Managing Cash FlowManaging Cash FlowExposuresExposures

Risk Management

Prof. Ali Nejadmalayeri,

a.k.a. “Dr NDr N”

Volatility-Minimizing Hedge Ratio

• Suppose that futures price for maturity T is given by Ft(T). Now, the basis is Bt = Ft – St. In general, the optimal hedge ratio, h, is:

TFVar

TFSCovh

t

ttt

,

Optimal Hedge w/ Quantity Risk

• Quantity risk dramatically changes the optimal hedge (i.e., volatility-minimizing hedge ratio). – Consider this: Swiss franc in 3 months is either $1.50

or $0.50 with equal likelihood. Futures price is $1 now. A firm’s income in 3 months us either SF0.5M or SF1.5M with equal probability. So how do we hedge income?

– We need to know how cash flow covaries with exchange rate, or:

$/SF ρ = 1 ρ = 0 ρ = −1

$1.5 1.5M 1.5M 0.5M

$1.5 1.5M 0.5M 0.5M

$0.5 0.5M 1.5M 1.5M

$0.5 0.5M 0.5M 1.5M

Optimal Hedge w/ Quantity Risk• In US$ then the income is distributed:

• So there are three possibilities:– Perfect positive correlation:Cov = 0.5(2.25M – 1.25M) (1.5 – 1) + 0.5(2.25M – 1.25M) (1.5 – 1) = 0.5M

Var = 0.5 (1.5 – 1)2 + 0.5 (0.5 – 1)2 = 0.25, so h = 0.5 / 0.25 = 2

– No correlation:Cov = 0.25(2.25M – 1.25M) (1.5 – 1) + 0.25(0.75M – 1.25M) (1.5 – 1)

+ 0.25(0.75M – 1.25M) (0.5 – 1) + 0.25(0.25M – 1M) (0.5 – 1) = 0.5M

So h = 0.25 / 0.25 = 1; short 1 contract to hedge

– Perfect negative correlation:Clearly the value stays constant, so h = 0; no hedge needed

$/SF ρ = 1 ρ = 0 ρ = −1

$1.5 $2.25M $2.25M $0.75M

$1.5 $2.25M $0.25M $0.75M

$0.5 $0.25M $2.25M $0.75M

$0.5 $0.25M $0.25M $0.75M

Exposure to Risk

• The cash flow exposure to risk factor is define as Change in Cash Flow per Unit Change in Cash Flow per Unit Change in Risk FactorChange in Risk Factor

– Pertinent Extra factors in play:• Time Horizon

– Present value sensitivity to time, interest rates, exchange rates, etc.

• Competitive exposures– Also called “Operating Exposures”, or

“Economic Exposure”

Competitive Exposure• When demand is not price sensitive:

• When market is very competitive:

Pro-forma based Exposure

• From accounting statements, we have:

Cash Flow = Sales – Cost of Good SoldCash Flow = Sales – Cost of Good Sold– Investments – – Investments –

Taxes Taxes

• To find out how cash flow changes with exchange rate, we need to know how each of the above components changes with exchange rate.

Pro-forma Exposure (Cont.)

• Imagine a car maker’ has a cost function:

Cost = 10M + 0.25Cost = 10M + 0.25(Quantity)(Quantity)22

• And quantity produced follows:

Quantity = SQuantity = S$/$/££ 40,000 40,000

• The cash flows are:

Cash Flow = Quantity Cash Flow = Quantity Car Price – Cost Car Price – Cost

= S= S$/$/££ 40,000 40,000 S S$/$/££ 20,000 20,000

– 10M + 0.25– 10M + 0.25(S(S$/$/££ 40,000) 40,000)22

= 400M 400M SS$/$/£ £ – 10M – 10M

Hedge Ratio

• So optimal hedge is:hh = Cov[400M = Cov[400M S S$/$/£ £ – 10M , S– 10M , S$/$/££]/Var[]/Var[SS$/$/££]]

– Volatility-minimizing hedge increase with exchange rate volatility, σ:

Exposure

• The delta measure of exposure with respect to a risk factor is given by:

Delta Risk Exposure Delta Risk Exposure

= Cash Flow Change per Unit Change = Cash Flow Change per Unit Change in in the Risk factor (small the Risk factor (small change)change)

– For our previous example of car maker:

Example

• Problem 8 in Chapter 8– The cash flow from the U.S. sales = 11,000 x

$20,000 (pound/dollar exchange rate) is normally distributed with a mean 110 M pounds and a standard deviation 22 M pounds. Precisely the same is true of the cash flow from sales in Sweden. Since the jointly normal increments model holds, and the U.S. and Swedish cash flows are uncorrelated, the total cash flow from the overseas sales is normally distributed with a mean 220 M pounds and a standard deviation 31.11 M pounds (square root of 222+222). Therefore, CaR = 1.65 x $31.1 = CaR = 1.65 x $31.1 = 51.36 M pounds.51.36 M pounds.