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What is a discounted factor (DF) and how to calculate

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    Benton Li
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A dollar today probably won’t be worth the same as a dollar 10 years later, and the future dollar probably worth less than today (Yenno, inflation)

e.g. In 2010, you can buy a dozen eggs for $2. But today, $2 can probably buy only half a dozen.

So although this green paper is the same at present and in the future, they have different values: present value (PV) and future value (FV)

When you lend money, you can compound interest over time, that is, compound rate maps PV to FV. The other way would be mapping FV to PV, and the rate here is called the discount factor.

  • Compounding rate: PV |→ FV
  • Discounting factor: FV |→ PV

DF depends on time and therefore is a function of time. So we denote DFtDF_t as the discount factor at time t.

You can think of it as the current price of a zero-coupon bond, which is a one-payment bond that doesn’t pay coupon, but $1 at time t. The price is probably less than $1 (e.g. $0.97). Mathematically,

DFt=PV/FVtDF_t = PV / FV_t

Before you read the following content, make sure you are familiar with yield to maturity (YTM)

Let’s look at the bond rate (for a vanilla bullet bond).

Suppose at time t, you receive your last payment, which is the principal + the last coupon payment. Your FV at t will be

FVt=1+rpFV_t = 1+r_p

But now, this payment is the bond price (PV) - the sum of discounted coupons between now and t-1. That is,

PV=B−∑i=1t−1rp∗DFi=B−rp∗∑i=1t−1DFiPV = B - \sum_{i=1}^{t-1} r_p*DF_i= B - r_p*\sum_{i=1}^{t-1}DF_i

Consequently,

DFt=B−rp∗∑i=1t−1DFi1+rpDF_t = \frac{B - r_p*\sum_{i=1}^{t-1}DF_i}{1+r_p}

See also:

Other DF formulas

Depo rate and Swap rate for t < 1 yr(α\alpha = accrual factor)

DFt=11+rp∗αDF_t = \frac{1}{1+r_p*\alpha}