Illinois State University, Mathematics 483, Fall 2014
Test No. 3, Tuesday, December 2, 2014
SOLUTIONS
1. Spring 2013 Casualty Actuarial Society Course 9 Examination, Problem No. 7
Given the following information on a currency swap:
• The swap has a remaining life of 15 months.
• The swap involves exchanging interest at 9% on 20 million British pounds for interest
at 5% on $30 million at the end of each year.
• Principal amounts will be exchanged when the swap term expires.
• The term structure of interest rates in both the United Kingdom and the United States is
currently flat and if the swap were negotiated today, the interest rates exchanged would
be 6% in British pounds and 4% in dollars.
• The current exchange rate (dollars per British pound) is 1.65.
• All interest rates are quoted with annual compounding.
Viewing the swap as a portfolio of forward contracts, calculate the current value of the
swap to the party paying British pounds, such value calculated in British pounds.
Solution.
There are two payment times remaining on the swap: 3 months from now and 15 months
from now. In 3 months, the party paying British pounds will need to pay 9% on 20
million British pounds, or 1.8 million pounds, and will receive 5% on $30 million, or
$1.5 million. This is equivalent to a forward contract for purchase of $1.5 million in three
months by paying 1.8 million pounds for it. Remember that we are doing calculations in
pounds. Current forward price (in pounds) for purchase of $1.5 million in 3 months is (in
pounds)
Accumulated with
interest in British pounds

3
$1,500,000
1.65


Spot price in pounds
1.06 12
3
12
≈ 913, 430.37.
1.04


Contract price without
dollar interest income,
i.e., "ex-dividend"
Instead, 1.8 million pounds will be paid. This amounts to
913,430.37 – 1,800,000.00 = –886,569.63
pounds in three months, and in present value
886,569.63
−
≈ −873, 749.38 pounds.
3
12
1.06
In 15 months, the party paying pounds will pay 1.8 million pounds plus 20 million
principal, and receive $1.5 million plus $30 million principal, effectively purchasing
$31.5 million for 21.8 million pounds. Current forward price for $31.5 million is
Accumulated with
interest in British pounds

$31,500,000
1.65




Spot price in pounds
1.06
15
12
15
12
≈ 19,550,923.10 pounds.
1.04


Contract price without
dollar interest income,
i.e., "ex-dividend"
Instead, 21.8 million pounds will be paid. The value of such transaction is
19,550,923.10 – 21,800,000.00 = –2,249,076.90
pounds in 15 months, and in today’s money, this is
2,249,076.90
−
≈ −2,091,086.40 pounds.
15
12
1.06
Total value, in pounds, to the party paying British pounds, is
−873,749.38 + −2,091,086.40 = –2,964,834.80.
(
) (
)
The problem specifically asks for the value in British pounds, but if you are interested in
the value in U.S. dollars, that value is calculated by multiplying the above by 1.65, and it
equals approximately –4,891,977.40.
2. You are given that the duration of a standard five-year discrete decreasing annuity
immediate (paying 5 at time 1, 4 at time 2, 3 at time 3, 2 at time 4, and 1 at time 6),
35
whose present value is ( Da )5 , is
, while the duration of a standard five-year discrete
15
increasing annuity immediate (paying 1 at time 1, 2 at time 2, 3 at time 3, 4 at time 4 and
55
5 at time 5), whose present value is ( Ia )5 , is
. The interest rate is i. Given that
15
information, find the duration of a standard level five-year annuity immediate (paying 1 a
times 1, 2, 3, 4 and 5), whose present value is a5 .
Solution.
With zero interest rate Macaulay duration (and duration, as well) of a standard decreasing
annuity is
1⋅5 + 2 ⋅ 4 + 3⋅3+ 4 ⋅ 2 + 5⋅1 35
= .
5 + 4 + 3+ 2 + 1
15
and Macaulay duration (as well as duration) of increasing annuity is
1⋅1+ 2 ⋅ 2 + 3⋅3+ 4 ⋅ 4 + 5⋅5 55
= .
1+ 2 + 3+ 4 + 5
15
Therefore, the interest rate is 0, and the Macaulay duration (as well as duration) of a unit
five year annuity immediate is
1⋅1+ 1⋅ 2 + 1⋅ 3 + 1⋅ 4 + 1⋅ 5 ( Ia )5 i=0% 1+ 2 + 3 + 4 + 5
=
=
= 3.
