102 A 2004

Exam © Faculty & Institute of Actuaries Unchecked solutions © Owen Kellie-Smith

If you have any comments on these solutions (especially if you disagree!) please email [email protected] See also examiner's report at

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Date Time Amount

31/12/2003 31/12/2004 31/12/2005 31/12/2006 1 2 3 4 4 4 4 109

120 100 80 60

Interest rate applying during year ending 31/12/2003 31/12/2004 31/12/2005 31/12/2006 n/a i2004 i2005 i2006

40 20 0 1

Accumulated amount = 4 * ( 1 + i2004)(1 + i2005)(1 + i2006) +4* ( 1 + i2005)(1 + i2006) +4* (1 + i2006) + 4 + 105

2

So expected accumulated amount = E(Accumulated amount) = E[ 4 * ( 1 + i2004)(1 + i2005)(1 + i2006) +4* ( 1 + i2005)(1 + i2006) +4* (1 + i2006) + 4 + 105 ] = Since E(A + B) is always E(A) + E(B), E[ 4 * ( 1 + i2004)(1 + i2005)(1 + i2006) ] + E[ 4 * ( 1 + i2005)(1 + i2006) ] + E[ 4 * (1 + i2006) ] + E[ 4 + 105 ] =, Since for independent random variables, E(AB) = E(A) x E(B), and E(constant) = constant 4 * ( 1 + E[i2004] )( 1 + E[i2005] )(1 + E[i2006] ) + 4* ( 1 + E[i2005])(1 + E[i2006]) +4* (1 + E[i2006]) + 4 + 105 = ,substituting from question, 4 * ( 1 + 5.5% )( 1 + 6% )(1 + 4.5% ) + 4* ( 1 + 6% )(1 + 4.5% ) +4* (1 + 4.5%) + 4 + 105 122.28529

See also

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4

Guess rolled up value: 109 plus about 2 yrs interest on payments of 12 = approx 109 + 12 * (1.055, say)^2 = 122

i2004, i2005, i2006 are random variables.

=

3

cf guess of 122 - OK session 15

Rough guess: s8 @ 5% = 9.5 (to make more accurate, use delta of 5% and make continuous - this would increase value)

14.0% 12.0% delta

10.0% 8.0% 6.0%

Average interest rate from t = 8 to t = 15 is about 10% So rolled up amount is about 50 x s8 x 1.10^(15-8) = 50 x 9.5 x 1.1^7 = 925

4.0% 2.0% 0.0% 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

time (t)

Exact calc £1 paid at time t rolls up to £1 x exp(total force of interest from t to 8) by time 8 ie it rolls up to £1 x exp[ 0.05 x (8 - t) ] Accumulated value at t = 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

5.0% 5.0% 5.0% 5.0% 5.0% 5.0% 5.0% 5.0% 5.0%

= Integral (from t = 0 to 8) of rate paid x Accumulation factor from t to 8 = Integral (from t = 0 to 8) of 50 x exp[ 0.05 x (8 - t) ] = 50 x exp [ 0.05 x 8 ] x Integral (from t = 0 to 8) of exp[ -0.05t ] = 50 x exp [ 0.05 x 8 ] x Integral (from t = 0 to 8) of exp[ -0.05t ] 6.6% 7.2% 8.0% 8.8% 9.8% 10.8% 11.8% 13.0%

= 50 x exp [ 0.05 x 8 ] x (1 - exp[ -0.05 x 8] )/0.05 9.8%

= 50 x =

1.491825 x 6.593599 491.8247

Total force of interest from t = 8 to t = 15 = Integral (from t = 8 to 15) of delta(t) = Integral (from t = 8 to 15) of .04 + .0004t^2 = 7 x .04 + .0004(15^3 - 8^3)/3 =

0.661733

So accumulated value at t = 15 = accumulated amount at t = 8 rolled up to t = 15 = 491.8247 * exp(total force of interest from t = 8 to t = 15) = 491.8247 * exp(0.661733) = 953.2292 cf guess of 925 - OK, is paid continuously.

