Quiz: 1
Course: M339W/M389W - Financial Math for Actuaries
Page: 1 of 3
University of Texas at Austin
Quiz # 1
Prerequisite material.
Provide your final answer only for the following problems.
Problem 1.1. (1 pt) Let a yield curve be defined by the following equation:
ik = 0.09 + 0.002k − 0.001k 2
where ik denotes the annual effective rate of return for zero coupon bonds with maturity of k years. Let
r0 (4, 5) be the one-year effective forward rate during year 5 that is implied by this yield curve.
Calculate f = r0 (4, 5).
(a) About 3.6%
(b) About 4.7%
(c) About 5.6%
(d) About 6.8%
(e) None of the above
Solution: (b)
Let f = r0 (4, 5). By definition, f should satisfy the equation
(1 + i5 )5
.
(1 + i4 )4
From the equation for the yield curve given in the problem, we get that i4 = 0.082 and i5 = 0.075. Plugging
these numbers into the above equation and solving for f , we obtain that f = 0.047.
1+f =
Problem 1.2. (1 pt) Assume that the price of a two-year zero-coupon bond is 0.881659 and that the price
of a three-year zero-coupon bond is 0.816298. The two bonds have the same redemption amounts of $1.
Denote by f = r0 (2, 3) the implied forward interest rate from year 2 to year 3. Then:
(a) 0 ≤ f ≤ 0.01
(b) 0.01 ≤ f ≤ 0.05
(c) 0.05 ≤ f ≤ 0.09
(d) 0.09 ≤ f ≤ 0.015
(e) None of the above
Solution: (c)
Let P (0, 2) denote the price of a two-year zero-coupon bond and let P (0, 3) denote the price of a two-year
zero-coupon bond. Then,
f=
0.881659
(1 + r3 )3
P (0, 2)
−1=
− 1 = 0.08.
−1=
2
(1 + r2 )
P (0, 3)
0.816298
Problem 1.3. (1 pt) A firm wishes to borrow money repayable in two years. The one-year and two-year
nominal annual spot rates are 7% and 9% respectively. Find the one-year one year deferred spot rate, i.e.,
the year one implied forward effective annual rate.
(a) About 7%
(b) About 9%
(c) About 11%
(d) About 13%
(e) None of the above
Instructor: Milica Čudina
Quiz: 1
Course: M339W/M389W - Financial Math for Actuaries
Page: 2 of 3
Solution: (c)
Let us denote the forward rate we seek by f . From the definition of forward rates:
(1 + 0.09)2 = (1 + 0.07)(1 + f ) ⇒ f ≈ 0.11.
Problem 1.4. (2 points) Source: SoA Sample FM (DM) Problem #23.
In our usual notation, you are given the following zero-coupon bond prices
P (0, 1) = 0.96154,
P (0, 2) = 0.91573,
P (0, 3) = 0.85770,
P (0, 4) = 0.78466,
P (0, 5) = 0.69656.
You enter into a 5-year interest rate swap (with a notional amount of $100,000) to pay a fixed rate and to
receive a floating rate based on future 1-year LIBOR rates. If the swap has annual payments, what is the
fixed rate you should pay?
(a) 0.052
(b) 0.057
(c) 0.067
(d) 0.072
(e) None of the above.
Solution: (d)
Let us denote the fixed rate by R. Then, in our usual notation,
P5
P (0, k)r0 (k − 1, k)
.
R = k=1P5
k=1 P (0, k)
We have
5
X
k=1
P (0, k)r0 (k − 1, k) =
5
X
P (0, k) ·
k=1
P (0, k − 1)
= 1 − P (0, 5) = 1 − 0.69656 = 0.30344.
P (0, k)
Also,
5
X
P (0, k) = 4.21619.
k=1
Finally, R = 0.30344/4.21619 ≈ 0.072.
Please, provide a complete solution to the following problem.
Problem 1.5. (5 points) Emmanuel entered an extra special kind of game with his friend Fischer. First,
they toss a fair coin. If the coin comes up heads, Emmanuel gives $5, 000 to Fischer. If the coin comes up
tails, Fischer gives $2, 000 to Emmanuel. Then, regardless of the outcome of the first cointoss, they toss
the same fair coin again. If it comes up heads, Emmanuel gives Fischer $4,000. If the coin comes up tails,
Fischer gives $3,000 to Emmanuel. What is the expected cashflow, i.e., what is the expected amount of
money that changes hands and who gives it to whom?
Solution: Let Xi be the cashflow from Emmanuel’s perspective after the ith cointoss for i = 1, 2. Then,
we are looking for
E[X1 + X2 ] = E[X1 ] + E[X2 ]
1
1
= (−5, 000 + 2, 000) + (−4, 000 + 3, 000) = −2, 000.
2
2
So, the expected cashflow is $2,000 from Emmanuel to Fischer.
Instructor: Milica Čudina
Quiz: 1
Course: M339W/M389W - Financial Math for Actuaries
Page: 3 of 3
Problem 1.6. (10 pts) For a two-period binomial model, you are given that:
(1) each period is one year;
(2) the current price of a non-dividend-paying stock S is S(0) = $20;
(3) u = 1.3, with u as in the standard notation for the binomial model;
(4) d = 0.9, with d as in the standard notation for the binomial model;
(5) the continuously compounded risk-free interest rate is r = 0.05.
Consider a special call option which pays the excess above the strike price K = 23 (if any!) at the end of
every binomial period.
Find the price of this option.
Solution: The risk-neutral probability is
e0.05 − 0.9
= 0.3782.
1.3 − 0.9
When one constructs the two-period binomial tree, one gets
Su = 26, Sd = 17,
p∗ =
Suu = 33.80, Sud = Sdd = 23.4, Sdd = 16.2.
So, the payoffs at the end of the first period are
Vu = 3, Vd = 0.
The payoffs at the end of the second period are
Vuu = 10.80,
Vud = 0.4,
Vdd = 0.
So, taking the expected value at time 0 of the payoff with respect to the risk-neutral probability, we get that
the price of this call should be
e−0.05 × Vu × p∗ + e−0.05×2 [Vuu × (p∗ )2 + Vud × 2p∗ (1 − p∗ )]
= e−0.05 × 3 × 0.3782 + e−0.1 [10.8 × 0.37822 + 0.4 × 2 × 0.3782 × (1 − 0.3782)]
= 1.079 + 1.568 = 2.647.
Instructor: Milica Čudina