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A pension fund has a liability of R400,000 due in ten years' time.
The pension fund has exactly enough funds to cover the liability based on an effective rate of interest of $8 \%$ per annum. This is also the interest rate at which current market prices are calculated and the rate earned on cash.

The pension fund wishes to hold $10 \%$ of its funds in cash, and to invest the balance in the following securities:
- A zero-coupon bond redeemable at par in 12 years' time
- A fixed-coupon bond paying half-yearly coupons of $8 \%$ per annum, in arrear, redeemable at $110 \%$ in 16 years' time
i. Determine the amount to invest in the zero-coupon bond and the fixed-coupon bond, respectively, for the pension fund's portfolio of assets and liabilities to satisfy Redington's first two conditions for immunisation.
ii. Explain, without further calculation, whether the pension fund would be immunised against small changes in the interest rate if the quantities of stock in i. are purchased.
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(i)
Let $\quad \mathrm{X}=$ nominal amount of zero-coupon bond
$\mathrm{Y}=$ nominal amount of fixed-coupon bond
$\mathrm{PV}(\mathrm{L})=400,000 v_{\mathrm{sis}}^{10}=185,277.3952$
$10 \%$ is in cash thus $18,527.73952$ and remainder $166,749.6557$ will be invested in 2 bonds
$18,527.73952+X v_{8 \mathrm{~s}}^{12}+0.08 Y a_{16 \mathrm{si}}^{(2)}+1.1 \mathrm{~V} v_{\mathrm{ss}}^{16}=185,277.3952$
$X v_{8 \mathrm{~s}}^{12}+Y 0.08 a_{14}^{(2)}+1.1 Y v_{s \mathrm{~s}}^{16}=166,749.6557 \ldots \ldots .1$
Numerator of DMT(L): $(10) 400,000 v_{s \mathrm{~s}}^{10}=1,852,773.952$
$i=8 \% \Rightarrow i^{(2)}=7.864096908 \% \Rightarrow \frac{i^{(2)}}{2}=3.923048454 \%$
Numerator of $\operatorname{DMT}(\mathrm{A})$ :
From (1)
$X=\frac{(166,749.6557-1.043078787 Y)}{0.397113759}$
sub into (2)
$Y=57,483.53505$
sub into $(1)$ and $X=268,914.8824$

(ii)
- Spread of asset around DMT is greater than spread of liabilities around DMT
- Convexity of assets $>$ Convexity of liabilities
- Immunized
by Platinum (91,198 points)

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