See also

http://myweb.tiscali.co.uk/kseducation/102/102course.htm

session 2

(i) 1/1/01 - 1/11/02 1/11/01 - 1/5/02 1/5/02 - 31/12/02

Manager held 120 157 221

Manager delivered 137 173 205

Cashflow 20 48

Growth factor in manager's hands = 137 / 120 = Amount delivered / Amount held = 173 / 157 = 205 / 221

TWRR factor over two years = product of growth factors = 137 / 120 * 173 / 157 * 205 / 221 = ie (1 + TWRR )^2 = 1.166937 so TWRR = 1.166937 ^0.5 - 1 = 8.025% (ii)

TWRR

+ avoids distortion from cashflows - requires more data than MWRR

See also

http://myweb.tiscali.co.uk/kseducation/102/102course.htm

session 9

1.166937

Guess: see what price the stock would be if the one-year-spot rate equalled the two=year spot rate = 4.15% it would cost 8 x 1.0415^-1 + (8 + 98) x 1.0415^-2 = 105.4021 - very close to 105.40 The rate that most affects the price is the 2-year rate (since most cash comes at t=2) - so expect 2-yr rate to be very close to 4.15% (& 1-yr rate close) Time Amount Amount

0 0 0

1 4.15 8

2 Price 104.15 100 106 105.4

2-yr par yield = 4.15 means a bond paying 4.15% annual coupon would be priced at £100% 8% stock is redeemed at 98, so it pays 98 + 8 at time 2 ie 106

120

So at t=0 it is true that 4.15v(1) + 104.15v(2) = 100

100 80 60 40

(A)

4.15v(1) + 104.15v(2) = 100

(B)

8

v(1) + 106

v(2) = 105.4

(par-yield) (8% coupon stock)

20

Solve the simultaneous equations

0 0

1

2

(C) = (A) x 8 / 4.15 (C) - (B)

8v(1) + 8/4.15 x 104.15v(2) = 8/4.15 x 100

( 8/4.15 x 104.15 - 106 ) v(2) = 8/4.15 x 100 - 105.4 => v(2) = 0.92191711

Substituting into (A) => v(1) =

( 100 - 104.15 x 0.92191711 ) / 4.15 =

Check price of 8% coupon stock = =

0.959598

8 x 0.959598 + 106 x 0.92191711 105.40000 OK

v(1) =

0.959598 => one year spot rate = v(1)^-1 - 1 =

4.2103%

v(2) =

0.921917 => one year spot rate = v(2)^-0.5 - 1 =

4.1488% v.close to 4.15% OK cf guess

See also

http://myweb.tiscali.co.uk/kseducation/102/102course.htm

session 10

Time Assets Liabilities

2 7.4

2

(i)

10

10

15

-10

-20

15

25 31.834

25

Discounted value of assets = 7.4v(2) + 31.834v(25) = 7.4 x 1.07^-2 + 31.834 x 1.07^-25 =

12.329 m

Discounted value of liabilities = 10v(10) + 20v(15) = 10 x 1.07^-10 + 20 x 1.07^-15 =

12.332 m, so PV(assets) = PV(liabilities) (to 4 significant figures)

Discounted mean term of assets = Average term weighted by value = 2 x 7.4v(2)/12.33 + 25 x 31.834v(25)/12.33 = 1/12.33 x ( 2 x 7.4 x 1.07^-2 + 25 x 31.834 x 1.07^-25 )

=

Discounted mean term of liabilities = Average term weighted by value = 10 x 10v(10)/12.33 + 15 x 20v(15)/12.33 = 1/12.33 x ( 10 x 10 x 1.07^-10 + 15 x 20 x 1.07^-15 )

=

12.94

12.94 so DMT assets = DMT liabs and therefore duration assets = duration liabilities

PV(assets) = PV(liabilities) AND duration(assets) = duration(liabilities) => 1st 2 conditions for immunisation hold. (ii)

If rate changes to 7.5% Discounted value of assets = 7.404v(2) + 31.834v(25) = 7.4 x 1.075^-2 + 31.834 x 1.075^-25 =

11.627 m

Discounted value of liabilities = 10v(10) + 20v(15) = 10 x 1.075^-10 + 20 x 1.075^-15 =

11.611 m, so PV(assets) > PV(liabilities)

Profit = 11.627 minus 11.611 = Fall in liabilities minus fall in assets = ( 12.332 11.611 ) minus (iii)

