I am currently studying about term structure and interest rates such as forward rates, swap rates, etc...

The following problem seems like an actual actuarial problem that I might see in the future and the question seems fairly open to me, and this is the first time I attempt to work on such problem so it would be great if I can get some advice from those who are (or are studying to become) actuaries.

The current term structure has the following nominal annual spot rates, $i^{(2)}$

6 month ... $8\%$ 1 year ... $10\%$ 1.5 year ... $x\%$

You predict that 6 months from now, the 6-month spot rate will be $10\%$. Construct a strategy to implement now, involving sale and purchase of zero coupon bonds that will make a profit for you if your prediction is correct.

(the $x\%$ was used for the first two questions which I already solved so it's robably safe to ignore)

I was able to so far understand that in the current situation, the forward rate from $t=1/2$ to $t=1$ is $\approx 12.07\%$.

So, if I predict that instead of that our rate is $10\%$, then the corresponding predicted spot rate for 1 year would have to be $\approx 8.995\%$

This is where I am stuck. How would an actuary utilize the discrepancy between these rates in order to profit?

My guess is that the present value in the predicted case is $84.18$ while the given present value is $79.62$, so the difference, $4.56$ can be reinvested somehow?


The strategy is to borrow by selling the 6-month bond $B_{0.5}$ and invest the proceeds in the 1-year bond $B_1$. Then after 6 months, when the loan is due ($B_{0.5}$ matures), sell $B_1$ to repay the loan. There will be a net profit as long as the 6-month spot rate in 6 months is less than the forward rate.

Suppose the face value of a bond is $100$. Let $P(t,T)$ denote the price of a zero-coupon bond at time $t$ maturing at time $T$.

At the present time $t=0$ the price of the bond $B_{0.5}$, maturing in 6 months, is

$$P(0,0.5) = \frac{100}{1+ 0.08/2}= \frac{100}{1.04}$$

and the price of the bond $B_1$, maturing in 1 year, is

$$P(0,1) = \frac{100}{1+ 0.10}=\frac{100}{1.10}.$$

The annualized forward rate is given by

$$\frac1{2}f(0.5,1) = \frac{P(0,0.5)}{P(0,1)}-1 \implies f(0.5,1) = 0.1154.$$

The predicted spot rate is less than the forward rate and the strategy is expected to make a profit.

Suppose we sell $1000$ $B_{0.5}$ bonds. The proceeds will be

$$1000 \cdot \frac{100}{1.04} \approx 96,154.$$

We can then purchase the following number of $B_1$ bonds

$$\frac{96,154}{\frac{100}{1.10}}\approx 1058.$$

After 6 months the $B_{0.5}$ bonds have a price of $100$ and the loan amount due is

$$1000 \cdot 100 = 100,000.$$

If the 6-month spot rate is $10 \%$ then the $B_1$ bonds are now worth

$$1058 \cdot P(0.5,1) =1058 \cdot \frac{100}{1+ 0.10/2} \approx 100,762,$$

and the net profit is $762$.


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