102 Session 5
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102 Session 5 net present value
© KSES Exam questions are copyright Faculty & Institute of Actuaries & are used with their permission Source: www.actuaries.org.uk 130
Specimen Q4
131
Specimen Q4
Approx £… pa going up at …% pa. Ie discount at net 8% - …% = …% a9999¬ @ …%= 1/…. = 50 Value roughly £… pa * … = ….
1
2
3
4
…
Treating half-years as unit of time, Value = … * (1.03v + 1.032v2 + 1.033v3 + 1.034v4 + … ) = 5 a9999¬ @ √1.08 / …… – 1 = 5 * 1/….. = ….* 111.6 = £…. (in area of guess)
132
Specimen Q4
Approx £10 pa going up at 6% pa. Ie discount at net 8% - 6% = 2% a9999¬ @ 2%= 1/0.02 = 50 Value roughly £10 pa * 50 = 500
1
2
3
4
…
Treating half-years as unit of time, Value = 5 * (1.03v + 1.032v2 + 1.033v3 + 1.034v4 + … ) = 5 a9999¬ @ √1.08 / 1.03 – 1 = 5 * 1/0.89616% = 5 * 111.6 = £558 (in area of guess)
133
Specimen Q4
134
Specimen Q15(i)
135
Specimen Q15(i)
Rough guess. Total cash in = ½ * (start+finish) * years paid = ……. Guess term cash in half project (15) PV cash in ≈ 1170 * 1.08-…. = 370 Cash out in two bits, near t = ½ & t=15. So, PV cash out ≈ 3 * ….. * 1.08-½ + …. * 1.08-15 = 366 => NPV ≈ 370 (in) – ….. (out) = 4k Exact out = ……. = 105 (1 + v½ + v) + 200v15 @ 8% Exact out = ……. = a1 [20v + 23v2 + 26v3 + 29v4{ 1 + a25 @ (1+i)/1.03-1} ] @ i = 8% NPV = in – out = £……. (near guess?)
136
Specimen Q15(i)
Rough guess. Total cash in = ½ * (start+finish) * years paid = 1170 Guess term cash in half project (15) PV cash in ≈ 1170 * 1.08-15 = 370 Cash out in two bits, near t = ½ & t=15. So, PV cash out ≈ 3 * 105 * 1.08-½ + 200 * 1.08-15 = 366 => NPV ≈ 370 (in) – 366 (out) = 4k Exact out = 366.3 = 105 (1 + v½ + v) + 200v15 @ 8% Exact out = 370.6 = a1 [20v + 23v2 + 26v3 + 29v4{ 1 + a25 @ (1+i)/1.03-1} ] @ i = 8% NPV = in – out = £4,300 (near guess)
137
Specimen Q15(i)
138
Specimen Q16(i)a
139
Specimen Q16(i)a
Sanity check: net yield > net coupon rate => capital gain => price < par 1
2
…
…
39
40
Work in half-years as the unit of time After tax value per £100 nominal = (1 – tax) * 5 a40¬ + 110 v40 = 75% * 5 * (1 – (1+j)-40)/j + 110 * 1.1-20 = 3.75 * 17.443 + 110 * 0.1486 = £81.76
@ j=(√1.10 – 1)%
140
Specimen Q16(i)a
Sanity check: net yield >/< net coupon rate => capital gain/loss => price >/< par 1
2
…
…
39
40
Work in half-years as the unit of time After tax value per £100 nominal = (1 – tax) * …a40¬ + ….. v40 = 75% * … * (1 – (1+j)-40)/j + 110 * 1.1-…. = 3.75 * …….. + 110 * 0.1486 = £81.76
@ j=(√1.10 – 1)%
141
Specimen Q16(i)a
142
April 2000 Q7
143
April 2000 Q7
Rough check. 8% and 7% increase in 1st 2 years, so value about ….% higher than if 5% increase for ALL years. 5% increase for all years => net interest rate of 7 – 5 = 2%. a9999 * 2% = 1/ ….. = 50. Guess = 100 *8p *50 *….. =420 Exact value (given the assumptions) = ….v (1 + 1.08v( 1 + 1.07v(1 + a9999 @ j =[ (1 + i) / 1.05 - 1 ] ) = …./1.07 * (1 + 1.08/1.07 * (1 + 1.07/1.07 * (1 + 1/1.905%) ) ) = 418.76
144
April 2000 Q7
Rough check. 8% and 7% increase in 1st 2 years, so value about 5% higher than if 5% increase for ALL years. 5% increase for all years => net interest rate of 7 – 5 = 2%. a9999 * 2% = 1/ 2% = 50. Guess = 100 *8p *50 *1.05 =420 Exact value (given the assumptions) = 8v (1 + 1.08v( 1 + 1.07v(1 + a9999 @ j =[ (1 + i) / 1.05 - 1 ] ) = 8/1.07 * (1 + 1.08/1.07 * (1 + 1.07/1.07 * (1 + 1/1.905%) ) ) = 418.76
145
April 2000 Q7
146
April 2000 Q10
147
April 2000 Q10(i)
Lender wants return above net coupon rate (7 * 0.75 = …..%), so requires a capital ….. to get 6%. Bond will be redeemed for …., in any case, so amount of capital …… is known, but is worth less the …….. it is delayed.
