Question

After finishing your risk management course, you are asked by your portfolio manager to calculate the VaR of the portfolio that is consisting of two asset classes: long-term government bonds issued in the United States and long-term government bonds

issued in the United Kingdom. The expected monthly return on US bonds is 0.85%

and the standard deviation is 3.20%, while the expected monthly return on UK bonds (in US

dollars) is 0.95% and the standard deviation is 5.26%. The correlation between the US

dollar returns of UK and US bonds is 35%. The portfolio market value is $100 million

and is equally weighted between the two asset classes. Using the analytical method, compute the following:

(A) 5% monthly VaR

(B) 1% monthly VaR

(C) 5% weekly VaR

(D) 1% weekly VaR

Answer 1

For a two assets portfolio,

Expected return =

Rp = w1 x R1 + w2 x R2 and

Std deviation =

Here we have been given monthly returns, hence:

Monthly expected return = R_P =  0.5 x 0.85% + 0.5 x 0.95% = 0.90%

and monthly std dev =

σp = [(50% x 3.2%)2 + (50% x 5.26%)2 + 2 x 50% x 50% x 35% x 3.20% x 5.26%]1/2 =

= 3.52%

Part (A)

5% monthly VaR = (R_p - 1.65σ_p) x V = (0.90% - 1.65 x 3.52%) x $ 100 million = - $ 4.92 million = - $ 4,915,519 (if you want the value in $ and not in $ million)

Part (B)

1% monthly VaR = (R_p - 2.33σ_p) x V = (0.90% - 2.33 x 3.52%) x $ 100 million = - $ 7.31 million = - $ 7,312,217
(if you want the value in $ and not in $ million)

Part (C)

Weekly return, Rp* = R_p x 12 months / 52 weeks = 0.90% x 12 / 52 = 0.2077%

Weekly std dev = σ_p* = σ_p x (12 / 52)^1/2 = 3.52% x (12 / 52)^1/2 = 1.6931%

5% weekly VaR = (R_p* - 1.65σ_p*) x V = (0.2077% - 1.65 x 1.6931%) x $ 100 million

= - $ 2.59 million = - $ 2,585,992 (if you want the value in $ and not in $ million)

Part (D)

1% weekly VaR = (R_p* - 2.33σ_p*) x V = (0.2077% - 2.33 x 1.6931%) x $ 100 million

= - $ 3.74 million = - $ 3,737,329 (if you want the value in $ and not in $ million)