0.016 m (

or, if you prefer 12.332 -

11.627 ) =

0.016 m

Assets are "spread out" over a longer time than liabilities - so they have greater convexity => ( along with other conditions in (i) ) => ( along with other conditions in (i) ) immunised against rate changes => rate change would cause a profit See also

http://myweb.tiscali.co.uk/kseducation/102/102course.htm

session 14

(i)

Things that pay the same cost the same. No risk-free profits / free lunch etc etc

(ii)

I assume 20p dividend means a quarterly dividend ie annual dividend yield approx 4 x 20p = 80p => dividend yield approx 80/450 = 18% dividend yield >> risk-free rate => future price < market price (otherwise you could get a risk-free 18% yield) Rough Guess Dividend increases roughly equal risk free rate, so, ignoring interest and increases, dividends make up about 20p x 3 years x 4 quarters = 240p of price So forward price has present value of about 450 - 240 = 210p and forward price is about 210 * 1.05^3 = 240p

t div redeem

Future dividends are 0 0.25 20.2

0.5 0.75 20.402 20.60602

1 20.81208

1.25 21.0202

1.5 1.75 2 2.25 2.5 21.2304 21.44271 21.65713 21.98199 22.31172

2.75 3 22.6464 22.98609 23.33088 Forward Price 300 ie once the forward is struck (which itself has no value, since neither the buyer nor the seller expects to win or lose on it), the share becomes like a bond, paying coupons = dividends and redeemed at the forward price. PV (price) = PV (receipts) ie 450 = 20 a8 @ j% + 20 x 1.01^8 * exp-(5% * 2) a4 @ k% + ForwardPrice exp-(5% * 3) where j% is net quarterly interest rate based on increases of 1% per quarter & k% is net quarterly interest rate based on increases of 1.5% per quarter ie 1 + j% = exp(5% * 0.25) / 1.01 = & 1 + k% = exp(5% * 0.25) / 1.015 = So a8 @ j% = So a4 @ k% =

1.0025529 0.9976142

( 1 - 1.0025529 ^-8 ) / ( 0.0025529 ) = ( 1 - 0.9976142 ^-4 ) / ( -0.0023858 ) =

7.908871 4.023972

ie 450 = 20 * 7.9088714 + 20 * 1.01^8 *exp-(5% * 2) * 4.023972 + ForwardPrice * exp-( 5% * 3) = = 237.0319 + ForwardPrice * exp-( 5% * 3) => ForwardPrice = ( 450 - 237.031928 ) x exp( 5% * 3) = 247.4336 ie

See also

http://myweb.tiscali.co.uk/kseducation/102/102course.htm

session 11

£2.47

(i)

Receipts were on

01/12/2000 01/06/2001 01/12/2001 01/06/2002

Relevant Relevant index date index (6 m earlier) from table 01/06/2000 102 01/12/2000 107 01/06/2001 111 01/12/2001 113

Base Base index Cashflow / £100 Indexation index date pre indexing (6m pre issue) Coupon Redemption 01/12/1999 100 1.5 x 102 / 100 = 01/12/1999 100 1.5 x 107 / 100 = 01/12/1999 100 1.5 x 111 / 100 = 01/12/1999 100 1.5 100 x 113 / 100 =

Cashflow / £100,000

Cashflow / £100

post indexing

post indexing

Coupon 1.53000 1.60500 1.66500 1.69500

Total £1,530.00 £1,605.00 £1,665.00 £114,695.00

Redemption

113.000

Coupon is 3% pa half-yearly => 3%/2 = 1.5% each 6 months (ii)a

(ii)b

Capital gains indexation doesn't have the time lag. Bond was bought at issue date, in June 2000 (when index was 102) Bond was redeemed in June 2002 (when index was 118) So purchase price (94) is indexed by 118/102 to 94 x 118 / 102 = (Indexed) Redemption amount =

108.74510 113 (113 / 100 x 100, above)

So capital gain per £100 nominal = 113 - 108.7451 = And so capital gain per £100000 nominal = 4.25490 x 1000 = on which 35% tax is payable ie 35% x £4,254.90 =

