Question

1. Compute the Value-at-Risk (VaR) of a six-month forward contract. The transaction requires the investor to deliver $12.7 million in 180 days and receive €10 million in exchange. Assume that the current spot rate is $1.26/1€ and the annualized interest rate is 4% on a six-month zero coupon bond and 3% on a six-month zero coupon Euro bond. Again, assume the variance/co-variance matrix (on daily returns) across those instruments are as follows: Six-Month $ Bond Six-Month $ Bond Six-Month $ Bond Six-Month $ Bond 0.0000314 Six-Month € Bond 0.0000043 0.0000260 Spot $/€ Rates 0.0000012 0.0000013 0.0000032 (a) Compute the value of the short position in the zero coupon dollar bond. (b) Compute the value of the long position in the zero coupon euro bond (in $ terms), holding spot rate fixed. (c) Compute the VaR for this forward Contract. (d) Compute the daily VaR for this forward contract assuming returns are normally distributed with a 90% confidential interval. .

Answer 1

Value at Risk can be defined as a fimancial tool to assess and measure the risk involved in any investment. It is a statistic that measures the probability of occurrence of defined loss of a risky asset or portfolio. VaR is a risk assessment model that is maily used by the investment & commercial banks to measure the extent & probability of potential losses of their institutional investment portfolios. For example, the VaR of a institutional portfolio is $50 million at one week, & at 95% confidence interval means that there is only 5% chance that the portfolio will lose in value of more than $50 million over any particular week.

From the data given in the problem, the valuation of the positions of the given standard instruments i.e. the zero coupon dollar bond, zero-coupon euro bond in dollar terms & holding spot rate fixed can be computed as follows :-

(a) The value of the short position in the zero coupon dollar bond is given by the following formula :-

Dollar Forward / (1+r_dollar)^t

where, dollar forward = value of the forward contract in dollar terms

r_dollar = annualized interest rate on the zero-coupon dollar bond

t = maturity period

here, the value of the forward in dollar terms = $12.7 million

r_dollar = 4%

t = six month matuirty i.e. 180/360 or 1/2

Therefore, the value of the short position in the zero coupon dollar bond in million :-

$12.7 / (1+0.04)^180/360  

= $12.7 / (1.04)^1/2

= $12.7 / 1.01980

= - $12.4534 million

The value of the short position in the zero coupon dollar bond = - $12.45 million.

(b)The value of long position in zero-coupon euro bond (in dollar terms) holding spot rate fixed is given by:

Spot $/Euro * {Euro forward / (1+r_euro)^t}

Here, it is given that the current spot $/Eu = 1.26,
The value of the forward contract in euro terms is = 10 million euro
t = six month zero coupon bond i.e. 180/360 or 1/2,
r_euro = the annualized interest rate on a zero coupon euro bond = 3%

Therefore the value of long position in zero - coupon euro bond (in dollar terms), holding the spot rate fixed :-

1.26 * { 10 million/ (1+0.03)^180/360}

= 1.26 * {10 million /(1.03)^1/2}

= 1.26 * (10 million / 1.015)

= 1.26 * 9.8522 million = 12.414 million

Hence,the value of long position in zero - coupon euro bond (in dollar terms), holding the spot rate fixed = 12.414 million

The Value of spot euro position in dollar terms holding euro rate fixed is given by :-

Spot $/Euro * {Euro forward / (1+r_euro)^t}

= 1.26 * {10 million /(1.03)^1/2

= $12.414 million

(c) The variance- covariance matrix (on daily returns) across the above instruments can be presented as follows:-

	Six-month $ bond	Six-month Euro bond	Spot $/Euro
Six-month $ bond	0.0000314
Six-month Euro bond	0.0000043	0.0000260
Spot $/Euro	0.0000012	0.0000013	0.0000032

Now, the Value at Risk (VaR) of this forward contract can be computed using the values of the positions of the aforementioned instruments and the variance-covariance among these instruments.

If we assume the position of an instrument j as X_j & the covariance between the two instruments i & j as _ij.

Therefore, the daily variance of the 6 month $/Euro forward contract is given by :-

(X₁₁ + X₂₂ + X₃₃)²

= X₁²₁² + X₂²₂²+ X₃²₃² + 2X₁X₂₁₂ + 2X₂X₃₂₃ + 2X₁X₃₁₃

Here, X₁ = The value of the short position in the zero coupon dollar bond = - $12.45 million.

X₂ = the value of long position in zero - coupon euro bond (in dollar terms), holding the spot rate fixed = 12.414 million

X₃ = The Value of spot euro position in dollar terms holding euro rate fixed is given by = 12.414 million

= (-12.4534)² (0.0000314) + (12.4145)² (0.0000260) + (12.4145)² (0.0000032) + 2(-12.4534)(12.4145) (0.0000043) + 2(12.4145)(12.4145)(0.0000013) + 2(-12.4534)(12.4145)(0.0000012)

= $0.0111021 million

Therefore, the daily variance of the forward contract is = $0.0111021 million

Hence, the daily standard variation of the forward contract is = square root of $0.0111021 million = $0.105367 million = $105,367

therefore the VaR of the forward contract = $105,367

(d) The daily VaR for this forward contract assuming returns are normally distributed with a 90% confidential interval is given by:-

We know the Z-score at 90% confidence interval = 1.645

The daily VaR of the forward contract at 90% confidence interval = $105,367 * 1.645 = $173,329