102 Session 7

102 Session 7 calculating i

© KSES Exam questions are copyright Faculty & Institute of Actuaries & are used with their permission Source: www.actuaries.org.uk 197

List the “aliases” for i Rate of r…..n Int….st r….e Rate at which cashflows are d……..nted Internal ….. of return M…..y-weighted …… of return Eff………ve rate of r…….n A.R Accum…..n r..te Red……..n y……d Inv…….ment r…….rn Net rate of re……n 198

List the “aliases” for i Rate of return Interest rate Rate at which cashflows are discounted Internal rate of return Money-weighted rate of return Effective rate of return APR Accumulation rate Redemption yield Investment return Net rate of return 199

Specimen Q5

200

Specimen Q5

703.84

703.84

1

Exactly, 1000 = ……. v

So 1 + i approx = (pay / loan)(1/…) = 1 + …..%

2

+

Roughly…. Total amount paid = approx 2 * …… = …… paid at time …

…… v2

Solve via quadratic formula or by linear interpolation or … by trial and error.

=> APR (which is i rolled down to lower 0.1%) = 26.1%

201

Specimen Q5

703.84

703.84

1

Exactly, 1000 = 703.84 v

So 1 + i approx = (1400 / 1000)(1/1.5) = 1 + 25%

2

+

Roughly…. Total amount paid = approx 2 * 700 = 1400 paid at time 1.5

703.84 v2

Solve via quadratic formula or by linear interpolation or … by trial and error. If i = 25.0%, value = 703.84 / 1.25 + 703.84 / 1.252 = 1014 (value too high) If i = 25 + 1.5%/1.5 = 26% value = 703.84 / 1.26 + 703.84 / 1.262 = 1002 (high) If i = 26.1% value = 703.84 / 1.261 + 703.84 / 1.2612 = 1001 (still too high) If i = 26.2% value = 703.84 / 1.261 + 703.84 / 1.2612 = 999.6 (value too low) => APR (which is i rolled down to lower 0.1%) = 26.1% 202

Specimen Q5

203

Specimen Q6

204

Specimen Q6

205

Apr 2002 Q4

206

Apr 2002 Q4

Roughly Total amount received (mostly after a year) =…+…+…=…

June

December

So 1 + i approx = … / 99 =1+…%

Working in periods of half years, relative to time price paid (Jan 2000), June 2000 is at t=1 and Dec 2000 is at t=2. So value = Price = … = … v + (… + …)v2 By quadratic formula, v = ……. => i = ……% per half year => 2 * ……. % = ……. % convertible half-yearly pa207 (tallies with rough guess?)

Apr 2002 Q4

Roughly Total amount received (mostly after a year) = 101 + 6.5 + 6.6 = 114.1

June

December

So 1 + i approx = 114.1 / 99 = 1 + 15.3%

Working in periods of half years, relative to time price paid (Jan 2000), June 2000 is at t=1 and Dec 2000 is at t=2. So value = Price = 99 = 6.5v + (101 + 6.6)v2 By quadratic formula, v = 1/1.0759 => i = 7.59% per half year => 2 * 7.59% = 15.18% convertible half-yearly pa (tallies with rough guess)

208

Apr 2002 Q4

209

Sep 2000 Q5

210

Sep 2000 Q5

GRY is i: discounted value = price

1

2

3

4

5

6

7

8

If yield were 4%, price would be ………. Guess duration about 2/3 term ie cash paid about at about t = … on average. Price actually …… lower than 100,so guess yield about …./duration = ….% ………. than 4%. Ie guess yield as …..%

9 10 11 12 13 14 15 16 17 18 19 20

Value = Price = ….. Get i by trial & interpolation GRY = ……..%

= …. a20¬ + …. v20 211

Sep 2000 Q5

GRY is i: discounted value = price

1

2

3

4

5

6

7

8

If yield were 4%, price would be par = 100. Guess duration about 2/3 term ie cash paid about at about t = 13 on average. Price actually 4% lower than 100,so guess yield about 4%/13 = 0.3% higher than 4%. Ie guess yield as 4.3%

9 10 11 12 13 14 15 16 17 18 19 20

Value = Price = 96 If i = 4.3%, value = 96.03 If i = 4.4%, value = 95.

= 4 a20¬ + 100 v20 4.3% is closer to actual price, so GRY = 4.3%

212

Sep 2000 Q5

213

Sep 2002 Q5(i)

214

Sep 2002 Q5(i)

215

Sep 2001 Q10(ii)b

216

Sep 2001 Q10(ii)b

217

Apr 2003 Q10

218

Apr 2003 Q10 (i-ii)

219

Apr 2003 Q10 (iii)

220

Apr 2002 Q7(i)

221

Apr 2002 Q7(i)

Average forward rate is 1/3 * ( ….+….+…. )% = …..%, so guess GR yield = …..% Interest rate not constant. v(1) = 1/……. v(2) = v(1) / ……… v(3) = v(2) / ……… 1

2

3

Value = Price = …. v(1) + …. v(2) + …… v(3) = …… (coupon >/< yield? so price >/< 100 ) Using constant i, Value = ………… = …a3 + …..v3. Get i by trial & interpolation (near guess?)

