1. Tutorial. Measuring Market Risk: VaR
SEEM 3580, 2018
Ji Jiahui

2. VaR Definition:
Measure the risk through maximum loss at certain probability level:
Pr ( Loss ≤ VaR ) = X%
For calculating it:
Specify a loss distribution
X%: 90%, 95%, etc.
Time scale: 1-day loss, 2-day loss, etc.
Loss = −Return

3. Return ~ N(-μ, σ 2 )
Return
Loss = −Return

6. VaR of Single Asset: (all variables are normally distributed)
Q = N × S (Quantity × Price)
∆Q = N × ∆S
= N × S × p (p, rate of change)
If you are told the σp, then σ∆Q = N × S × σp = Q × σp
This is why DEAR = Q × DEAR_U:
DEAR = σ∆Q 𝑍 𝑋% = Q × σp × 𝑍 𝑋% = Q × DEAR_U

7. The same idea about why 𝑵-day VaR = 𝑵 × DEAR:
N-day loss = day-1 loss + day-2 loss + … + day-N loss
They are identically independently distributed: N(0, 𝜎 2 ).
So Variance of N-day loss = N 𝜎 2
So, 𝜎 of N-day loss = 𝑁 𝜎

8. Exercise
XXX has a position in 1000 shares of one stock with price $10 today. If the daily price change rate has mean 0 and standard deviation 100bp (1%), what is the 5-day 90% VaR?

9. Exercise
XXX has a position in 1000 shares of one stock with price $10 today. If the daily price change rate has mean 0 and standard deviation 100bp (1%), what is the 5-day 90% VaR?
DEAR_U = 1% × 𝑍 90% = 0.01 × 1.28 =
DEAR = 1000 × 10 × DEAR_U = 128
5-day VaR = 5 × DEAR =
What if the mean is not 0, e.g., 50bp? (Apply the formula in Page 4)
Now DEAR_U = − % × 𝑍 90% =

10. VaR of Portfolio: Q = Q 1 + Q 2 + Q 3 ∆Q = ∆Q 1 + ∆Q 2 + ∆Q 3
So σ∆Q = ?
Because D(X+Y) = D(X) + D(Y) + 2cov(X,Y) (D -- variance)
DEAR Q 2 = DEAR Q DEAR Q DEAR Q 𝜌 12 DEAR Q 1 DEAR Q 𝜌 13 DEAR Q 1 DEAR Q 𝜌 23 DEAR Q 2 DEAR Q 3
A slight difference from that in the lecture notes.

11. See the Excel How to get μ, σ, ρ: or a(%) Sort a(%) 𝜇= 1 𝑛 𝑎 𝑖
Date
Close Price
4-Mar-16
20,176.70
3-Mar-16
19,941.76
2-Mar-16
20,003.49
1-Mar-16
19,407.46
29-Feb-16
19,111.93
26-Feb-16
19,364.15
25-Feb-16
18,888.75
24-Feb-16
19,192.45
23-Feb-16
19,414.78
22-Feb-16
19,464.09
19-Feb-16
19,285.50
18-Feb-16
19,363.08
17-Feb-16
18,924.57
16-Feb-16
19,122.08
15-Feb-16
18,918.14
12-Feb-16
18,319.58
11-Feb-16
18,545.80
5-Feb-16
19,288.17
4-Feb-16
19,183.09
3-Feb-16
18,991.59
2-Feb-16
19,446.84
1-Feb-16
19,595.50
1.16
-0.31
2.98
1.52
-1.32
2.46
-1.61
-1.16
-0.25
0.92
-0.40
2.26
-1.04
1.07
3.16
-1.23
-4.00
0.54
1.00
-2.40
-0.76 -4
-2.4
-1.61
-1.32
-1.23
-1.16
-1.04
-0.76
-0.4
-0.31
-0.25
0.54
0.92 1
1.07
1.16
1.52
2.26
2.46
2.98
3.16
𝜇= 1 𝑛 𝑎 𝑖
𝜎= 1 𝑛− 𝑎 𝑖 −𝜇 2
𝜌= … ……
Lose
by a% or
VaR through historical data

12. Thanks!
Q&A