4.25490 £4,254.90 £1,489.22

I assume the net effective yield means the nominal after-tax yield (ie no attempt to turn into a "real" yield) Net yield solves the after-tax equation of value Equation of value is Price = PV after-tax coupon payments + PV after-tax redemption payment ie 94 = (from table in part i) (1 - 25% income tax) x ( 1.53v^0.5 + 1.605v + 1.665v^1.5 + 1.895v^2) + (113 - 1.48922 CGT)v^2 Initial guess for i: after-tax coupon rate is 3% x 75% = 2.25%. Capital gain is about 19% (after tax), ie about 9% pa, so try 2.25 + 9 = 11%pa as first guess i

Equation (1 - 25% ) x ( 1.53 x 1.11^-0.5 + 1.605x1.11^-1 + 1.665x1.11^-1.5 + 1.895x1.11^-2) + (113 - 1.48922)x1.11^-2 = (1 - 25% ) x ( 1.53 x 1.12^-0.5 + 1.605x1.12^-1 + 1.665x1.12^-1.5 + 1.895x1.12^-2) + (113 - 1.48922)x1.12^-2 =

11% 12%

Value Too high 94.77785 Try higher I 93.12174

Interpolate 94.77785 By similar triangles, 94 93.121743

=> x% = 11%

11.47%

x% - 11% 94.7785 - 94

x%

12% - 11% 94.7785 - 93.121743 11.47%

12%

(1 - 25% ) x ( 1.53 x 1.1147^-0.5 + 1.605x1.1147^-1 + 1.665x1.1147^-1.5 + 1.895x1.1147^-2) + (113 - 1.48922)x1.1147^-2 =

so net yield pa is 11.47%

See also

=

http://myweb.tiscali.co.uk/kseducation/102/102course.htm

sessions 5 & 7

93.9939552 v.close to 94

'(i)

Guess

say stock bought at nearly 100. After tax coupon rate is a little over (1 - 25%) x 8% = 6% Net capital gain is about 8%, less than 1% pa when spread over 25 years. So guess net yield is about 6.25% (coupons) + 0.25% capital gain = 6.5% and real net yield is about 6.5% - 3% inflation = 3.5%

Exact calc Coupon payments are made on 1 July and 1 Jan, ie at times 0.5, 1, 1.5, 2, …, 24.5, 25 Tax on coupons is paid on 1 April ie at time 1.25, 2.25, 3.25, …, 25.25 Redemption payment is made at time 25 Capital gains tax is paid at time 25.25

ie

Equation of value is Price = PV coupons - PV income tax + PV redemption - PV CGT 25 - 30% x (110 - 99) x v25.25 8 a(2)25¬ - v0.25 x 25% x 8 a25¬ + 110 x v 99 =

where v = 1/(1+i)

By trial and error, interpolation etc i Value 6.50% 8( 1 - 1.065^-25 )/[(1.065^0.5 - 1) x 2] -1.065^-0.25 x 25% x 8( 1-1.065^-25 ) / 0.065 + 110x 1.065^-25 - 30% x(110-99)x 1.065^-25.25 =97.241 6.30% 8( 1 - 1.063^-25 )/[(1.063^0.5 - 1) x 2] -1.063^-0.25 x 25% x 8( 1-1.063^-25 ) / 0.063 + 110x 1.063^-25 - 30% x(110-99)x 1.063^-25.25 =99.656 6.35% 8( 1 - 1.0635^-25 )/[(1.0635^0.5 - 1) x 2] -1.0635^-0.25 x 25% x 8( 1-1.0635^-25 ) / 0.0635 + 110x 1.0635^-25 - 30% x(110-99)x 1.0635^-25.25 =99.044 Real net yield r: (1 + r) * (1 + inflation) = (1 + i ) So real net yield = 1.0635 / 1.03 - 1 = (ii)

3.25%

OK vs guess

If tax is collected on 1 June, all tax will be paid later, so after tax yield will rise. (Mathematically, present value of tax payments falls, whereas PV of receipts unchanged. So to keep present value of whole investment the same (94), must use higher yield.) (Financially: it's preferable to pay later rather than sooner as you can get more interest on whatever you were going to pay the tax with.) "More preferable" => higher yield.

