Studying for FM/2 and ran into this problem dealing with bonds;
A 1,000 par value 3 year bond with annual coupons of 50 for the first year, 70 for the second year, and 90 for the third year is bought to yield a force of interest
$$\delta_t=\frac{2t-1}{2(t^2-t+1)}, t\ge0.$$
Calculate the price of the bond.
So, $\delta_1=\frac{1}{2}, \delta_2=\frac{1}{2},$ and $\delta_3=\frac{5}{14}.$ And now, where do I go from here?
This is how I'd approach it. The price of the bond is just the present value of the coupon payments and redemption value.
But note that $$\delta_t = \frac{2t-1}{2(t^2-t+1)} = \frac{\frac{1}{2}(2t-1)}{t^2-t+1}.$$ This tells us that the accumulation function, $a(t)$, is given by $$ a(t)=(t^2-t+1)^{1/2}$$ because $$\delta_t = \frac{a'(t)}{a(t)}.$$ Now the coupon payments are as follows: $50$ at time $t=1$, $70$ at time $t=2$, $90$ at time $t=3$ and the redemption value(= par value because bond is redeemed at par) of $1000$ at time $t=3$. Hence $$PV = \frac{50}{a(1)}+\frac{70}{a(2)}+\frac{90}{a(3)} + \frac{1000}{a(3)}.$$