A 91 J 91 O 91 A 92 O 92 A 93 O 93 …
…
A 09 O 09 A 10
Exact Price = £93.85% = 1.063/12 *[ …../2 * (1 – 25%) a38 + …… v38 ]
So lender must assume …… possible redemption. Guess duration = 10. Price for net yield of 5.25% would be 100. Guess price = (100 – 10 * (6-5.25)%) = …….
@ j = √ 1.06 - 1
148
April 2000 Q10(i)
Lender wants return above net coupon rate (7 * 0.75 = 5.25%), so requires a capital gain to get 6%. Bond will be redeemed for 100, in any case, so amount of capital gain is known, but is worth less the longer it is delayed.
A 91 J 91 O 91 A 92 O 92 A 93 O 93 …
…
A 09 O 09 A 10
Exact Price = £93.85% = 1.063/12 *[ 7/2 * (1 – 25%) a38 + 100 v38 ]
So lender must assume latest possible redemption. Guess duration = 10. Price for net yield of 5.25% would be 100. Guess price = (100 – 10 * (6-5.25)%) = 92.50
@ j = √ 1.06 - 1
149
April 2000 Q10(i)
150
April 2000 Q10(ii)
Lender gets redemption yield …….. net coupon rate (7 * 0.75 = ……%), so faces a capital ……. Bond will be redeemed for ….., in any case, so amount of capital ……. is known, and is more painful the …………. it is realised. So lender must assume ……… possible redemption. A 99
O 99
A 00
O 00
…
…
A 03
O 03
A 04
Exact Price = £101.36% (near guess? Not very!) = 7/2 * (1 – ……%) a10 + 100 v….. @ j = √ 1.05 - 1
Guess duration = 3. Price for net yield of 5.25% would be 100. Guess price = (100 – 3 * (5-5.25)%) = …… 151
April 2000 Q10(ii)
Lender gets redemption yield below net coupon rate (7 * 0.75 = 5.25%), so faces a capital loss. Bond will be redeemed for 100, in any case, so amount of capital loss is known, and is more painful the sooner it is realised. So lender must assume earliest possible redemption. A 99
O 99
A 00
O 00
…
…
A 03
O 03
A 04
Exact Price = £101.36% (near guess? Not very!) = 7/2 * (1 – 25%) a10 + 100 v10 @ j = √ 1.05 - 1
Guess duration = 3. Price for net yield of 5.25% would be 100. Guess price = (100 – 3 * (5-5.25)%) = 100.75 152
April 2000 Q10(ii)
153
Sep 2000 Q12(i)
154
Sep 2000 Q12(i)
0
1
2
3
4
…
23
24
25
Rough check. If i for NPV = 6% then NPV = 25 * 25 – 3 * 180/1.06 = 116. At higher interest rate, NPV falls by say 1% * 13 * 25 * 25 = 81, to say 35 as long term receipts fall faster in value than short-term outgoings.
NPV = -180 ä3 @…..% + …..ā25 @ 1.07 / 1.06 –1 = -……… +………… = +51.7
155
Sep 2000 Q12(i)
0
1
2
3
4
…
23
24
25
Rough check. If i for NPV = 6% then NPV = 25 * 25 – 3 * 180/1.06 = 116. At higher interest rate, NPV falls by say 1% * 13 * 25 * 25 = 81, to say 35 as long term receipts fall faster in value than short-term outgoings.
NPV = -180 ä3 @7% + 25ā25 @ 1.07 / 1.06 –1 = -505.4 +557.1 = +51.7
156
Sep 2000 Q12(i)
157
Key questions Get 100% on Sep 2002 # 7 and April 2003 # 3. It doesn’t matter how many times you see the answers. Cover the answers up & do them again until you can explain the solutions to someone else.
