Thankyou so much David !! That was really helpful !!!!Hi @ann123456 Yes you are correct that bond price returns VaR is the product of a change in yield and modified duration! However, Yield VaR is a change in yield. Let's take Jorion's own example, see below, for the 7-year term. The yield VaR is 0.484%. I don't think he anywere decomposes the yield VaR but Yield VaR = (Yield volatility) * (deviate); e.g., if yields are normal, then if the yield volatility is 1.0%, then the 95.0% yield VaR = 1.0% * 1.645 = 1.645%. So probably his 7-year yield VaR of 0.484% = 0.294% yield volatility * 1.645 = 0.484%. This is just a way of scaling volatility: "if the yield volatility is 294 basis points, then our worst expected change with 95% confidence is 484 basis points." In this way, this is a measure of yield change. Therefore, 7-year returns var = yield VaR * modified duration = 0.484% * 7/(1+6.07%) = 3.192 because the 0.484% itself is Δy*z(α), so we have returns VaR = Δy*z(a)*[Mac_duration/(1+y)] = 0.294%*1.645*(7/1.067) = 3.173%. I hope that helps!
Hi @Pflik,
Since VaR returns =|D*| x VaR(dY) we can solve for VaR yield
VaR change in yield =VaR(dP/P)/|D*|
Where D* is modified duration
Check out this video of David's
EDIT: I forgot to include that you can calculate Yield var by doing a historic simulation on the daily yield observations of the yield curve pertaining to that bond.
On another note (on this topic), I'm a bit confused by the mapping example table (below) in the study notes for Jorian chapter 11. I can't understand how each risk factor column is aggregated.
Hi David, I’m not sure if I’m missing something but how do we calculate the 7 year yield var itself? Also, are we always supposed to assume 95% on these questions? Conf level is never actually specified in the videos or notes.Hi @ann123456 Yes you are correct that bond price returns VaR is the product of a change in yield and modified duration! However, Yield VaR is a change in yield. Let's take Jorion's own example, see below, for the 7-year term. The yield VaR is 0.484%. I don't think he anywere decomposes the yield VaR but Yield VaR = (Yield volatility) * (deviate); e.g., if yields are normal, then if the yield volatility is 1.0%, then the 95.0% yield VaR = 1.0% * 1.645 = 1.645%. So probably his 7-year yield VaR of 0.484% = 0.294% yield volatility * 1.645 = 0.484%. This is just a way of scaling volatility: "if the yield volatility is 294 basis points, then our worst expected change with 95% confidence is 484 basis points." In this way, this is a measure of yield change. Therefore, 7-year returns var = yield VaR * modified duration = 0.484% * 7/(1+6.07%) = 3.192 because the 0.484% itself is Δy*z(α), so we have returns VaR = Δy*z(a)*[Mac_duration/(1+y)] = 0.294%*1.645*(7/1.067) = 3.173%. I hope that helps!
"For instance, suppose that you sold a 6 × 12 FRA on $100 million. This is equivalent to borrowing $100 million for 6 months and investing the proceeds for 12 months. When the FRA expires in 6 months, assume that the prevailing 6-month spot rate is higher than the locked-in forward rate. The seller then pays the buyer the difference between the spot and forward rates applied to the principal. In effect, this payment offsets the higher return that the investor otherwise would receive, thus guaranteeing a return equal to the forward rate. Therefore, an FRA can be decomposed into two zero-coupon building blocks.
Long 6 × 12 FRA = long 6-month bill + short 12-month bill" -- Philippe Jorion. Value at Risk, 3rd Ed.: The New Benchmark for Managing Financial Risk (pp. 294-295). Kindle Edition.
David, great thanks for your answer. I like entirely understand what I am reading. I have read carefully this chapter and the passage you cited and I think that the table presented in Jorion’s chapter for FRA concerns short FRA (negative CF in 6 months (180 days), positive CF in 12 months (360 days)). Yes, you are right that in mapping FRA it does not matter whether we calculate short or long FRA because signs are different but PV(CF) are the same. I also agree that long 6M bill (purchase) is outflow (-) and short (sell) 12 M bill is inflow(+) when open. But we are concerned about what is happening in 6M and 12M, not today. Today both CF cancel each other. If (as in Jorion‘s chapter) we have (-) in 6M and (+) in 12M it means, I think, that we pay (close short) in 6M and receive (close long) in 12 M. So it should be:@wojtek See Jorion's explanation below. if we are long (purchase) the 6-month bill, the purchase is treated a cash outflow; if we sell (short) the 12-month bill, we receive cash and it's treated as an inflow. I think the important thing for mapping is they are different: i don't think it matters greatly (it may not matter at all!) if we instead map a short 6 × 12 FRA with a positive CF for shorting the 6-month plus a negative CF for purchase of 12-month bill. Thanks,