| Exam Board | OCR MEI |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modelling and Hypothesis Testing |
| Type | Investment and asset allocation decisions |
| Difficulty | Moderate -0.5 This is a standard decision mathematics question involving decision trees, EMV calculations, and utility functions. While it has multiple parts and requires careful bookkeeping of probabilities and exchange rates, each step follows routine procedures taught in D2. The calculations are straightforward applications of expected value formulas and utility theory—no novel problem-solving insight is required, making it slightly easier than average for A-level. |
| Spec | 7.05a Critical path analysis: activity on arc networks7.05b Forward and backward pass: earliest/latest times, critical activities7.05d Latest start and earliest finish: independent and interfering float7.05e Cascade charts: scheduling and effect of delays |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Decision tree constructed correctly | M1 | |
| Chance nodes correct (0.6: 75, 0.4: 55 for Greece) | A1 | chance nodes |
| Choice node correct, value 67 | A1 | choice node |
| UK values: \(1.20 \times 35 \times 1.6 = 67.2\); \(1.10 \times 35 \times 1.6 = 61.6\); node value 64.4 | B1\(\checkmark\) | invest in Greece |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| New chance nodes added correctly | M1 | |
| 64.855 or 0.86 or 0.85 calculated | A1 | |
| UK node values: \(1.2\times35\times1.7=71.4\); \(1.2\times35\times1.5=63\); \(1.10\times35\times1.7=65.45\); \(1.10\times35\times1.5=57.75\) | ||
| Nodes 68.88 and 60.83 correct | ||
| Decision: invest in Greece | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Utilities calculated correctly | M1 | utilities |
| \(75^{0.75}=25.49\); \(55^{0.75}=20.20\); Greece node = 23.37 | A1 | 23.37 and 28.14 |
| UK utilities: \(71.4^{0.8}=30.41\); \(63^{0.8}=27.51\); \(65.45^{0.8}=28.36\); \(57.75^{0.8}=25.66\) | ||
| Nodes 29.54 and 26.74; overall node = 28.14 | ||
| Decision: invest in UK | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Require \(\dfrac{1.2+1.1}{2} \times 35 \times x = 67\) | M1 | |
| \(x = 1.665\) | A1 cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Require \(\dfrac{(1.2 \times 35 \times y)^{0.8} + (1.1 \times 35 \times y)^{0.8}}{2} = 23.37\) | M1 M1 | M1 cash, M1 house |
| One bracket evaluated correctly | A1 | |
| Trying \(y=1.277\): \((1.2\times35\times1.277)^{0.8}=24.185\); \((1.1\times35\times1.277)^{0.8}=22.559\); \((24.185+22.559)/2=23.37\) | A1 |
# Question 2:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Decision tree constructed correctly | M1 | |
| Chance nodes correct (0.6: 75, 0.4: 55 for Greece) | A1 | chance nodes |
| Choice node correct, value 67 | A1 | choice node |
| UK values: $1.20 \times 35 \times 1.6 = 67.2$; $1.10 \times 35 \times 1.6 = 61.6$; node value 64.4 | B1$\checkmark$ | invest in Greece |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| New chance nodes added correctly | M1 | |
| 64.855 or 0.86 or 0.85 calculated | A1 | |
| UK node values: $1.2\times35\times1.7=71.4$; $1.2\times35\times1.5=63$; $1.10\times35\times1.7=65.45$; $1.10\times35\times1.5=57.75$ | | |
| Nodes 68.88 and 60.83 correct | | |
| Decision: invest in Greece | B1 | |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Utilities calculated correctly | M1 | utilities |
| $75^{0.75}=25.49$; $55^{0.75}=20.20$; Greece node = 23.37 | A1 | 23.37 and 28.14 |
| UK utilities: $71.4^{0.8}=30.41$; $63^{0.8}=27.51$; $65.45^{0.8}=28.36$; $57.75^{0.8}=25.66$ | | |
| Nodes 29.54 and 26.74; overall node = 28.14 | | |
| Decision: invest in UK | B1 | |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Require $\dfrac{1.2+1.1}{2} \times 35 \times x = 67$ | M1 | |
| $x = 1.665$ | A1 cao | |
## Part (v)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Require $\dfrac{(1.2 \times 35 \times y)^{0.8} + (1.1 \times 35 \times y)^{0.8}}{2} = 23.37$ | M1 M1 | M1 cash, M1 house |
| One bracket evaluated correctly | A1 | |
| Trying $y=1.277$: $(1.2\times35\times1.277)^{0.8}=24.185$; $(1.1\times35\times1.277)^{0.8}=22.559$; $(24.185+22.559)/2=23.37$ | A1 | |
---2 Karl is considering investing in a villa in Greece. It will cost him 56000 euros ( € 56000 ). His alternative is to invest his money, $\pounds 35000$, in the United Kingdom.
He is concerned with what will happen over the next 5 years. He estimates that there is a $60 \%$ chance that a house currently worth $€ 56000$ will appreciate to be worth $€ 75000$ in that time, but that there is a $40 \%$ chance that it will be worth only $€ 55000$.
If he invests in the United Kingdom then there is a $50 \%$ chance that there will be $20 \%$ growth over the 5 years, and a $50 \%$ chance that there will be $10 \%$ growth.\\
(i) Given that $\pounds 1$ is worth $€ 1.60$, draw a decision tree for Karl, and advise him what to do, using the EMV of his investment (in thousands of euros) as his criterion.
In fact the $\pounds / €$ exchange rate is not fixed. It is estimated that at the end of 5 years, if there has been $20 \%$ growth in the UK then there is a $70 \%$ chance that the exchange rate will stand at 1.70 euros per pound, and a $30 \%$ chance that it will be 1.50 . If growth has been $10 \%$ then there is a $40 \%$ chance that the exchange rate will stand at 1.70 and a $60 \%$ chance that it will be 1.50 .\\
(ii) Produce a revised decision tree incorporating this information, and give appropriate advice.
A financial analyst asks Karl a number of questions to determine his utility function. He estimates that for $x$ in cash (in thousands of euros) Karl's utility is $x ^ { 0.8 }$, and that for $y$ in property (in thousands of euros), Karl's utility is $y ^ { 0.75 }$.\\
(iii) Repeat your computations from part (ii) using utility instead of the EMV of his investment. Does this change your advice?\\
(iv) Using EMVs, find the exchange rate (number of euros per pound) which will make Karl indifferent between investing in the UK and investing in a villa in Greece.\\
(v) Show that, using Karl's utility function, the exchange rate would have to drop to 1.277 euros per pound to make Karl indifferent between investing in the UK and investing in a villa in Greece.
\hfill \mbox{\textit{OCR MEI D2 2005 Q2 [16]}}