222

Apr 2002 Q7(i)

Average forward rate is 1/3 * ( 4 +4.5 + 4.8 )% = 4.4%, so guess GR yield = 4.4% Interest rate not constant. v(1) = 1/1.04 v(2) = v(1) / 1.045 v(3) = v(2) / 1.048 1

2

3

Value = Price = 5 v(1) + 5 v(2) + 105 v(3) = 101.597

(coupon > yield => capital loss => price > 100 )

Using constant i, Value = 101.597 = 5a3 + 100v3. By trial & interpolation, i = 4.42% (near guess)223

Apr 2002 Q7(i)

224

Sep 2001 Q7(i)

225

Sep 2001 Q7(i)

Average forward rate is ¼ * ( … +…. + …. + …. )% =….. so guess GR yield = …..% Interest rate not constant. v(1) = 1/……. v(2) = v(1) / ……. v(3) = v(2) / ……. v(4) = v(3) / ……. 1

2

3

4

Value = Price = …. v(1) +…. v(2) + …. v(3) + …… v(4) = …… Using constant i, Value = ……. = ….a4 + ….v^…. Get I by trial & interpolation (near guess?)226

Sep 2001 Q7(i)

Average forward rate is ¼ * ( 8 + 7 + 6 + 5 )% = 6.5% so guess GR yield = 6.5% Interest rate not constant. v(1) = 1/1.08 v(2) = v(1) / 1.07 v(3) = v(2) / 1.06 v(4) = v(3) / 1.05 1

2

3

4

Value = Price = 5 v(1) + 5 v(2) + 5 v(3) + 105 v(4) = 94.68 Using constant i, 227 Value = 94.68 = 5a4 + 100v^4. By trial & interpolation, i = 6.55% (near guess)

Sep 2001 Q7(i)

228

Sep 2000 Q7

229

Sep 2000 Q7

Div yield (d) = div/price = …%. Dividend growth (g) = ….. Inflation (inf) = ….. ?Nominal return (i) = d + g = …. ?Real return = i - inf =….

NOW

- 9/12

0

1

2

3

Value = Price = ….. Get I by trial and interpolation

4

1 + exact real return = (1 + i) / (1 + inflation)

= [v (…..) @ i ] * …. ä999999¬ @ (i – …. / …..)

So 1 + real yield = …… / ……. = ………

230

Sep 2000 Q7

Dividend yield about 10/250 = 4%. Dividend growth = 5% Inflation = 3% ?Nominal return = 4 + 5 = 9% ?Real return = 9 - 3 = 6%

NOW

- 9/12

0

1

2

3

4

1 + exact real return = (1 + i) / (1 + inflation)

Value = Price = 250 = [v (9/12) @ i ] * 10 ä999999¬ @ (i – 5% / 1.05) If i (nominal) = 9.0%, value = 255.4 (2% too high, try 0.2% higher yield) If i (nominal) = 9.2%, value = 243.4 Interpolate, i = 9.09% 231 So 1 + real yield = 1.0909 / 1.03 = 1.0591

Sep 2000 Q7

232

Sep 2003 Q6

233

Sep 2003 Q6

234

Apr 2001 Q5

235

Apr 2001 Q5 (i)

236

Apr 2001 Q5 (ii)

237

Sep 2002 Q6(i)

238

Sep 2002 Q6(i)

1

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7

9

11

13

15

17

Total cashflows are approx £m …..* 2 + ….. * 15 = …………Discount the cash out for 1 yr & the cash in for about 7 years Guess NPV -…./1.1 + …./1.9 = …

NPV = Value = -…..a2 (“a bar 2”) + v2[ … a10 + ….. (Da)10] + v…. (Da)… @ 10% = = £……….m (near guess?) 239

Sep 2002 Q6(i)

1

3

5

7

9

11

13

15

17

Total cashflows are approx £m -10 * 2 + 8/2 * 15 = -20 + 60 = 40 Discount the cash out for 1 yr & the cash in for about 7 years Guess NPV -20/1.1 + 60/1.9 = 13

NPV = Value = -10a2 (“a bar 2”) + v2[ 3 a10 + 0.5 (Da)10] + v12 (Da)3 @ 10% = -10 * 1.8209 + 0.8264 * [3 * 6.145 + 0.5 * 38.55 ] + 0.3186 * 5.131 = £14.59m (not a million miles from guess) 240

Sep 2002 Q6(ii)

241

Sep 2002 Q6(ii)

IRR is interest rate at which NPV = ……. NPV was ……. on rate of 10%. 1

3

5

7

9

11

13

15

17

If interest rate rises, it will have only slight effect on value of …….. term cashflows (the money ….), but it will have big effect on the …….. term cashflows (the money ….). So as yield rises, the NPV will ….. So (to get NPV of zero) the IRR must … 242 to be ……. than 10%.

Sep 2002 Q6(ii)

IRR is interest rate at which NPV = 0. NPV was 14.59 on rate of 10%. 1

3

5

7

9

11

13

15

17

If interest rate rises, it will have only slight effect on value of short term cashflows (the money out), but it will have big effect on the longer term cashflows (the money in). So as yield rises, the NPV will fall. So (to get NPV of zero) the IRR must rise 243 to be more than 10%.

Sep 2002 Q6

244

Sep 2001 Q8

245

Sep 2001 Q8 (start)

246

Sep 2001 Q8 (middle)

247

Sep 2001 Q8 (end)

248

Key question Get 100% on Apr 2003 Q10. It doesn’t matter how many times you see the answers. Cover the answers up & do it blind to the point where you can explain it out loud.

249

Next session: payback period

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