102 Session 13

102 Session 13 Calculate duration

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

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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.”

interest rate of 5% 14

ān¬

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

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nv^(n/2)

4 2 0

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

Cashflows unweighted (= weighted at zero yield)

Longer cashflows given less weight (at high yield)

Discounted mean term = ∑ Term to cashflow * value of cashflow / total value 374

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.

Price = value

Why care about DMT?

Low yield Higher yield

Term

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”.

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. 375

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 reminder Duration 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

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Rules of thumb Duration 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 and term/2 * v(term/4) for a continuous annuity calculated using constant i = 5%.

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 and term*2/3 * v(term/6) for a continuously increasing annuity.

Annuity paid continuously (Constant i) Duration Term/2 v^(term/4) * Term/2

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Continuously increasing annuity (constant i)

Duration Term * 2/3

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v^(term/6)*ter m 2/3 381

Q: Really long bonds People 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 bonds People 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 funds Duration is also known as volatility. If you borrow long-term and save short term, how volatile is your portfolio?

Value = Deficit 2

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?

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A: Pension funds Duration is also known as volatility. If you borrow long-term and save short term, how volatile is your portfolio?

Value = Deficit 2

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%

Deficit 3.2 up 60%

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Specimen 16(i)b

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Specimen 16(i)b

120 100 80 60 40 20 0 0.5

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Guess duration of annuity (excluding redemption payment) is about half the term with adjustment fo interest 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%) = £3.75 Exactly, 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 = 9.802 So duration = v * 9.802 = 9.802 / 1.1 = 8.91 (in area of guess)

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Specimen 16(i)b

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Specimen 16(ii)a

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

? (t)

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

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) 389

Specimen 16(ii)a

? (t)

10

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Apr 2000 14(ii)a

Guess duration approx 2/3 of term * v(n/6) = 2/3 * 20 * 1.07^-20/6 = 10.6

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PV approx 150 * 20 / 1.0710.6 = £1.46m

Exactly,...

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Apr 2000 14(ii)a

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Sep 2000 13(i)b

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Sep 2000 13(i)b

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Apr 2001 10(i)

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Apr 2001 10(i)

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Sep 2001 12(i)

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Apr 2002 8(ii)&(iii)

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Apr 2002 9(iii)

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Sep 2002 10(ii)

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Sep 2003 11(i)

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Key questions Get 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.

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Next session: Describe convexity and immunisation

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