to cashflow (weighted by value).

Duration = v * DMT

You can just remember this (or the way it was derived) or (for example) think about a zero coupon 100 year bond. Would you expect its sensitivity to a change in yield to be constant, or to have some relation with the yield already in operation?

Yield rises from zero

1st Qtr

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Rules of thumbDuration of annuity is about half the term?

(See what you think by experimenting with the graph on the right, which shows

duration and term/2 andterm/2 * v(term/4)

for a continuous annuity calculated using constant i = 5%.

Annuity paid continuously (Constant i)

1 3 5 7 9 11 13 15 17 19

Term

DurationTerm/2v^(term/4) * Term/2

Continuously increasing annuity (constant i)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Term

Duration

Term * 2/3

v^(term/6)*term 2/3

Duration of increasing annuity is about 2/3 of the term?

(Experiment with the graph on the right, which shows

duration and term*2/3 andterm*2/3 * v(term/6)

for a continuously increasing annuity.

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Q: Really long bondsPeople trying to match long liabilities may complain that there aren’t long enough bonds in the market.

However, even if all you could buy or sell were cash and a ten year bond, how could you create a portfolio with a duration of about 90 years?What would be the discounted mean term of this portfolio?

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A: Really long bondsPeople trying to match long liabilities may complain that there aren’t long enough bonds.

However, even if all you could buy or sell were cash and a ten year bond, how could you create a portfolio with a duration of about 90 years?What would be the discounted mean term of this portfolio?

Answer: borrow cash and buy 10 year bonds.

E.g. you could create an portfolio worth £10 made up of an overdraft of £80 plus 10 year bonds worth £90.

The value of the ten year bonds would change by 9 * the value of a holding of those bonds worth £10 (just because there are more of them in the portfolio).

Discounted mean term = ∑ Term to cashflow * value of cashflow / total value

= ( 0 years (overdraft)* -£80 + 10 years * £90 ) / £10 (value) = 90 years

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Q: Pension fundsDuration is also known as volatility.

If you borrow long-term and save short term, how volatile is your portfolio?

E.g. suppose you are a pension fund in deficit.

You have assets worth £8 (with duration 10 years).You have liabilities worth £10 (duration 20 years).

What is the discounted mean term of your fund?

Value = Deficit 2

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A: Pension fundsDuration is also known as volatility.

If you borrow long-term and save short term, how volatile is your portfolio?

E.g. suppose you are a pension fund in deficit.

You have assets worth £8 (with duration 10 years).You have liabilities worth £10 (duration 20 years).

What is the discounted mean term of your fund?

Answer: Discounted mean term = ∑ Term to cashflow * value of cashflow / total value

= 10 years * £8 - 20 years * £10 / (£8 - £10) = 60 years …. Volatile????

Yields down 1% so Assets up 10% Liabilities up 20%

Value = Deficit 2

Deficit 3.2 up 60%

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Specimen 16(i)b

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Specimen 16(i)b

0

20

40

60

80

100

120

0.5 1 1.5 … 19.5 20

Guess duration of annuity (excluding redemption payment) is about half the term with adjustment fointerest ie 10v(10/4) = 10 / 1.1^5 = 6.2. Duration of redemption payment is v20 = 20/1.1 = 18.Redemption payment is worth 1/5 total value.So guess total duration is about 6.2 * 4/5 + 18 * 1/5 = 8 ½

Net Coupon (paid each 6 months) is 10/2 * (1 – 25%) = £3.75Exactly, discounted mean term is ( ½ x 3.75 v½ + 1x 3.75 v +… + 20 x 3.75 v20 + 20x110v20 ) / P= (½ x 3.75 x (Ia)40¬

@√1.10-1 + 20x110v20 @10%) / 81.76 = 9.802So duration = v * 9.802 = 9.802 / 1.1 = 8.91 (in area of guess)

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Specimen 16(i)b

0

20

40

60

80

100

120

0.5 1 1.5 … 19.5 20

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Specimen 16(ii)a

? (t) 10

Guess DMT of assets = Duration * (1 + i)= 8.91 * 1.1 = 9.8

So roughly, need one asset with term 0.2 above, and another one 0.2 below (suggesting that one with term 9.6 will do).

Duration = v * DMT must equal asset duration

i.e. 8.91 = v * (t * 1.1-t + 10 * 1.1-10 ) / (1.1-t + 1.1-10)

If t = 9.61, duration is indeed 8.91 (substitute into formula)

Present value of £1 nominal of unknown bond = vt = 1.1-t)

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Specimen 16(ii)a

? (t) 10

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Apr 2000 14(ii)a

0

50

100

150

200

250

1 2 3 ... 19 20

Guess duration approx 2/3 of term * v(n/6)= 2/3 * 20 * 1.07^-20/6

= 10.6

PV approx 150 * 20 / 1.0710.6

= £1.46m

Exactly,...

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Apr 2000 14(ii)a

393

Sep 2000 13(i)b

394

Sep 2000 13(i)b

395

Apr 2001 10(i)

396

Apr 2001 10(i)

397

Sep 2001 12(i)

398

Apr 2002 8(ii)&(iii)

399

Apr 2002 9(iii)

400

Sep 2002 10(ii)

401

Sep 2003 11(i)

402

Key questionsGet 100% on Apr 2002 8(ii)&(iii); Sep 2001 12(i); Sep 2003 11(i)

Cover the answers up & do them again until you can explain the solutions to someone else.

403

Next session: Describe convexity and

immunisation

END