Explain what a discount factor is and use a discount function to calculate present and future values.

FRM® Part I · Valuation and Risk Models · Pricing Conventions, Discounting, and Arbitrage · FRM1.VR.9.a

The idea

A discount factor d(t) is the present value today of one unit of currency to be received at date t, so it is below 1 when interest rates are positive. The collection of d(t) across maturities is the discount function. The present value of a cash flow is the amount times d(t). A bond's value is the sum of each cash flow times the factor for its own date. Running it the other way, the amount at date t that is worth a given present value is Amount = PV ÷ d(t). To move a sum from date s to date t, use Value at t = Value at s × d(s) ÷ d(t). A semiannual spot rate r converts to a factor as Discount factor = (1 + r ÷ 2)^(−2t). The usual mistakes are multiplying where you should divide and using one factor for every cash flow. On the exam, check which date each factor belongs to before using it.

Formulas

Present value of a cash flow from its discount factor

PV = Amount × d(t)

Amount: the cash flow received at date t
d(t): the discount factor for date t, the value today of one unit paid then

Holds for any default-free cash flow. Going the other way, Amount = PV ÷ d(t); multiplying where you should divide is the common slip.

Moving a value from date s to date t

Vt = Vs × d(s) ÷ d(t)

Vs: the value at date s
d(s): the discount factor for date s
d(t): the discount factor for date t

Works in either direction along the discount function. Check which date each factor belongs to before using it.

Discount factor from a semiannual spot rate

d(t) = (1 + r ÷ 2)^(−2t)

r: the semiannually compounded spot rate for date t, as a decimal
t: the time to the cash flow, in years

Semiannual compounding: halve the rate and count 2t half-year periods. Using annual compounding on a semiannual quote gives the wrong factor.

A practice question

From the question bank, with the reason each wrong option is wrong.

What does the discount factor d(t) represent?

The annualised yield earned on a zero-coupon bond that matures at date tWhy not: Confuses the discount factor, which is a price, with the spot rate, which is a yield.
The amount to which one unit of currency invested today grows by date tWhy not: Describes the accumulation factor, 1 / d(t), which runs in the opposite direction.
The present value today of one unit of currency to be received at date t - correct
The percentage by which a bond's price sits below its par valueWhy not: Confuses the discount factor with a bond trading at a discount to par.

d(t) is the price today of a claim to one unit of currency at date t. Present value is amount x d(t), and the accumulation factor is its inverse, 1 / d(t).

A harder one

A bond with face value 100 pays a 4% annual coupon in semiannual instalments, so its cash flows are 2.00, 2.00 and 102.00 at 0.5, 1.0 and 1.5 years. The discount function is d(0.5) = 0.9900, d(1.0) = 0.9780 and d(1.5) = 0.9650. The price per 100 face value is closest to:

5.87Why not: Omits the principal and values only the three coupons.
100.44Why not: Values the last flow as 100, leaving out the final coupon of 2.00.
108.23Why not: Pays the full 4.00 annual coupon at every half-year date instead of 2.00.
102.37 - correct

Price = 2 x 0.9900 + 2 x 0.9780 + 102 x 0.9650 = 1.980 + 1.956 + 98.430 = 102.366. Each flow takes the factor for its own date.

The hardest tier

A zero-coupon bond maturing in two years has a discount factor d(2) = 0.9157. The corresponding semiannually compounded spot rate is closest to:

4.50%Why not: Uses annual compounding, d(2) = (1 + r)^-2, rather than semiannual.
4.60%Why not: Uses simple interest, (1 / d - 1) / 2, ignoring compounding altogether.
4.45% - correct
9.00%Why not: Takes the square root of 1 / d(2) and doubles it, treating the two years as two periods before annualising.

With semiannual compounding d(2) = (1 + r/2)^-4, so r = 2 x (0.9157^(-1/4) - 1) = 2 x 0.02226 = 4.45%. Four half-year periods are compounded.

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