5
a5 i=0%
5
3. You are the investment actuary of an insurance firm, which is subject to new
regulatory requirements in Freedonia. Your company has 10 million freebies (currency of
Freedonia) worth of assets, half of which is invested in stocks and half in bonds. You are
given that the one year effective rate of return on stocks follows the normal distribution
with mean 0.10 and standard deviation 0.20, while the one year effective rate of return on
bonds follows the normal distribution with mean 0.05 and standard deviation 0.02. The
new regulation requires you to find the 1-st percentile of the joint distribution of the value
of your asset portfolio of stocks and bonds assuming that the joint distribution is formed
by applying the Gaussian copula to the distributions of stocks and bonds value at the end
of the period. That 1-st percentile is the maximum amount of premium that the company
is allowed to collect from business sold next year, if it ever exceeds it, it must stop sales
for the year. The correlation of stocks and bonds returns is 0.25. Find that crucial first
percentile.
Solution.
You should know immediately that applying the Gaussian copula to two normal
distributions will result in a joint bivariate normal distribution, with the prescribed
correlation parameter. The value of stocks in the portfolio at year end will have normal
distribution with mean 5 million times 1.10, i.e., 5.5 million, and standard deviation of 5
million times 0.20, i.e., 1 million. The value of bonds in the portfolio at year end will
have normal distribution with mean 5 million times 1.05, i.e., 5.25 million, and standard
deviation of 5 million times 0.02, i.e., 0.1 million. Since the joint distribution is bivariate
normal, their sum (i.e., the whole portfolio) will have a normal distribution with mean
5.5 + 5.25 = 10.75
million and standard deviation
12 + 0.12 + 2 ⋅ 0.25 ⋅1⋅ 0.1 = 1.06.
Since the 99-th percentile of the standard normal distribution is approximately 2.33 (from
the table), the 1-st percentile sought is
10.75 − 2.33⋅1.0.6 = 8.2802.
4. You are given the following with respect to a one-period securities market model:
(i) The time 0 prices of the three securities in the market are
S ( 0 ) = [ 0.8 6 1].
(ii) The time 1 payoffs of the three securities are described in the matrix:
⎡1 14 0 ⎤
S (1) = ⎢⎢1 6 0 ⎥⎥ .
⎣⎢1 7 5 ⎦⎥
Find the risk neutral probabilities of the three states of the future, or show that the riskneutral probabilities do not exist.
Solution.
Let us write
[ψ 1 ψ 2 ψ 3 ]
for the state price vector, if one exists. We see that
⎡ 1 ⎤
⎡ 1 ⎤
⎡ 0 ⎤
⎡ 0 ⎤
⎢ ⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢ 1 ⎥ = 1⋅ ⎢ 0 ⎥ + 1⋅ ⎢ 1 ⎥ + 1⋅ ⎢ 0 ⎥
⎢⎣ 1 ⎥⎦
⎣⎢ 0 ⎦⎥
⎣⎢ 0 ⎦⎥
⎣⎢ 1 ⎦⎥
so that
0.8 = 1⋅ ψ 1 + 1⋅ ψ 2 + 1⋅ ψ 3 .
Similarly, 6 = 14 ⋅ ψ 1 + 6 ⋅ ψ 2 + 7 ⋅ ψ 3 , and 1 = 0 ⋅ ψ 1 + 0 ⋅ ψ 2 + 5 ⋅ ψ 3 . Thus we must have
⎧ψ 1 + ψ 2 + ψ 3 = 0.8,
⎪
⎨14ψ 1 + 6ψ 2 + 7ψ 3 = 6,
⎪ 5ψ = 1.
⎩ 3
Therefore ψ 3 = 0.2, and
⎧⎪ ψ 1 + ψ 2 + 0.2 = 0.8,
⎨
⎪⎩ 14ψ 1 + 6ψ 2 + 1.4 = 6.
Consequently, ψ 2 = 0.6 − ψ 1 , so that 14ψ 1 + 6 ( 0.6 − ψ 1 ) + 1.4 = 6, and we get
ψ 1 = 0.125, ψ 2 = 0.475. We get
1 19 1
[ψ 1 ψ 2 ψ 3 ] = [ 0.125 0.475 0.2 ] = ⎡⎢ 8 40 5 ⎤⎥ .
