5. EXERCISE SESSION 1
5.1 INTEREST RATES
Exercise 1
An investor receives €1 100 in 1 year in return for an investment of €1000 now. Compounding:
Calculate the percentage return per annum with:
( )
m× t
Rm
1. Annual compounding: m = 1 1+
1100 m
1100=1000. ( 1+ R )=¿ R= −1=0.1=10 %
1000
2. Semiannual compounding: m = 2
( ) ( √ 1100 −1 )=9.5323 %
365
R 365
1100=1000. 1+ =¿ R=365 ×
365 1000
5. Continuous compounding: m = ∞
1100=1000× e =¿ R=ln
R
( 1100
1000 )
=9.5310 %
Exercise 2
Given zero coupon interest rates in quarterly compounding:
1. Compute the discount factors using the quarterly compounding rates:
1%
=0.25 % = quarterly rate
4
, 3 years: RC =4 × ln 1+
0.03
4 [=2.988806 %
]
4 years: RC =4 × ln 1+
0.04
4 [=3.98013 %
]
3. Compute the discount factors starting from the rates in continuous compounding:
We’ll get the exact same answers as question 1, because we computed the equivalent rates.
4. Compute the forward rate for the period year 2 and 3 in continuous compounding:
R 2C × 2 f 2,3 ×1 R3 C ×3
e ×e =e
¿> f 2,3 =R 3 C × 3−R2 C ×2=2.988806 % ×3−1.99502% ×2=4.976378 %
Exercise 3
Suppose that the forward SOFR rate for the period between time 1.5 years and time 2 years in the future is
5% (with semiannual compounding) and that some time ago a company entered into an FRA where it will
receive 5.8% (with semiannual compounding) and pay SOFR on a principal of $100 million for the period.
The 2-year SOFR risk-free rate is 4% (with continuous compounding). What is the value of the FRA?
FRA → FR A 0=PV [ τ ( R K −R F ) L ] of PV [τ ( RF −R K ) L]
τ =time , L=principal , ( R F−R K )∨( R K −R F )=difference between the ¿∧floating rate
5.2 RISK MEASUREMENT
Exercise 1
Consider a position consisting of a €300 000 investment in gold and a €500 000 investment in silver.
Suppose that the daily volatilities of these 2 assets are 1.8% and 1.2% respectively, and that the coefficient
of correlation between their returns is 0.6.
We make the assumptions: normal distribution and zero means.
1. What is the 10-day 99% VaR for the portfolio?
−1
VaR=σ . N ( X ) . ¿ ¿
Va R portfolio =σ portfolio . N−1 ( X ) .¿ ¿
We still need to compute the standard deviation of the total portfolio.
Cov ( X ,Y )
Var ( aX +bY ) =a2 Var ( X )+ b2 Var ( Y ) +2 abCov (X , Y ) & ρ=
σ X σY
2. By how much does diversification reduce the VaR?
ρ gs =1
Joint VaR=Va R1+ Va R 2=0.018 .2.326 . 300 000+0.012 . 2.326 .500 000
10−day VaR=( 0.018 . 2.326 .300 000+ 0.012. 2.326 .500 000 ) . √ 10=83 852.217
The benefits of diversification are €83 852.217 - €75 025.67.
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