See also

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session 5

(i) Guess When will annuity have paid off debt? a14 @ 7% = 8.7. 8.7 x 14k = 122k, so guess discounted payback period is about 14 years Try 13.5 years. PV of receipts = 14000 x

a(2)13.5¬

= 14000 x ( 1 - 1.07^-13.5 )/[(1.07^0.5 - 1) x 2] = 14000 x 8.70201658542704 =

121828.232 too much

Try 13.0 years. PV of receipts = 14000 x

a(2)13.5¬

- 7000 v13.5 7000 x 1.07-13.5

=

121828.232 -

=

119020.103 too little, so the payment at t = 13.5 is necessary to get back into credit.

Hence, the discounted payback period is 13.5 years (ii)

Guess Discounted payback period is 13.5 years, so the rest (approx 11 years) is profit. So accumulated profit is about 11 x 14000 x 1.05^6 (about half remaining term) = 210k Exact calc. Loan is paid off at t=13.5, when the investor has asset with present value of 121828 - 120000 = ie actual amount investor is in credit by is accumulated value:

1828.232 x 1.07^13.5 =

Investor can save this at 5%, plus the rest of the annuity payments. (25 - 13.5) 4557.34967 x 1.05

So rolled up profit =

= 4557.35 x 1.75258 =

See also

+ 14000 s(2)(25 - 13.5)¬

@ 5%

+ 14000 x ( 1.05^11.5 - 1 )/[(1.05^0.5 - 1) x 2]

£221,310

http://myweb.tiscali.co.uk/kseducation/102/102course.htm

session 8

1828.232 4557.35

(i)

First find parameters of LogN distribution, using formulae tables for mean and variance of LogN distribution 2 Guess μ approx 4% and σ approx 2% If X ~ LogN(μ,σ2) 2 then from tables E(X) = exp( μ+ 1/2 σ ) , and Var(X) = [ exp( 2μ+ σ2) ] [ exp(σ2)-1 ] 0.02 =

Var(X) = E(X)2 [ exp(σ2)-1 ] = 1.042 [ exp(σ2)-1 ]

=>

[ exp(σ2)-1 ]

=>

σ2

= 0.02 / 1.042 = ln( 1 + 0.02 / 1.042 ) =

0.0183220

2

Substitute σ into formula for E(X) 1.04 = E(X) = exp( μ+ 1/2 σ2) = exp( μ+ 0.0183222 / 2) =>

μ

= ln( 1.04 ) - 0.0183222 / 2 =

0.0300600

Calculate cash needed, say £X Cash will roll up to X (1 + i1)(1 + i2) .. (1 + i5) Probability that

X (1 + i1)(1 + i2) .. (1 + i5) > 5000

= Probability that

log [

= Probability that

log X + log (1 + i1) + log (1 + i2) + … + log (1 + i5) > log(5000)

X (1 + i1)(1 + i2) .. (1 + i5) ] > log [ 5000 ]

log (1 + in) is distributed as N ( 0.03006, 0.018322 ) So, since sum of independent normals is normal (with sum of means and variances) log (1 + i1) + log (1 + i2) + … + log (1 + i5) is distributed as N( 0.03006 x 5, 0.018322 x 5) ie as N( 0.1503, 0.09161) So probability that

log X + log (1 + i1) + log (1 + i2) + … + log (1 + i5) > log(5000)

= probability that

log X + N( 0.1503, 0.09161 ) > log(5000)

= probability that

N( 0.1503, 0.09161 ) > log(5000) - logX

= probability that

N( 0,1 ) > ( log(5000) - logX - 0.1503 ) / ( 0.09161

= 1 - probability that

0.5

)

N( 0,1 ) < ( log(5000) - logX - 0.1503 ) / ( 0.091610.5 )

= 1 - BigPhi[ log(5000) - logX - 0.1503 ) / ( 0.091610.5 ) ] We want this probability to equal 99%, BigPhi[ log(5000) - logX - 0.1503 ) / ( 0.091610.5 ) ] must equal 1 - 99% = 1% ie BigPhi[ - { log(5000) - logX - 0.1503 ) / ( 0.091610.5 ) } ] = 99% So find x in the table of "BigPhi" such that BigPhi(x) = 99% (ie 2.326) Hence