⎣
⎦
Note that the risk-free interest rate is determined from the relationship
0.8 ⋅ (1 + i ) = 1,
so that
1
1+ i =
= 1.25,
0.8
and i = 25%. The risk-neutral probabilities are
1 5 5
⋅ =
,
8 4 32
19 5 19
q2 = ψ 2 ⋅ (1 + i ) =
⋅ =
,
40 4 32
1 5 1 8
q3 = ψ 3 ⋅ (1 + i ) = ⋅ = =
.
5 4 4 32
q1 = ψ 1 ⋅ (1 + i ) =
5. Consider a position consisting of $1,000,000 investment in asset X and $1,000,000
investment in asset Y. Assume that the daily volatilities of both assets are 0.1% and that
the correlation coefficient between their returns is 0.30. What is the 5-day 99% Value at
Risk for this portfolio, assuming a parametric model with zero expected return? The 99-th
percentile of the standard normal distribution is 2.326.
Solution.
The standard deviation of the daily dollar change in the value of each asset is $1,000. The
variance of the portfolio’s daily change, using the formula:
Var (U + V ) = Var (U ) + Var (V ) + 2 ρU ,V ⋅ σ U ⋅ σ V
is:
1000 2 + 1000 2 + 2 ⋅ 0.3⋅1000 ⋅1000 = 2,600,000.
The standard deviation of the portfolio’s daily change in value is the square root of
2,600,000, i.e., $1,612.45. The standard deviation of the five-day change in the portfolio
value is: $1,612.45 ⋅ 5 = $3,605.55. The 99-th percentile of the standard normal
distribution is 2.326. Therefore (assuming zero mean), the five-day 99% Value at Risk is:
2.326 ⋅$3, 605.55 ≈ $8386.51.
6. Regulated Insurance is a company domiciled in Freedonia, a country that implemented
a risk-based capital requirement system for regulation of its insurance companies. Riskbased capital requirement is the amount equal to the 95-th percentile of random amount
of the loss that the company experiences in one year. The company has liabilities of
1,000,000 freebies (the currency of Freedonia) in life annuities reserves, and assets of
$1,200,000 freebies, invested 40% in stocks and 60% in bonds. The liabilities of the
company will increase by 2% over the next year. The effective annual rate of return on
bonds follows a normal distribution with mean 4% and standard deviation of 2%, while
the effective annual rate of return on stocks follows a normal distribution with mean 9%
and standard deviation of 12%. The correlation of returns of stocks and bonds is 0.6, and
the joint distribution of those returns is bivariate normal. Find the ratio of the capital held
by Regulated Insurance to the capital required by regulation in Freedonia. The 95-th
percentile of the standard normal distribution is 1.645.
Solution.
The capital held by Regulated Insurance is equal to the excess of its assets over its
liabilities, i.e., 1,200,000 – 1,000,000 = 200,000 freebies. Let X be the random rate of
return of stocks, and Y be the random rate of return on bonds. The loss of Regulated
Insurance in one year is
0.02 ⋅1000000 − X ⋅ 0.4 ⋅1200000 − Y ⋅ 0.6 ⋅1200000 =
= 20000 − 480000X − 720000Y .
This is a normal random variable with mean
E ( 20000 − 480000X − 720000Y ) = 20000 − 480000E ( X ) − 720000E (Y ) =
= 20000 − 480000 ⋅ 0.09 − 720000 ⋅ 0.04 = 20000 − 43200 − 28800 = −52000,
and variance
Var ( 30000 − 480000X − 720000Y ) =
= Var ( 480000X + 720000Y ) =
= Var ( 480000X ) + Var ( 720000Y ) + 2 ρ X ,Y ⋅ Var ( 480000X ) Var ( 720000Y ) =
= 480000 2 ⋅ 0.12 2 + 720000 2 ⋅ 0.02 2 + 2 ⋅ 0.6 ⋅ 480000 ⋅ 0.12 ⋅ 720000 ⋅ 0.02 =
= 3317760000 + 207360000 + 995328000 =
= 4520448000.
The 95-th percentile of the distribution of the loss is
−52000 + 1.645 ⋅ 4520448000 ≈ 58600.3856.
The ratio sought is
200000
≈ 3.41294683.
58600.3856
7. You are in investment manager and you purchase a catastrophe bond, which pays oneyear LIBOR plus 2.5% every year in which Truerisk Catastrophe Index of insured losses
is below $3 billion. However, if Truerisk Catastrophe Index of insured losses exceeds $3
billion, the payment is reduced by ten basis point for each billion dollars of losses in
excess of $3 billion, and if that reduction results in a negative value, bondholders’s
principal is appropriately reduced (e.g., payment of negative 1% means that principal is
reduced by that 1%). If the current one-year LIBOR is 0.25%, what level of Truerisk
Catastrophe Index of insured losses would result in payment to bondholder dropping to
zero?
