Cheet Sheet

(1). The present value of annuities

Arⁿ = (1/((1+r)¹))+(1/((1+r)²))+...+(1/((1+r)ⁿ)) = (1/r)(1-(1/((1+r)ⁿ)))

where r is the discount rate for one period, and n is the number of periods.

(2) The future value of annuities:

Frⁿ = (1/r)((1+r)ⁿ-1) = Arⁿ×(1+r)ⁿ

where r is the discount rate for one period, and n is the number of periods.

3) Present value of a bond:

P = (C/((1+r)¹))+(C/((1+r)²))+...+(C/((1+r)ⁿ))+(M/((1+r)ⁿ)) = C×Arⁿ+(M/((1+r)ⁿ))

where C is the half year coupon value, M is the face value of debt, and r is the discount rate for half a year.

(4) T-bills are quoted as follows:

bank discount yield = ((100-price)/(100))×((360)/n),

or

price=100×(1-bank discount yield×(n/(360))),

where price is the actual price, and n is the number of days from today to maturity.

(5) Duration: MD=(($ duration)/P)

Macauly Duration=MD×(1+(Y/2))

where MD is modified duration, P is price, and Y is the yield to maturity.

(6) Convexity:

convexity=(($ convexity)/P)

where $ convexity is dollar convesity, and P is price..

(7) Duration and convexity:

dP ≈ -$ duration×dY+(1/2)×$ convexity×(dY)² ≈ -MD×dY×P+(1/2)×convexity×P×(dY)²

((dP)/P)≈-MD×dY+(1/2)×convexity×(dY)²

where dP represents the change in price, dY is the change in yield, P is the price of the bond, and MD is the modified duration.

(8) Effective annual rate = (1+(Y/m))m-1, where Y is the annualized m-compounding rate.

(9) The present value of $1, if it is paid after t years (assuming semi-annual compounding), is

zt=(1/(1+Rt/2))2×t,

where Rt is the spot rate for year t.

(10) Forward rate: (1+r∗)T∗= (1+r)T×(1+FT,T∗)(T∗-T),

where r∗ is the spot rate per period for maturity T∗ (in periods), r is the spot rate for maturity T (in periods), and FT,T∗ is the forward rate between T and T∗ per period.