158
Sep 2002 Q7
159
April 2003 Q3
160
Next session: Calculate mortgage schedule
END 161
© KSES Exam questions are copyright Faculty & Institute of Actuaries & are used with their permission Source: www.actuaries.org.uk 130
Specimen Q4
131
Specimen Q4
Approx £… pa going up at …% pa. Ie discount at net 8% - …% = …% a9999¬ @ …%= 1/…. = 50 Value roughly £… pa * … = ….
1
2
3
4
…
Treating half-years as unit of time, Value = … * (1.03v + 1.032v2 + 1.033v3 + 1.034v4 + … ) = 5 a9999¬ @ √1.08 / …… – 1 = 5 * 1/….. = ….* 111.6 = £…. (in area of guess)
132
Specimen Q4
Approx £10 pa going up at 6% pa. Ie discount at net 8% - 6% = 2% a9999¬ @ 2%= 1/0.02 = 50 Value roughly £10 pa * 50 = 500
1
2
3
4
…
Treating half-years as unit of time, Value = 5 * (1.03v + 1.032v2 + 1.033v3 + 1.034v4 + … ) = 5 a9999¬ @ √1.08 / 1.03 – 1 = 5 * 1/0.89616% = 5 * 111.6 = £558 (in area of guess)
133
Specimen Q4
134
Specimen Q15(i)
135
Specimen Q15(i)
Rough guess. Total cash in = ½ * (start+finish) * years paid = ……. Guess term cash in half project (15) PV cash in ≈ 1170 * 1.08-…. = 370 Cash out in two bits, near t = ½ & t=15. So, PV cash out ≈ 3 * ….. * 1.08-½ + …. * 1.08-15 = 366 => NPV ≈ 370 (in) – ….. (out) = 4k Exact out = ……. = 105 (1 + v½ + v) + 200v15 @ 8% Exact out = ……. = a1 [20v + 23v2 + 26v3 + 29v4{ 1 + a25 @ (1+i)/1.03-1} ] @ i = 8% NPV = in – out = £……. (near guess?)
136
Specimen Q15(i)
Rough guess. Total cash in = ½ * (start+finish) * years paid = 1170 Guess term cash in half project (15) PV cash in ≈ 1170 * 1.08-15 = 370 Cash out in two bits, near t = ½ & t=15. So, PV cash out ≈ 3 * 105 * 1.08-½ + 200 * 1.08-15 = 366 => NPV ≈ 370 (in) – 366 (out) = 4k Exact out = 366.3 = 105 (1 + v½ + v) + 200v15 @ 8% Exact out = 370.6 = a1 [20v + 23v2 + 26v3 + 29v4{ 1 + a25 @ (1+i)/1.03-1} ] @ i = 8% NPV = in – out = £4,300 (near guess)
137
Specimen Q15(i)
138
Specimen Q16(i)a
139
Specimen Q16(i)a
Sanity check: net yield > net coupon rate => capital gain => price < par 1
2
…
…
39
40
Work in half-years as the unit of time After tax value per £100 nominal = (1 – tax) * 5 a40¬ + 110 v40 = 75% * 5 * (1 – (1+j)-40)/j + 110 * 1.1-20 = 3.75 * 17.443 + 110 * 0.1486 = £81.76
@ j=(√1.10 – 1)%
140
Specimen Q16(i)a
Sanity check: net yield >/< net coupon rate => capital gain/loss => price >/< par 1
2
…
…
39
40
Work in half-years as the unit of time After tax value per £100 nominal = (1 – tax) * …a40¬ + ….. v40 = 75% * … * (1 – (1+j)-40)/j + 110 * 1.1-…. = 3.75 * …….. + 110 * 0.1486 = £81.76
@ j=(√1.10 – 1)%
141
Specimen Q16(i)a
142
April 2000 Q7
143
April 2000 Q7
Rough check. 8% and 7% increase in 1st 2 years, so value about ….% higher than if 5% increase for ALL years. 5% increase for all years => net interest rate of 7 – 5 = 2%. a9999 * 2% = 1/ ….. = 50. Guess = 100 *8p *50 *….. =420 Exact value (given the assumptions) = ….v (1 + 1.08v( 1 + 1.07v(1 + a9999 @ j =[ (1 + i) / 1.05 - 1 ] ) = …./1.07 * (1 + 1.08/1.07 * (1 + 1.07/1.07 * (1 + 1/1.905%) ) ) = 418.76
144
April 2000 Q7
Rough check. 8% and 7% increase in 1st 2 years, so value about 5% higher than if 5% increase for ALL years. 5% increase for all years => net interest rate of 7 – 5 = 2%. a9999 * 2% = 1/ 2% = 50. Guess = 100 *8p *50 *1.05 =420 Exact value (given the assumptions) = 8v (1 + 1.08v( 1 + 1.07v(1 + a9999 @ j =[ (1 + i) / 1.05 - 1 ] ) = 8/1.07 * (1 + 1.08/1.07 * (1 + 1.07/1.07 * (1 + 1/1.905%) ) ) = 418.76
145
April 2000 Q7
146
April 2000 Q10
147
April 2000 Q10(i)
Lender wants return above net coupon rate (7 * 0.75 = …..%), so requires a capital ….. to get 6%. Bond will be redeemed for …., in any case, so amount of capital …… is known, but is worth less the …….. it is delayed.