- { log(5000) - logX - 0.1503 ) / ( 0.091610.5 ) }

=>

{ log(5000) - logX - 0.1503 ) }

=>

- logX

=>

X = cash needed

= 2.326

= -2.326 x 0.091610.5

= -2.326 x 0.091610.5 + 0.1503 - ln(5000) = = -exp(9.070907) =

-9.0709070

£8,699

Check variance of 2% pa => variance of about 10% over 5 years, or standard deviation of 0.1^0.5 = 30% To have 99% chance of meeting liability, you must have margin of at least 2 standard deviations (about 2 x 30% = 60%) - OK (ii)

She has to have a large margin of over 70% (8699 / 5000 - 1) to have a 99% chance of meeting her liability in 5 years time. (If the liability were payable, in, say, 10 years, she would need even more to be 99% certain of paying it.)

See also

http://myweb.tiscali.co.uk/kseducation/102/102course.htm

session 15

(i)

Payments are at "odd" times: July, November, March, ie at times 10/12, 1 2/12, 1 6/12, 1 10/12, 2 2/12, …. 5 6/12 So we can't just calculate them all in one annuity Can treat them as 3 groups of payments PV (July payments: 5 payments in 99, 00, 01, 02, 03) = 1000 * v^(10/12) * ( 1 + 1.05^3 v + 1.05^6 v^2 + 1.05^9 v^3 + 1.05^12 v^4 ) PV (November payments: 5 in 99, 00, 01, 02, 03) = 1000 * 1.05 * v^(14/12) * ( 1 + 1.05^3 v + 1.05^6 v^2 + 1.05^9 v^3 + 1.05^12 v^4 ) PV (March payments: 5 in 00, 01, 02, 03, 04) = 1000 * 1.05^2 * v^(18/12) * ( 1 + 1.05^3 v + 1.05^6 v^2 + 1.05^9 v^3 + 1.05^12 v^4 ) Factorising, PV of total payments = 1000 * [ v^(10/12) + 1.05 * v^(14/12) + 1.05^2 * v^(18/12) ] * [ 1 + 1.05^3 v + 1.05^6 v^2 + 1.05^9 v^3 + 1.05^12 v^4 ] which you can work out using annuities with adjusted interest rates, or using geometric formula, or just directly: given v = 1.06^-1 = 0.9433962 Amount of loan = PV payments = 1000 * 2.9438243 *

(ii)

6.009791 =

£17,692

Initial loan = £17,692 Effective interest rate is 6% pa So loan rolled forward to 1 July 1999 (after 10 months) is 1.06^(10/12) * 17692 = ie interest added =

18572.034 minus

£17,692 =

18572.03

£880.27

Amount paid was £1000, so capital repayment was the non-interest part of payment = 1000 - 880.27 = (iii)

£119.73

First calculate amount outstanding after 6th payment and then do similar process to part (ii) Amount outstanding after 6th payment (in March 2001) is present value of remaining 9 payments as at March 2001 PV (July payments: 3 payments in 01, 02, 03) = 1000 * v^(4/12) * ( 1.05^6 + 1.05^9 v + 1.05^12 v^2 ) PV (November payments: 3 in 01, 02, 03) = 1000 * 1.05 * v^(8/12) * (1.05^6 + 1.05^9 v + 1.05^12 v^2 ) PV (March payments: 3 in 02, 03, 04) = 1000 * 1.05^2 * v * (1.05^6 + 1.05^9 v^ + 1.05^12 v^2 ) So total outstanding just after the sixth payment = 1000 * ( v^(4/12) + 1.05 * v^(8/12) + 1.05^2 * v ) * ( 1.05^6 + 1.05^9 v + 1.05^12 v^2 ) = 1000 * 3.0308526 * 4.401919 = £13,341.57 Effective interest rate is 6% pa So loan rolled forward to 7th payment date (in July 2001, 4 months later) is 1.06^(4/12) * 13,341,57= ie interest added =

£13,603.23 minus

£13,342 =

£261.67

Amount paid was £1000 x 1.05^6 = 1340.10, so capital repayment was the non-interest part of payment = 1340.10 -198.55 = £1,078.43

See also

http://myweb.tiscali.co.uk/kseducation/102/102course.htm

session 6

£13,603.23