Solution.
With LIBOR at 0.25%, non-reduced payment is 2.5% + 0.25% = 2.75%. The difference
between 0% and 2.75% is 275 basis points, so the Truerisk Catastrophe Index would have
to exceed $3 billion by $27.5 billion, and reach the level of $30.5 billion.
8. In the country of Softwareland, there are now only two securities traded: risk-free
government bond earning 5% a year, and the stock of the only corporation in that nation:
Megasoft, currently trading at 100, and in a year possibly trading at one of these three
prices: 90, 105 and 120. You are the Minister of Finance for that nation, and propose to
complete the market in Softwareland by adding a third security, paying 1 when Megasoft
trades below 100, and 0 otherwise. Will adding that security make the market complete?
If the security is issued, it will only be issue if it completes the market, and it will be
issued by the government through an initial public offering of 1 billion units of that
security with 5% of proceeds paid to brokers marketing the security, and the rest
collected by the Treasury of the government of Softwareland. Estimate the proceeds to
the Treasury if the security is issued.
Solution.
The current market is
⎡1.05 90 ⎤
S ( 0 ) = [1 100 ],
S (1) = ⎢⎢1.05 105 ⎥⎥ .
⎢⎣1.05 120 ⎥⎦
It is, clearly incomplete. Let us see how it looks with respect to being arbitrage-free. If
the state price vector ⎡ ψ 1 ψ 2 ψ 3 ⎤ exists then it must satisfy the equations:
⎣
⎦
⎧1.05ψ 1 + 1.05ψ 2 + 1.05ψ 3 = 1,
⎨
⎩90ψ 1 + 105ψ 2 + 120ψ 3 = 100.
20
− ψ 1 − ψ 2 and
21
12 ⎛ 20
⎞
0.9ψ 1 + 1.05ψ 2 + ⋅ ⎜ − ψ 1 − ψ 2 ⎟ = 1,
⎠
10 ⎝ 21
8
− 1 = 1.2ψ 1 + 1.2ψ 2 − 0.9ψ 1 − 1.05ψ 2 ,
7
Hence ψ 3 = 1.05 −1 − ψ 1 − ψ 2 =
or
20
− 2ψ 1 .
21
If the asset proposed is added to the market, its price must be ψ 1 , and we get the market
as follows:
⎡ 1.05 90 1 ⎤
⎡
⎤
S (1) = ⎢ 1.05 105 0 ⎥ .
S ( 0 ) = 1 100 ψ 1 ,
⎢
⎥
⎣
⎦
⎢⎣ 1.05 120 0 ⎥⎦
The matrix S(1) is of rank 3 because its determinant is
1⋅1.05 ⋅ (120 − 105 ) = 15.75.
The conditions that must be satisfied by the parameters ψ 1 , ψ 2 , and ψ 3 are
ψ 1 > 0,
20
ψ2 =
− 2ψ 1 > 0,
21
20
20
⎛ 20
⎞
ψ3 =
−ψ 1 −ψ 2 =
− ψ 1 − ⎜ − 2ψ 1 ⎟ = ψ 1 > 0.
⎝ 21
⎠
21
21
This means that
10
0 <ψ1 < .
21
10
10
We can choose any number between 0 and
for the price of the new security, with
21
21
being the unobtainable maximum. With 1 billion units issues, the total proceeds will be
10
⋅1 billion ≈ 476,190, 476.
21
But the Treasury will receive only 95% of these proceeds, or
10
0.95 ⋅ ⋅1 billion ≈ 452, 380,952.
21
ψ2 =
9. You are in charge of hedging the property-casualty loss exposure at your company
using the Property Claims Service spreads, which consist of a long call of lower exercise
price, and short call of a higher exercise price. The premium for the spreads is quoted in
points, and each point is worth $200. Suppose that you are given that the 200/250 spread
provides protection for the underlying index values at the rate of $100 000 000 per point,
and the spread acts like a long 200 points call combined with short 250 points call, with
the actual payment equal to number of points times $200. You have bought 500 contracts,
and the index settled at $25 billion. What is the payoff to your company?
Solution.
$25 billion corresponds to
$25000000000
= 250 points.