A 91 J 91 O 91 A 92 O 92 A 93 O 93 …
…
A 09 O 09 A 10
Exact Price = £93.85% = 1.063/12 *[ …../2 * (1 – 25%) a38 + …… v38 ]
So lender must assume …… possible redemption. Guess duration = 10. Price for net yield of 5.25% would be 100. Guess price = (100 – 10 * (6-5.25)%) = …….
@ j = √ 1.06 - 1
148
April 2000 Q10(i)
Lender wants return above net coupon rate (7 * 0.75 = 5.25%), so requires a capital gain to get 6%. Bond will be redeemed for 100, in any case, so amount of capital gain is known, but is worth less the longer it is delayed.
A 91 J 91 O 91 A 92 O 92 A 93 O 93 …
…
A 09 O 09 A 10
Exact Price = £93.85% = 1.063/12 *[ 7/2 * (1 – 25%) a38 + 100 v38 ]
So lender must assume latest possible redemption. Guess duration = 10. Price for net yield of 5.25% would be 100. Guess price = (100 – 10 * (6-5.25)%) = 92.50
@ j = √ 1.06 - 1
149
April 2000 Q10(i)
150
April 2000 Q10(ii)
Lender gets redemption yield …….. net coupon rate (7 * 0.75 = ……%), so faces a capital ……. Bond will be redeemed for ….., in any case, so amount of capital ……. is known, and is more painful the …………. it is realised. So lender must assume ……… possible redemption. A 99
O 99
A 00
O 00
…
…
A 03
O 03
A 04
Exact Price = £101.36% (near guess? Not very!) = 7/2 * (1 – ……%) a10 + 100 v….. @ j = √ 1.05 - 1
Guess duration = 3. Price for net yield of 5.25% would be 100. Guess price = (100 – 3 * (5-5.25)%) = …… 151
April 2000 Q10(ii)
Lender gets redemption yield below net coupon rate (7 * 0.75 = 5.25%), so faces a capital loss. Bond will be redeemed for 100, in any case, so amount of capital loss is known, and is more painful the sooner it is realised. So lender must assume earliest possible redemption. A 99
O 99
A 00
O 00
…
…
A 03
O 03
A 04
Exact Price = £101.36% (near guess? Not very!) = 7/2 * (1 – 25%) a10 + 100 v10 @ j = √ 1.05 - 1
Guess duration = 3. Price for net yield of 5.25% would be 100. Guess price = (100 – 3 * (5-5.25)%) = 100.75 152
April 2000 Q10(ii)
153
Sep 2000 Q12(i)
154
Sep 2000 Q12(i)
0
1
2
3
4
…
23
24
25
Rough check. If i for NPV = 6% then NPV = 25 * 25 – 3 * 180/1.06 = 116. At higher interest rate, NPV falls by say 1% * 13 * 25 * 25 = 81, to say 35 as long term receipts fall faster in value than short-term outgoings.
NPV = -180 ä3 @…..% + …..ā25 @ 1.07 / 1.06 –1 = -……… +………… = +51.7
155
Sep 2000 Q12(i)
0
1
2
3
4
…
23
24
25
Rough check. If i for NPV = 6% then NPV = 25 * 25 – 3 * 180/1.06 = 116. At higher interest rate, NPV falls by say 1% * 13 * 25 * 25 = 81, to say 35 as long term receipts fall faster in value than short-term outgoings.
NPV = -180 ä3 @7% + 25ā25 @ 1.07 / 1.06 –1 = -505.4 +557.1 = +51.7
156
Sep 2000 Q12(i)
157
Key questions Get 100% on Sep 2002 # 7 and April 2003 # 3. It doesn’t matter how many times you see the answers. Cover the answers up & do them again until you can explain the solutions to someone else.
158
Sep 2002 Q7
159
April 2003 Q3
160
Next session: Calculate mortgage schedule
END 161
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