NO.PZ2025040202000060
问题如下:
Question Six months ago, an investor entered into a receive-fixed 4.5%, pay-equity index 3-year annual reset swap, where both legs have a notional value of $1,000,000. The current present value factors for the appropriate spot interest rate maturities are as follows:
If the value of the underlying equity index decreased from 100 to 98 over the recent 6-month period, the current value of the equity swap is closest to:
选项:
A.A.–$6,721. B.B.$20,000. C.C.$33,687.解释:
A Incorrect because it assumes the
underlying equity index rose from 98 to 100 over the six months, rather than
fell from 100 to 98, yielding:
VEQ,t = $1,013,687 –
(100/98) × $1,000,000 ≈ –$6,721
B Incorrect because it uses the fair value
of the implied fixed-rate bond at initiation (equal to par = $1,000,000) rather
than the value after six months, computed with the current present value
factors:
VEQ,t = $1,000,000 –
(98/100) × $1,000,000 = $20,000
C Correct because
finding the value of an equity swap after the swap is initiated, say at Time t
(so, VEQ,t), is similar to valuing an interest rate swap except that
rather than adjusting the floating-rate bond for the last floating rate
observed (remember, advanced set), we adjust the value of the notional amount
of equity, as shown below:
VEQ,t = VFIX(C0) – (St/St–1)NAE –
PV(Par – NAE),
where VFIX(C0) denotes the value at Time t of a
fixed-rate bond initiated with coupon C0 at Time 0, St denotes
the current equity price, St–1 denotes the equity price
observed at the last reset date, and PV() denotes the present value function
from the swap maturity date to Time t.
Hence, the fair value of this swap is found
by solving for the fair value of the implied fixed-rate bond. We then adjust
for the equity value. The fixed rate of 4.5% results in fixed cash flows of
0.045 × $1,000,000 = $45,000 at each settlement. Applying the respective
present value factors gives us the following:
Therefore, the fair value of the implied
fixed-rate bond, VFIX(C0), is $1,013,687 and the fair
value of the equity swap is:
VEQ,t = $1,013,687 –
(98/100) × $1,000,000 – 0 ≈ $33,687,
where the last term is zero since the
notional amount of equity and the bond par value are assumed equal.
如何理解实际含义。很容易误以为向上箭头盈利13686.92,向下箭头亏损20000,轧差选成了6k的选项