$100000000
You have a 200 call, which is now worth 50 points, or 50 times $200 = $10000, and you
are short a 250 call, which expires worthless. Thus your payoff is $10000 per contract, or
$5000000 for 500 contracts.
10. The market price of a security is $50. Assume that Capital Asset Pricing Model holds.
The expected rate of return of this security is 14%. The risk-free rate is 6%, and the
market risk premium (over the risk-free rate) is 8.50%. What will be the market price of
the security if the covariance of its returns with the market doubles, while no other
parameters are changed? Assume that the security is a perpetuity of a constant dividend.
Solution.
Basic CAPM formula is
E ( r ) − rf = β E ( rM ) − rf
(
where
)
Cov ( r, rM )
.
Var ( rM )
When the covariance doubles, beta doubles. We need to find the original beta first. It
must satisfy
14% − 6% = β ( 8.50%)
so that
80 16
β=
= .
85 17
After beta doubles,
32
β= .
17
Therefore, new rate of return is
32
rnew = 6% + ( 8.50%) = 22%.
17
What is the dividend D of this security? At 14% expected rate, perpetuity of D is worth
D
, so that D = $7. At 22% expected return, perpetuity of $7
$50, and this must equal
0.14
is worth $31.82.
β=
11. You are the investment actuary for a life insurance company. Your company has of a
bond portfolio with the value of $3 billion dollars. The portfolio is invested only in two
securities: a 3-year bond with annual coupons of 5% and a 10-year zero coupon bond,
with 27.981823% invested in the first security, and the rest in the second security. The
continuously compounded risk-free interest rate is 4%. Calculate the 99% 10 day Value
at Risk (VaR) of this portfolio, assuming a standard parametric model, and using only
duration of the portfolio, and given that the annual yield volatility is 0.5% (yield volatility
defined here as the standard deviation of Δy, where y is the yield). Assume 252 days in a
year for the calculation, i.e., only use trading days, and you can also assume that the 99-th
percentile of the standard normal distribution is 2.33.
Solution.
The Macaulay duration of the zero coupon bond is its maturity, i.e. 10 years. The coupon
paying bond has a Macaulay duration of
0.05e−0.04 + 2 ⋅ 0.05e−2⋅0.04 + 3⋅1.05e−3⋅0.04
≈ 2.86129686.
0.05e−0.04 + 0.05e−2⋅0.04 + 1.05e−3⋅0.04
Using the allocations given, this produces the portfolio Macaulay duration of
0.27981823⋅ 2.86129686 + (1− 0.27981823) ⋅10 ≈ 8.00246072,
and duration of approximately
8.00246072,
≈ 7.68867975.
e0.04
0.005
The daily volatility of yield is
and the daily volatility of the portfolio is therefore
252
estimated as
0.005
σ P = D ⋅ P ⋅ σ y ≈ 7.68867975 ⋅ 3,000,000,000 ⋅
≈ 7265119.48.
252
Since the 99-th percentile of the standard normal distribution is 2.33, the 10-day 99%
Value at Risk is estimated as
2.33⋅ 10 ⋅ σ P ≈ 2.33⋅ 10 ⋅ 7265119.48 ≈ 53530177.30,
or approximately 535.30 million (that’s about 1.78% of the portfolio).
12. You are given the following with respect to an option-free bond portfolio worth ten
million dollars and held by the insurance company in which you work as an investment
actuary:
· The value of the bond portfolio using the current interest rate of 2% is 800,000,
· The value of the bond portfolio using the current interest rate plus 20 basis points is
788,000,
· The value of the bond portfolio using the current interest rate minus 20 basis points is
813,000.
You are also given that your company’s only liability is a single payment of 800,000
three years from now, with the present value of 753,857.87. Calculate the convexity of
your company’s surplus.
Solution.
The convexity estimate for the asset portfolio is
P ( i − Δi ) − 2P ( i ) + P ( i + Δi )
C≈
=
2
P ( i ) ⋅ ( Δi )
813,000 − 2 ⋅ 800,000 + 788,000
1
=
= 312.5.
2
800,000 ⋅ 0.002
0.0032
On the other hand, the liability has convexity of
t t +1
3⋅ 4
=
≈ 11.5340254.
2
1.022
1+ i
=
(
(
)
)
Therefore, the convexity of your company’s portfolio is estimated as
800,000
753,857.87
⋅ 312.5 −
⋅11.5340254 ≈ 5229.60192.
800,000 − 753,857.87
800,000 − 753,857.87