tutorial questions set 1

8/3/2019 Tutorial Questions Set 1

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Investment Management

INVESTMENT MANAGEMENT

TUTORIAL SET 1

QUSETION 1

Consider two investments A and B with the following annual returns

Year A’s Return B’s Return1 4% 14%2 5% 12%3 6% 10%

Compute the correlation between the return on A and B.

AnswerThe correlation between return on A and B is

AB AB

A B

σρ =

σ σ.

R A R B (R A-E(R A))2 (R B-E(R B))2 (R A-E(R A))(R B-E(R B))

4 14 1 4 -2

5 12 0 0 0

6 10 1 4 -2

15 36 2 8 -4

15 36

5 123 3

A B E( R ) , E( R )= = = =

2 22 8 4

3 3 3 A B AB , ,σ = σ = σ = −

4

31

2 8

3 3

AB AB

A B

−σ

∴ρ = = = −σ σ

×

QUSETION 2

Consider two investments A and B with the following annual returns

Year A’s Return B’s Return1 7% 14%2 5% 10%3 3% 6%

Compute the covariance and correlation between the return on A and B.

Tutorial Questions Set 1 1




Answer

The correlation between return on A and B is

AB AB

A B

σρ =

σ σ .

RA RB (RA

-E(RA))2

(RB -E(RB))2 (RA -E(RA)) (RB

-E(RB))

7 14 4 16 2x4=8

5 10 0 0 0x0=0

3 6 4 16 -2x-4=8

15 30 8 32 16

15 305 10

3 3

8 32

3 3

16

3

A B

A B

A B

E( R ) %, E( R ) %

Var( R ) , Var( R ) ,

Cov( R ,R ) ,

= = = =

= =

=

16

3 18 32

3 3

A B

A B

Cov( R ,R )∴ ρ = = =

σ σ×

.

QUSETION 3

Asset price inUSD

USD\SGD rate of exchange

Start of January 10.00 2.00 End of June 11.00 2.00 End of December 15.00 1.60

Compute the continuously compounded SGD per annum rate of return, for aSingaporean investor, investing in the above US asset over the one year interval.

Show that this per annum rate of return,

(i) can be decomposed into (1) a rate of return on the asset and (2) arate of return on currency and

(ii) is consistent with the rate of return achieved in the first six-months of the year and the second six-months of the year.





Answer

The continuously compounded return is computed as

=

before

now

P

Plnr

(i)

01 1 1

0 0 0 1

01

0 1

overall

equity foreign exchange

E P E P r ln ln( )

P E P E

E P ln ln r r

P E

= = ×

= + = +

(ii) Consider the return in USD over the year

4055.010

15lnr =

=

The two year continuously compounded increase in the SGD value is

(+18,23%) is equal to the USD increase (+40.55%) plus the appreciation of

the USD against the SGD (-22.31%)

Asset price

in USD

USD\SGD

rate of exchange

Asset price

in SGD

Return

Start of January 10.00 2.00 20.00

End of June 11.00 2.00 22.00 9.53%

End of 15.00 1.60 24.00 8.70%

Whole year 40.55% -22.31% 18.23% 18.23%

The annual continuously compounded increase in the SGD value is (+18.23%)

is equal to the first year change (9.53%) plus the second year change (8.70%).





QUSETION 4

Suppose immediately after you purchased a 5 year 10% (paid semi-annual) coupon

bond, yields fell from a flat 8% to a flat 6% and remained there for 5 years of your investment.

Compute realized HPY on

a) quarterly-annual compound basis and,b) continuously compound basis.

Answer

b) 5 year 10% pa (semi-annually) coupon bond.

(1) HPY on quarterly-annual compound basis1

20

5 41 1 4

4

buy

buy

r FV FV P ( ) HPY

P

×

= + → = − ×

10

10

10

10 0 03

0 10100 5

2

1

1 1 0 04 100 1005 5 8 111 108 12

0 04 1 04 1 48

1 1

1 03 1100 100 5 100 57 32 157 32

0 03

buy nn i

n

sell coupon n i

.

.c

F P ca

( i )

( . ). .

. ( . ) .

( i ) FV P S S

i

( . )cS . .

.

−

= × =

= ++

− += × + = × + =

+ −= + =

−= + = + × = + =

1

20157 321 4 7 57

108 12

1 89

. HPY . % p.a.

.

. % percoupon period

∴ = − × =

=

(2) HPY on continuously compounded basis

1

1 157 327 50

5 108 12

rt

buy

buy

FV FV P e HPY ln

t P

. HPY ln . % p.a.

.

= → =

= × =


8%→6%5 yrs=10 periods




QUSETION 5

Consider the yields

Week Yield

0 10.0% p.a.

1 11.0% p.a.

Compute an annualized continuously compounded holding period return for an

investment entered into week 0 and exited week 1 if yields refer to

1) 90 day bank bill,

2) 10 year 10% semi-annual coupon bond.

AnswerThe annualised continuously compounded holding period return is derived from the

equation

1rt sell sell buy

buy

P P P e . HPY r ln .

t P = ∴ = = ×

(1) 90 day discount bank bill

Recall that 11

FV FV P( rt ) P

rt = + → =

+and assume that FV = 100, then

10097 5936

901 0 10

365

10097 5597

831 0 11

365

buy

sell

P .

.

P .

.

= =+ ×

= =+ ×

1 52 97 5597 1 811 97 5936

sell

buy

P . HPY ln ln . % p.a.t P .

∴ = × = × = −

(2) 10 year 10% p.a. semi annual coupon bond.

100buy P FV ( par bond ).= =


10% 11%

90 days7 days083 days

10% 11%

period 20

(10 years)

1week

period 1

(26 weeks)

019 periods




To evaluate sell P , find

(Step 1:)

19

19

0 10 100 52

1 1

1

0 111 1

10025 94 1962

0 11 0 11

12 2

n

next coupon date nn i n i

next coupon date

.c ,

FV ( i ) P ca where a .

( i ) i

.

P .. .

−

−

= × =

− += + =

+

− + = + =

+

(Step 2:)Add coupon to the PV of the bond at the next coupon date:

5 + 94.1962 = 99.1962

and (Step 3:)Discount back to week 1 (sale date):

25

26

99 196294 2187

0 111 2

sell

. P .

.

= =

+ Then

1 52 94 21873 096 309 6

1 100

sell

buy

P . HPY ln ln . . % p.a.

t P = × = = − = −


tutorial questions set 1

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