| Exam Guide
Question 2: Bond Valuation and Analysis
(a) (i) From the following formula:
DP
P (D Dr)
=- ′ D Dr .DP = ′- ′
P
( ′-6 )( ()r 10 <0
120 ()′Dr -110 ′-12 ′D)-
we can see: D r < D 0.0167
(where D r = interest-rate change and D = modified duration of bonds (years))
In other words, the surplus will be negative if interest rates fall 1.67 percentage
points or more.
(ii) From the following equation:
[(80 ′-D ′Dr)+(40 ′-4 ′ r -(110 )
( )()D)′- 12 ′Dr ]
12 024 +2 004 . -10
.
we can see: D = 14.5
(where D = modified duration of bonds)
In other words, the modified duration of the bonds should be 14.5 years.
(iii) From the following equation:
(120 ′ –D ′D r) – (110 ′ –12) ′D r) = 0
we can see: D = 11.0
In other words, the modified duration of the bonds should be 11 years.
(iv) Even if the bond weighting is small, the duration of the bonds can be
lengthened so that the duration of pension assets as a whole matches the
duration of pension liabilities, which reduces interest-rate risk. In doing this,
the equity component can be increased to cover the reduction in bonds, which
will increase the expected long-term return on the asset side. However, this also
increases risk from stock price movements, a different risk than the interest-rate
risk considered here.
(b) (i) Inflation will result in an increase in nominal interest rates and therefore lower
the market price of bonds (assets). Pension liabilities are virtually unchanged
because the increase in future benefit amounts is offset by the increase in the
discount rate (nominal interest rate).
(ii) When pension benefit amounts are indexed to inflation, it is essentially the
same as benefit amounts being fixed in real terms, so the liability (present
value) is found by discounting the benefit amount using real interest rates.
(iii) Pension liabilities essentially move dependent on real interest rates, so that an
investment in “CPI-linked bonds” for which prices change according to real
yields will result in a matching of bond and liability durations in terms of real
interest rates. If these bonds are not available, bearing in mind that short-term
instruments provide a good hedge against inflation, short-term bonds or
variable rate interest-bearing bonds should be used.
Exam Guide
Question 3: Derivative Valuation and Analysis
(a) (V – 16)/16 = 0.03 + 1.2 ′ [(T – 1,600)/1,600 – 0.03]
Rearranging this: V = 0.012T – 3.296 (￥ billion)
(b) (i) Assuming X is the strike price, Y the number of contracts, payoff at expiration
is:
V + 0.00001Y max [X – T,0]
= 0.012T – 3.296 if T > X
or (0.012 – 0.00001Y)T + 0.00001XY – 3.296 if T < X
Therefore, from 0.12 – 0.0001Y = 0
Y = 1,200 contracts
From 0.00001XY – 3.296 = 16
X = 1,608
(ii) Cost of hedge = put price ′ no. of contracts ′ TPOIX option: index
76.3 ′ 1,200 ′ 10,000
= ￥915,600,000
(c) (i) Relationship between change of TOPIX and the price of puts:
D P = –0.406 ′D T
Relationship between change in TOPIX and futures price:
D F = (1 + 0.03/12) D T
Therefore, you need to sell
1,200 ′ [0.406/(1 + 0.03/12)] = 486 contracts
(ii) You now need to sell
1,200 ′ [0.355/(1 + 0.03/12)] = 425 contracts
So you need to buy back
–425 – (–486) = 61 contracts
(iii) When volatility is high, the number of contracts you need to sell for hedging
becomes larger. In this case, you will suffer a loss because the number of futures
you have sold is smaller than is necessary. |
